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T (n > N) such that
Un 0
22
Chapter 1.
Basic Concepts and Facts
2° The critical point has one zero eigenvalue. We shall show this type of bifurcation for some one dimensional vector fields. Example 4. Consider the following systems,
x -- A - X 2 , x = AX - X2 ,
(1.2.4)
X3 ,
(1.2.6)
X = AX -
(1.2.5)
in which X E lR, A E lR. Denote the right-hand side of each equation above by f(x, A). Then f(O, 0) = 0, ~(O, 0) = 0, in each case. X = 0 is a critical point as A = 0 with zero eigenvalue. The bifurcation diagrams are depicted in Fig. 1.2.4( a), (b), (c), respectively. For (1. 2.4), on one side of A = 0 there is no critical point and on the other side of A = 0, there are two critical points: one is stable (represented by the solid branch of the parabola A = x 2 in Fig. 1.2.4(a)) and the other is unstable (represented by the dotted branch of the parabola). This type of bifurcation is referred to as a saddle-node bifurcation. x
---1~--t--+-A
o
/
./
(a)
(b)
(c)
Fig. 1.2.4 The critical points of (1.2.5) are given by x = 0 and x = A and are plotted in Fig. 1.2.4(b). Hence, for A < 0, there are two critical points: x = 0 is stable and x = A is unstable. These two critical points coalesce at A = 0 and, for A > 0, x = 0 is unstable and x = A is stable. Then, an exchange of stability has occured at A = o. This type of bifurcation is called a trans critical bifurcation.
1.2.
Structural Stability and Bifurcation
23
For system (1.2.6), if A < 0, then there is one critical point, x = 0, which is stable. For A > 0, x = is still a critical point, but two new critical points have been created at A = and are given by x 2 = A. In the process, x = has become unstable for A > 0, with the other two critical points stable. This type of bifurcation is called a pitchfork bifurcation. There are many complicated bifurcation phenomena that appeared in this case, among which the Bogdanov-Takens bifurcation with fundamental importance will be discussed later. 2) The bifurcation system has at least one nonhyperbolic closed orbit. It also includes several important subcases. 10 The closed orbit is isolated, that is a multiple limit cycle. This multiple cycle may split into several limit cycles in its neighborhood or disappear under small perturbations of the system.
°
°
°
Example 5. Consider the system
x = -y -
x(x 2 + y2 - 1)2 + AX, (1.2.7)
iJ = x - y(x 2 + y2 - 1)2 + Ay.
(1.2.7) has a multiple-two limit cycle x 2+y2 = 1 as A = 0, has no limit cycle as A < 0, and has two simple limit cycles near by x 2+y2 = 1 and are given by the equations x 2+y2 = 1 ± ~ as A > and small enough. The phase portraits for different A'S are depicted in Fig. 1.2.5.
°
,\ < 0
'\=0
Fig. 1.2.5
.\>0
Chapter 1.
24
Basic Concepts and Facts
If we look at the Poincare map near a multiple limit cycle, then the similar bifurcation diagrams can be obtained with those represented in Example 4. Writing system (1.2.7) in polar coordinates yields
r = -r{(r2 -
1)2 - A},
0=1.
(1.2.8)
Consider the Poincare map P(r, A) of the limit cycle La : x 2 + y2 along the ray 0 = from the origin,
°
=
1
= {(r, 0) : r > 0,0 = O}.
~
We have the successive function
d(r, A)
=
P(r, A) - r.
The bifurcation diagram is given by the graph of the relation d(r, A) = 0, that is, (r2 - 1)2 - A = in the (A, r)-plane. The points rt = J1 ± v0.. are fixed points of the Poincare map. For A > 0, they correspond to two one-parameter families of periodic orbits
°
Lt: x.x(t) = J1 ± v0..cos t,
y.x(t) = J1 ± v0..sin t,
with parameter A. For A = 0, there is a multiple two limit cycle La represented by (xo(t), Yo( t)) = (cos t, sin t). The bifurcation diagram is shown in Fig. 1.2.6, which is similar to Fig. 1.2.4(a) showing the saddle-node bifurcation. Note that the dotted parts of the A-axis represent the unstable critical point at 0 in Figs. 1.2.6-1.2.8. Example 6. Consider the system
x = -y -
x(x 2 + y2 _ 1)(x 2 + y2 - 1 - A),
iJ = x - y(x 2 + y2 - 1)(x2 + y2 - 1 - A),
(1.2.9)
or in polar coordinates
r = -r(r 2 - 1)(r2 -
e=
1 - A),
(1.2.10) 1.
1.2.
Structural Stability and Bifurcation
25
The points r).. = -If+): and r).. = 1 are fixed points of the Poincare map P(r, A) of La : r = 1. Let d(r, A) = P(r, A) - r. We have
d(l, A) = 0,
for all A E JR,
and d(~,A)=O,
for all A > -1.
That is, for all A E JR, system (1.2.10) has a one-parameter family of periodic orbits r = 1, and for A > -1, there is another one-parameter· family of periodic orbits represented by
The bifurcation diagram is shown in Fig. 1.2.7 (similar to Fig. 1.2.4(b)) in (A, r )-plane, which shows the transcritical bifurcation that occurs at multiple-two limit cycle La of (1.2.9) at the bifurcation value A = 0
/'
-
.,.,- rt
/ I
o
,\
Fig. 1.2.6 r
;'
/ I
o
Fig. 1.2.7
Fig. 1.2.8
Chapter 1.
26
Basic Concepts and Facts
Example 7. Consider the system x = -y - X(X 2 + y2 _ 1){(x 2 + y2 - 1)2 - A},
if
= x - Y(X 2 + y2 - 1){(x 2 + y2 - 1)2 - A}
(1.2.11)
which has a multiple three limit cycle Lo : x 2 + y2 = 1 as A = 1. By taking polar coordinates it is easy to show that for all A E IR, (1.2.11) has a one-parameter family of periodic orbits given by x 2 + y2 = 1, and for A > 0, there is another family (with two branches) represented by
(x~( t), y~( t))
=
,/1 ± V,\( cos t, sin t).
The Poincare map of Lo has a pitchfork bifurcation at (A, r) = (0,1) as shown in Fig. 1.2.8 (similar to Fig. 1.2.4(c)). 20 The bifurcation system has a family of closed orbits; while most of them are broken under small perturbations some may be preserved to become isolated closed orbits (limit cycles). Such kind of bifurcations is called Poincare bifurcation. A well-known example is the van der Pol system
x = y, if = -x + Ay(1 - x 2 ).
(1.2.12)
(1.2.12) has a family of closed orbits x 2 + y2 = h, and for A =1= 0 it has a unique limit cycle L)... The limit position of L).. is not the critical point 0, but the circle x 2 + y2 = 4, as A -+ O. 3) The bifurcation system has nontransversal intersection of some stable and unstable manifolds of certain critical elements. For planar vector fields, this appears to be an orbit connecting one or more saddle points (or saddle-nodes), which is referred to as a homo clinic or heteroclinic orbit as in Sec. 1.1. We call such type of bifurcation homoclinic or heteroclinic bifurcation. Some examples of such bifurcations are depicted in Fig. 1.2.9. We will meet specific systems which display these behavior later. They
1.3.
Codimension and Unfoldings
27
Fig. 1.2.9 play an important role in the study of global behavior of dynamical systems. The bifurcations mentioned above will be discussed in more detail in Chapters 2 and 3 for two dimensional systems. In the higher dimensional cases, there might be degenerate critical point with both zero and pure imaginary eigenvalues and more complicated bifurcations might occur, such as bifurcation to invariant torus, which will be discussed in Chapters 4 and 5. The homo clinic and heteroclinic bifurcations may cause chaotic dynamics and some related phenomena which will be studied in Chapters 5 and 6.
1.3.
Co dimension and Unfoldings
1.3.1.
Definition and codimension-l examples
As mentioned before, for a system with nonhyperbolic critical elements or with non-transversal intersections of certain stable and unstable manifolds, small perturbations may cause a change of the topological structure of orbits. Thus one may hope to know all the
Chapter 1.
28
Basic Concepts and Facts
possible change of a bifurcation system. This leads to the following concepts. Definition 1.3.1. Let X be a bifurcation vector field with a structurally unstable local phase portrait a (such as a degenerate critical point, a closed orbit, or a homo clinic or heteroclinic closed orbit). Any perturbation system of X is called an unfolding. If there is a particular unfolding (usually with a number of parameters) which represents all the possible variations of the phase portraits near a, then we call this unfolding universal. The smallest number of parameters contained in a universal unfolding is called the codimension of
X. For example, the Example 3 in Sec. 1.2.3 gives the codimensionone bifurcations of one-dimensional vector fields. In the following we shall discuss some fundamental bifurcations of planar vector fields with co dimensions one and two. Example 1. The system (1.3.1) y=y
has the origin 0 as a saddle-node point with a zero eigenvalue. Consider the following unfolding with one parameter,
X=A+X 2 ,
y=
(1.3.2)
y.
(1.3.2) has no critical points near 0 when A > 0, and has two critical points, a saddle (-V-A, 0) and a node (V-A, 0), when A < o. Figure 1.3.1 presents all the possible change of orbit structure near a saddlenode O. (1.3.2) is a universal unfolding of (1.3.1) with one parameter. The Example 2 in Sec. 1.2.3 shows a first order fine focus bifurcation, in which, by suitable perturbations, 0 can become a strong focus with the same stability or a strong focus with different stability
1.3.
29
Codimension and Unfoldings
~ . ~
,
J \
(a) A > 0
J
(c) A < 0
(b) A = 0 Fig. 1.3.1
and a small limit cycle around it at the same time. This is also a codimension-one bifurcation of critical points. The bifurcation of multiple-two limit cycle is a codimension-one bifurcation of closed orbits, which is shown in Example 4, Sec. 1.2.3. We give now an example to show a codimension-one bifurcation system with a homoclinic loop.
o
~ 0 I
(
Fig. 1.3.2 Example 2. Consider the system
x=
-y + dx
iJ = x,
+ mxy + y2,
(1.3.3)
which is a particular case of type I of the quadratic systems studied in [188], Sec. 12. For fixed mo there exists a certain do such that (1.3.3) has a homoclinic loop passing through the saddle N(O, 1), then (1.3.3)d=d o is a codimension-one bifurcation system. In fact, it is not difficult to show (which will be discussed in more detail in Sec. 2.4)
Chapter 1.
30
Basic Concepts and Facts
that there is a bifurcation curve l: d = d(m) on the (m, d)-plane with the following properties: for (m, d) E l, (1.3.3) has the same phase portrait as that for d = do, and for (m, d) in the different sides, (1.3.3) has different phase portraits as shown in Fig. 1.3.2, with a limit cycle in one side. 1.3.2.
Bogdanov-Takens bifurcation
We now introduce a typical codimension-two bifurcation studied independently by I. Bogdanov [17] and F. Takens, which plays an important role for the study of bifurcation theory in recent years.
Fig. 1.3.3 The bifurcation system is taken as
=
y,
y=
x2
x
+ xy,
(1.3.4)
for which the linear part has 0 as its multiple-two eigenvalue. 0 is a degenerate critical point of (1.3.4) with the phase portrait as shown in Fig. 1.3.3. To get the possible changes of phase portraits near 0, one may take the following unfolding of (1.3.4)
x = y, (1.3.5) It is easy to find bifurcation curves in which (1.3.5) undergoes saddlenode and Hopf bifurcations. We first seek the critical points, which
1.3.
Codimension and Unfoldings
31
are given by
(x,y) = (±v-)'l, 0) == (x±,O),
°
(x+,O) is a saddle for ).1 < and all ).2, while (x_,O) is a source ).2 > J-).l, ).1 < 0, and a sink for ).2 < J-).l, ).1 < 0. Thus
for
is a Hopf bifurcation, for which we study the stability of fine focus (x_,O). Bring the point (x_,O) to the origin and put the system into the standard form, we obtain ).2 = J-).l
x = -J-2x_y + xy + v_1x_y2, if = J-2x_x. By using the results of [188J, Sec. 12, we know that 0 is an unstable first order fine focus. If the parameter ().1, ).2) moves to ).2 < J-)., then an unstable limit cycle appears around (x_, 0), while 0 becomes stable. There is another homoclinic loop bifurcation curve situated in the part of ).1 < 0, ).2 > 0. To show this we choose a suitable change of variables such that the two critical points (x±,O) move to (0,0) and (0,1) respectively. Then we get the equivalent form of (1.3.5) by scaling: x = y, (1.3.6) 2
if
=
x- x
+ E(8y + ,xy),
which may be studied by applying a perturbation to the Hamiltonian system as E = O. Using the result in §2.3.3, we may obtain the approximation of homo clinic loop bifurcation curve (1.3.7) which is located below the Hopf bifurcation curve. Furthermore, the uniqueness of the limit cycles creating from the homoclinic loop may be proved rigorously. Figure 1.3.4 shows that there are four bifurcation curves through O. One Hopf bifurcation (HFB), one homo clinic loop bifurcation (HocB), and the saddle-node bifurcations (positive and negative y-axis) with four different regions divided by them, for which the unfoldings have different phase portraits near O. It is also
Chapter 1.
32
Basic Concepts and Facts
proved that for any small perturbation on (1.3.4), the local phase portrait near 0 is topologically equivalent to one of the pictures in Fig. 1.3.4. Thus, (1.3.5) is a universal unfolding of (1.3.4).
HacB S/VB
..\1
--------------~O+-------------
Fig. 1.3.4 It has been generalized for studying the following general cusp bifurcation system,
x = y,
iJ =
X2
+ Al + L:L1 Ai+1Xi-1y ± xly + O(lxJ, lyl)l+l.
(1.3.8)
If I = 1, (1.3.8) is the system (1.3.5). I = 2 corresponds to a codimension-3 bifurcation, which is investigated in [42]. For general I, the last author proves that there are at most I limit cycles bifurcated from O. There are also a number of papers dealing with the bifurcations of higher co dimension (2: 3), see for example [43], [210].
1.4.
Center Manifold Theory
1.4.
33
Center Manifold Theory
Center manifold theory can often be used to simplify dynamical systems. In this section, we introduce some main results of the center manifold theory.
1.4.1.
Main theorems on center manifolds
As mentioned in Sec. 1.1, for linear system (1.1.2) there are invariant su bspaces ES, EU, and EC, corresponding to the eigenspaces spanned by eigenvectors which in turn correspond to eigenvalues with negative, positive, and zero real parts respectively (resp. for maps, with modulus less than, greater than, and equal to one). If we suppose that EU = 0, then we find that any orbit of the system will decay exponentially to E C as t --t +00. Thus for nonlinear systems, if we are interested in long-time behavior, we need only to study systems restricted to E C • That is one of the main reason to introduce the concept of center manifolds. Consider now the system in the form of
x
=
Ax + f (x, y),
iJ = By + g(x, y),
(1.4.1)
in which (x,y) E IRc x IRs, A, B are constant matrices such that all the eigenvalues of A have zero real parts and all those of B have negative real parts, and f, 9 are CT (r 2: 2) vector functions with
f(O,O) = 0,
Df(O,O) = 0,
g(O,O) = 0,
Dg(O,O) = 0,
(1.4.2)
Definition 4.1.1. A local invariant manifold is called a local center manifold for (1.4.1) if it can be represented by
WC(O) = {(x, y) E IRc x IRS I y = hex), for small 8 > O.
Ixl < 8, h(O) = Dh(O) = O},
Chapter 1.
34
Basic Concepts and Facts
In the following, we simply use the term center manifold in place of local center manifold if the meaning is clear. We introduce the following three theorems, for the proof please see
[19J. The existence of center manifolds is given by Theorem 1.4.1. There exists a CT center manifold WC(O) for (1.4.1). The flow restricted to W C( 0) is governed by the c-dimensional system it = Au + f(u, h(u)) (1.4.3) for
lui
sufficiently small.
The second theorem shows that the flow of (1.4.1) near (x,y) = (0,0) is characterized by the flow of (1.4.3) near u = o. Theorem 1.4.2. i) Suppose that the zero solution of (1.4.3) is stable (resp. asymptotically stable, or unstable), then the zero solution of (1.4.1) is also stable (resp. asymptotically stable or unstable); ii) suppose that the zero solution of (1.4.3) is stable, then for the solution (x(t), y(t)) of (1.4.1) with initial values (x(O), y(O)) near (0,0), there exists a solution u(t) of (1.4.3) such that as t ~ 00
+ O( e-'Y t ), h(u(t)) + O(e-'Yt),
x( t) = u( t) y(t) =
(1.4.4)
where 'Y > 0 is a constant depending only on B.
If we substitute y(t) then obtain Dh(x){Ax
=
h(x(t)) into the second equation in (1.4.1),
+ f(x, h(x))} =
Bh(x)
+ g(x, h(x)),
(1.4.5)
or N(h(x)) = Dh(x){Ax+ f(x, h(x))}-Bh(x)-g(x, h(x)) = O. (1.4.6)
Equation (1.4.6) with the conditions h(O) = 0, Dh(O) = 0 is the system to be solved for the center manifold. In fact, it is equivalent
1.4.
Center Manifold Theory
35
to the original problem. The next theorem shows that the center manifold, however, can be approximated to any desired degree of accuracy. Theorem 1.4.3. Let be a C 1 mapping of a neighborhood of o in IRe onto a neighborhood of 0 in IRs with (O) = D(O) = O. Suppose that as x --t 0, N((x)) = O(lxl q ) for some q > 1, then
as x
--t
O.
Example 1. Consider the planar vector field
x=
x 2y - x 5 ,
(1.4.7)
iJ = -y + x 2 ,
where (x, y) E IR 2 • 0 is a degenerate critical point of (1.4.7). The question is whether or not 0 is stable. The linearized system has eigenvalues 0 and -1. We now use the theorems above to answer the question. From Theorem 1.4.1, there exists a center manifold of (1.4.7) which can be locally represented as
WC(O) for
= {(x, y)
E IR21 y
= h(x), Ixl < 8,
h(O)
= Dh(O) = O}
(1.4.8)
181
sufficiently small. We now want to compute it. Assume that h( x) has the form
(1.4.9) and substitute (1.4.9) into (1.4.6). Comparing equal powers of x, we can compute h( x) to any desired order of accuracy. In practice, computing only a few terms is usually sufficient to answer the question of stability of O. The center manifold is given by (1.4.6). Now we have
A = 0, f(x,y) = x 2 y - x 5 ,
B = -1, g(x,y) = x 2 •
(1.4.10)
36
Chapter 1.
Basic Concepts and Facts
We get
+ 3bx 2 + ... )(ax4 + bx 5 + ax 2 + bx 3 - x 2 + ... = o.
N(h(x)) =(2ax
x 5 + ... )
(1.4.11)
In order for (1.4.11) to hold, the coefficients of each power of x must be zero. Thus we have (1.4.12) Using (1.4.12) along with Theorem 1.4.1, the vector field restricted to the center manifold is given by (1.4.13) For x sufficiently small, x = 0 is unstable in (1.4.13). Hence, by Theorem 1.4.1, (x, y) = (0,0) is unstable in (1.4.7), see Fig. 1.4.1. y
W'(O)
--------~~~~~----___
o
x
E'
Fig. 1.4.1
Remark 1. From this example we should note the failure of the tangent space approximation. For (1.4.7), one might expect that the y components of the orbits starting near 0 should decay to zero exponentially. Then the question of stability of the origin should reduce to a study of the x components of the orbits starting near O. Suppose we set y = 0 in (1.4.7) and look at the reduced equation (1.4.14)
1.4.
Center Manifold Theory
37
°
This corresponds to approximating WC(O) by EC. However, as x = is stable for (1.4.14) we would arrive at the wrong conclusion that (x,y) = (0,0) is stable for (1.4.7).
Remark 2. There are similar concept and theorems on center manifolds for systems given by maps. We omitted the details here. 1.4.2.
Properties of center manifolds
We now give some important remarks on center manifolds. (1) In general the center manifold of the system (1.4.1) umque.
IS
not
Example 2. Consider the planar vector field
if
(1.4.15) = -yo
Let c be a real number and let
. {ce-X2 /2
h(x,c) = .0,
'
x> 0, x ::; 0.
For any c, the curve y = h(x, c) is an invariant manifold of (1.4.15) and by Definition 1.4.1, a center manifold because h(O, c)
= 0,
oh
ax (0, c) =
0.
Figure 1.4.2 shows the phase portrait near O. However, by Theorem 1.4.2, for any two center manifolds y = h 1 (x), y = h 2 (x), we have as x
---t
°
for all q > 1. It means that the Taylor series expansions of any two center manifolds agree to all orders. (2) Theorem 1.4.1 says that if j, g in (1.4.1) are C r (r 2: 2), then h is But if j, g are analytic, then in general (1.4.1) does not have an analytic center manifold.
cr.
Chapter 1.
38
Basic Concepts and Facts
Fig. 1.4.2 Example 3. Consider the planar system
x = -x3,
(1.4.16)
Suppose (1.4.16) has a center manifold y = h(x), where h is analytic at x = O. Then It is easy to see that a2n+l
= 0,
n = 1,2""
a2 = 1, a n+2 = nan,
n = 2,4""
.
This implies that h( x) cannot be analytic. The next example shows that (1.4.1) does not have a Coo center manifold, even if /, 9 are analytic.
Example 4. Consider the 3-dimensional system
x = -EX iJ
= -y
E = O.
x3,
+ x2,
(1.4.17)
1.4.
Center Manifold Theory
39
Suppose that (1.4.17) has a Coo center manifold y = hex, E), for Ixl < 8, lEI < 8. Choose n > 8- 1. Since hex, (2n)-1) is Coo in x, there are constants aI, a2,' .. ,a2n such that
hex, (2n)-1) =
2n
L
ai xi
+ O(x2n+1)
i=l
for
Ixl
=0
sufficiently small. One can show that ai
if i is odd and, if
n> 1, [1 - (2i)(2n tl]a2i
= (2i - 2)a2i-2,
i
= 2, ... ,n
and a2 =I- o. It is easy to show that this contradicts the hypothesis that h is CT. (3) Center manifold need not be unique, but it can be shown that due to its attractive nature, certain orbits that remain close to the critical point for all time must be on every center manifold of the critical point, for example, critical point, periodic orbits, homo clinic and heteroclinic orbits. 1.4.3.
Center manifolds depending on parameters
Suppose the system depends on a number of parameters, say ,X E IRk. In this case we write (1.4.1) in the form
x = Ax + f (x, y, ,x), iJ
= By
+ g(x, y, 'x),
(1.4.18)
where
f(O, 0, 0) = 0,
D f(O, 0, 0) = 0,
g(O, 0, 0) = 0,
Dg(O, 0, 0) = 0,
and we have the same assumptions on A and B as in (1.4.1), with f, 9 also being CT (r ~ 2) in some neighborhood of (x, y,,x) = (0,0,0). The way in which we will handle parametrized systems is to include the parameters ,x as the new independent variables as follows:
Chapter 1.
40
Basic Concepts and Facts
x = Ax + f(x, y, >'), >.
= 0,
if
= By
(1.4.19)
+ g(x, y, >'),
where (x, >., y) E IRc X IRk X IRs. (1.4.19) has a critical point at (x, >., y) = (0,0,0). The matrix associated with the linearization of (1.4.19) about 0 has (c+k) eigenvalues with zero real parts and s eigenvalues with negative real parts. Now let us apply center manifold theory to the above. Modifying Definition 1.4.1, a center manifold will be represented as a graph over the x and>' variables, i.e., the graph of h(x, >.) for lxi, 1>'1 sufficiently small. Theorems 1.4.1-1.4.3 still apply. The vector field reduced to the center manifold is given by
u = Au + f(u, h(u, >'), >'),
(1.4.20)
>. = 0,
Thus, adding the parameters as new independent variables merely acts to argument the matrix A in (1.4.1) by adding k new center directions, and the theory goes through just the same. From the Theorem 1.4.1, we have
WC(O) = {(x, >., y) E IRc X IRk
x IRS I y =
h(x, >'),
Ixl < 8,
(1.4.21)
1>'1 .) under the system (1.4.19) we have
if
=
Since
D>.h(x, >.)x
+ D>.h(x, >.),x =
x=
Bh(x, >.)
+ g(x, h(x, >'), >').
Ax + f(x, h(x, >'), >'),
>. = 0,
(1.4.22)
(1.4.23)
1.4.
41
Center Manifold Theory
we obtain that h(x, A) must satisfy
N(h(x, A)) = Dxh(x, A){Ax
+ f(x, h(x, A), A)}
(1.4.24)
-Bh(x, A) - g(x, h(x, A), A) = 0, which is similar to (1.4.6).
Remark 3. We explain why we do not allow the matrices A and B to depend on A. In fact, by considering A as new variables, terms such as or
YiAj,
1:Si:Ss,1:Sj:Sk,
become nonlinear terms. In this case, the parts of matrices A and B depending on A are now viewed as nonlinear terms and are included in the terms of f and 9 respectively.
Example 5. Consider the system
x=
AX - x 3
+ xy,
(1.4.25)
iJ = -Y + y2 - x 2,
where A is a real parameter. The aim is to study small solutions of (1.4.25) for small IAI. We write (1.4.25) in the equivalent form
x = AX iJ
= -y
A=
x 3 + xy,
+ y2 -
x 2,
(1.4.26)
o.
When it is considered as an equation on IR3 the AX term in (1.4.26) is nonlinear. Thus the linearization of (1.4.26) has eigenvalues -1,0, O. (1.4.26) has a two-dimensional center manifold y = h(x, A), Ixl < 81 , IAI < 82 , To find an approximation to h set
N((x, >.)) = x(x, A){AX - x 3 + x(x, A)} + (x, A)
+ x2 -
2(X, A).
Chapter 1.
42
Basic Concepts and Facts
Then, if q>(x, A) = -x 2, N(
+ O(lulc(u, A)),
(1.4.27)
A = O. The zero solution (u, A) = (0,0) of (1.4.27) is stable, so the representation of solutions given by Theorem 1.4.2 applies here. For -82 < A < 0, the solution u = 0 of the first equation in (1.4.27) is asymptotically stable and so does the zero solution of (1.4.25). For 0 < A < 82 , solutions of the first equation in (1.4.27) consist of two orbits connecting the origin to two small critical points. Hence, the stable manifold of the origin for (1.4.25) forms a separatrix, and the unstable manifold consists of two stable orbits connecting the origin to the critical points. The next example shows the application of the theory to a singular perturbation problem. Some further discussion will be developed in Chapter 6. Example 6. Consider the system (see [19])
iJ
=
-y+ (y+c)z,
AZ = Y - (y
+ l)z,
(1.4.28)
where A > 0 is small and 0 < c < 1. If A = 0, then (1.4.28) degenerates into an algebraic equation and a differential equation. Solve the algebraic one we get y z=-(1.4.29)
y+ l'
and substituting it into the first equation of (1.4.28) leads to . Y
-JLY
= 1 + Y'
(1.4.30)
where JL = 1- c. By the methods of singular perturbation, it is known that for A small enough, under certain conditions, the solutions of
1.4.
Center Manifold Theory
43
(1.4.28) are close to those of the degenerate systems (1.4.29), (1.4.30). We now show how center manifolds can be applied to obtain a similar result. Let t = )..7. Denote the differentiation with respect to 7 by a prime, then we obtain the equivalent form of (1.4.28):
y' = )..f(y, w),
w'
= -w + y2 -
yw + )..f(y, w),
(1.4.31)
)..' = 0,
in which f(y, w) = -y + (y + c)(y - w) and w =y - z. (1.4.31) has a center manifold w = h(y, )..). To find an approximation to h set
N(ip(y, )..))
=
)"y(Y, )..)f(y, '1 < E, such that it has exactly m limit cycles in UoJ 0).
°
Proof. Let the succession function of (2.1.1) be ak =1=
0,
k
= 2m + 1.
By the continuity of solutions with respect to the initial values and parameters, the corresponding succession function of the perturbed system has the form (2.1.7) as 1>'1 small enough, where o'i(>') are all small, and o'k(>') =1= 0. Thus we may choose Eo, Do > 0, such that there are at most k = 2m + 1 roots in (-Do, Do) of the equation d(r, >.) = when 1>'1 < Eo. If the perturbed system with 1>'1 < Eo has more than m, say (m + 1), limit cycles in U8JO), each of which meets () = and () = 7r at points with radicals rl, r2 E (0, Do), respectively, that is,
°
°
By (2.1.4), it means that rl, -r2 are two zeros of d. Corresponding to (m + 1) limit cycles, there are 2(m + 1) zeros of d, which contradicts the conclusion above. i) is thus proved.
Chapter 2.
48
Bifurcation of 2-Dimensional Systems
To prove conclusion ii), we give a series of perturbations in the following. Denote the right hand sides of (2.1.1) by Po(x,y), Qo(x,y) respectively. Consider the perturbed system in the following form
x = Po(x, y) + AoX + AIX(x2 + y2) + ... + Am_IX(X 2 + y2)m-1 == P(x, y, Ao,'" ,Am-d,
if = Qo(x, y) + AoY + AIy(x2 + y2) + ... + Am_IY(X 2 + y2)m-1 == Q(x, y, Ao ,' .• ,Am-I), (2.1.8) for which the succession function near 0 is denoted by
d(r,A o,'" ,Am-I)
°
with
d(r,O, ... ,0) =d(r).
°
For E > 0, there exist < A* < E, < r* < 80 , such that for IAil < A*, i = 0"" ,m - 1, d is defined for all r < r*, and every zero if < r* corresponds to a periodic orbit of (2.1.8) entirely contained in UhJO). Since d(2m+I)(0) =1= 0, we may assume that it is positive. Choose < rl < r* such that
°
d(rl) = d(rl' 0"" ,0) > 0. Let Ao
= ... = Am-2 =
° °< lAm-II < A* and
such that
dl(rl, 0"" ,0, Am-d > 0, where dl is the value of d at Am-I =1= 0. We now show that for suitable choice of Ao,'" ,Am-I, (2.1.8) has m limit cycles in UhJ 0). Compute the focal value of (2.1.8) as Ao = ... = Am-2 = 0, and Am-I =1= 0, it is easy to obtain
dl2m- I)(0) =
27f(2m - 1)!Am-l.
That is,
dl(r) = 27fA m_Ir 2m - 1 + o(r 2m - l ) < 0,
°
if we take Am-l < and r small enough. Hence 0 becomes a fine focus of order (m - 1), the stability changes from unstable to stable, and
2.1.
Generalized Hopf Bifurcation
49
there is an unstable limit cycle around it at the same time. Proceeding this way, we decrease the order of fine focus of 0 and change the stability at the same time step by step, then obtain (m + 1) numbers rl > r2 > ... > rm+l > such that the succession function dm for system (2.1.8) as Ao , •.• ,Am-l are slightly varied from zero satisfies
°
-
-
-
{
0, dm(r2) < 0"" ,dm(rm+d >
°°
for odd m £ lor even m.
Thus, there exists exactly one zero in each interval of (rI' r2), (r2' r3), ... (rm, rm+l)' The corresponding system (2.1.8) has exactly m limit cycles in UoJO). 0 Remark 1. The values to, 00 should be chosen so small that any system (2.0.1) with IAI < to has a unique critical point in UdO), which is a focus. Then the succession function is well defined. Remark 2. In the case that there are m limit cycles near 0, each of them should be a simple limit cycle, i.e., its characteristic exponential is not 0. 2.1.2.
Examples
We show some examples of quadratic systems in Ye's classification, see [188]. Example 1. Consider the special type II of quadratic systems,
x= iJ
-y + dx
= x
+ lx 2 + mxy,
(2.1.9)
+ x 2•
If d = 0, then 0 is a fine focus of (2.1.9). The sign of the next focal value is determined by l(m - 2). Assume that l > for definiteness, then 0 is a I-st order fine focus as m =1= 2, which is stable if m < 2 and unstable if m > 2. For m = 2, 0 is a stable second order fine focus. Thus if we fix d = and increase m from 2, then 0 changes its stability and a stable limit cycle Ll appears at the same time. Change
°
°
Chapter 2.
50
Bifurcation of 2-Dimensional Systems
the stability of 0 again by decreasing d from 0, and an unstable limit cycle L2 appears inside of L I . Let d continue to decrease, then LI shrinks inward, while L2 expands outward and they coincide to become a multiple-two limit cycle, and then disappear (since (2.1.11) forms a generalized rotated vector fields with respect to parameter d, we can say that the limit cycles expand or contract monotonically). For fixed l the bifurcation diagram is shown in Fig. 2.1.1 (M2LCBM ultiple-2 limit cycle bifurcation). d
ILO
m
o 2LO', noLO MzLOB
Fig. 2.1.1 Example 2. For the complete type II of quadratic systems :i; = -y
+ dx + lx 2 + mxy + ny2,
iJ = x + x 2 ,
(2.1.10)
the focal values of 0 are given in [188], Sec. 12 by
+ n) - 2l, W 2 = m(5 - m)[n(l + n)2 - (2l + n)], W3 = m[2 + n(l + 2n)][n(l + n)2 - (2l + n)].
d = 0,
If d
=
WI
=
W2
WI = m(l
= 0, d
W3
= 0,
i= 0,
(2.1.11)
i.e.,
m=5,
3l
+ 5n = 0,
then 0 is a 3-rd order fine focus, and is stable if W3 = 5n(2 + n 2 /3)( 4n 2 /9 + 7/3) < 0, and unstable if W3 > 0. Starting from this
2.1.
Generalized Hopf Bifurcation
51
system (suppose W3 < 0, i.e., n < 0), we can obtain three small limit cycles by the following steps: First, change m from 5 and l slightly such that WI :::;: 0, W 2 > 0, then 0 becomes an unstable fine focus of order 2, while a stable limit cycle Ll appears around O. Second, change l again slightly such that WI < 0, then 0 becomes a stable fine focus of order 1, while another limit cycle L2 appears inside L 1 . Third, increase d from zero to positive, then 0 becomes an unstable strong focus, and a third limit cycle L3 appears inside L 2. Let d continue to increase, then Ll continues to expand, and L3 expands outward while L2 shrinks inwards. There must appear a multiple-2 limit cycle bifurcation, for which L 3, L2 coincide. Note that l, d are varied so little in the second and third steps that L 1 , L2 do not disappear. Instead of increasing, decreasing d in the third step does not change the stability of 0 and no L3 appears at all. In this case Ll contracts inward while L2 expands outward to become a multipletwo limit cycle then disappers, which corresponds to a multiple two limit cycle bifurcation. We may draw the bifurcation diagram on a sectional plane including the d-axis in (l, m, d)-space, see Fig. 2.1.2 for the fixed l case. d MzLOB ILO
\. 2£0',
/ noLO
Fig. 2.1.2 There are many interesting limit cycle bifurcation diagrams for system (2.1.10) as given in [184]-[186]. Various diagrams for quadratic systems with different orders of fine focus are given in [107].
52
Chapter 2.
Bifurcation of 2-Dimensional Systems
Example 3. Consider the polynomial Lienard system x=-y+F(x), y
(2.1.12)
= x,
in which (2.1.13) [103] proved that system (2.1.12) with m = 1 has at most one limit cycle. Rychkov showed that (2.1.12) with F(x) = alx + a3x3 + a5x5 has at most two limit cycles. For general m, (2.1.12) has at most m small limit cycles, and there are coefficients with aI, a3, ... ,a2m+1 alternating in sign such that (2.1.12) has exactly m small limit cycles, see [14]. [132] gave a theorem about the system (2.1.12) with
and lEI sufficiently small. This system has exactly m limit cycles which are asymptotic to the circles x 2 + y2 = N, N = 1, ... ,m as E - t 0 if and only if the mth degree equation al
2
+
3a3 5a5 2 8 P + 16 P +
c 2m+2a2m+1 m+l
m_
22m+2P - 0
(2.1.14)
has m positive roots P = N 2 , N = 1"" ,m. This allows us to construct polynomial systems with as many limit cycles as we like. For example, if we wish to find a polynomial Lienard system above with exactly two limit cycles asymptotic to circles x 2 + y2 = 1 and x 2 + y2 = 2. To do this, we simply set the polynomial (p - 12)(p - 22) equal to that in (2.1.14) with m = 2, we get 2
P - 5p+4
5
2
3
1
= 16a5P + Sa3P+ 2al'
2.1.
Generalized HopE Bifurcation
This implies that a5 enough, the system
=
16/5,
53
=
a3
-40/3,
al
=
8. For
lEI i=
0 small
x = -y + E ( 8x - ~o x 3 + 156 x 5) , y=x
has exactly two limit cycles. 2.1.3.
Applications to Hilbert's 16th problem
As well known, the world-famous mathematician D. Hilbert presented a list of 23 outstanding mathematical problems to the ICM1900. The second half of his 16th problem was: What is the maximum number of limit cycles for the polynomial system n
X=
L
aijxiyj,
i+j=O
(2.1.15)
n
iJ =
L
bijxiyj,
i+j=O
and what are the relative positions? From the Dulac theorem proved rigorously by Il'yashenko [89] and Ecalle etc, the maximum number of limit cycles of nth degree system (2.1.15) is finite. We denote this number by H(n), the so called Hilbert number. We discuss the problem related to Theorem 2.1.2, i.e., suppose that (2.1.15) has 0 as a critical point of the center type and consider the number of small limit cycles near o. This number presents the local cyclicity of 0, which is denoted by Ho(n). The local cyclicity is finite for any polynomial system, but Example 3 shows that Ho( n) can go to infinity as n - t 00, and also that there are many limit cycles that can be created from a center point. Obviously, Ho(l) = o. For n = 2, the quadratic system case, N. V. Bautin proved that if the first three focal values of a fine focus are all zero, then all the focal values must be zero, and then it is indeed a center. Moreover, he proved also that Ho(2) = 3, even if
54
o is a
Chapter 2.
Bifurcation of 2-Dimensional Systems
center. The idea of using the bifurcation technique to create many small limit cycles from 0 was originally from him (the so-called Bautin technique, which we have already used many times above). In the case n 2: 3, it is quite difficult to evaluate the number Ho(n). For n = 3, there are also many works devoted, such as, to finding some necessary and sufficient conditions under which the cubic system has the origin as a center, and to present the focal values by the coefficients of polynomials on the right side of the system. This is a very complicated work, and computer calculations have been used to get many limit cycles from fine focus of higher orders. Up to now, there are 8 small limit cycles found for some cubic system by Lloyd etc., and he conjectures that Ho(3) = 8. Some other mathematicians believe the number of Ho(3) is greater than 8. However, it is still an open problem. Also, it is interesting to think about such a question: whether or not the second conclusion of Theorem 2.1.2 can be realized in polynomial systems with the same degree as the unperturbed system? When the order m of the fine focus is greater than the degree n of the system, the perturbation (2.1.8) will be unapplicable to answer this question since its degree is m. In general, we can say the answer is yes if the perturbed polynomial systems include enough parameters (Le., coefficients of the polynomials P, Q) to form a codimension-m unfolding. For the quadratic case, we already see in Example 2 of last paragraph that a quadratic system with a multiple-3 fine focus can be perturbed three times in quadratic polynomials to get 3 limit cycles. But for an nth degree system (n 2: 3), such steps are very difficult to be achieve. The problem is that the calculations of the successive focal values are much more complicated (the higher the order, the more complicated is the calculation), and the expressions of the focal values in terms of the coefficients of the system are as complicated as the order is high. That is the main reason that Lloyd has to use computer algebra to get the 8 limit cycles of a cubic system.
2.2.
Bifurcation of Multiple Limit Cycles
2.2.
55
Bifurcation of Multiple Limit Cycles
To study the problem of the number of limit cycles bifurcated from a multiple limit cycle, we must construct the Poincare map near this limit cycle and then investigate the succession function. Thus, both the methods used and the results obtained are entirely analogous to those of limit cycles created from a fine focus. We now discuss this problem in the following.
2.2.1.
Main theorems
Consider the system (2.0.1). Let Lo be a closed orbit of (2.0.1) for A = 0, represented by
x
= for Inl < n*. We choose < nl < n* such that
°
°
d(nl' 0"" ,0) = d(nr)
> 0. Then
> 0.
(2.2.5)
Now suppose that Al = ... = Ak-2 = 0, Ak-l i= 0, IAk-11 sufficiently small and consider the succession function dl(n) = d(n, 0"" ,0, Ak-r) of the system. It is easy to see that (2.2.6) where Cl is a nonzero constant. From (2.2.5) we may choose n* so small that if IAk-ll < A* then
d1 (nl) > 0. If we take Cl > 0, Ak-l < 0, then by (2.2.6), for Inl sufficiently small, d1 (n) < 0. Choose one of these numbers smaller than ni and denote it by n2. Thus (2.2.7) Along the same lines and after the (k -l)-th step, we obtain a system (2.2.4) and the numbers nl,'" ,nk, with
°
0) and has exactly one limit cycle for A > 0 (or < 0). Example 1. Consider the quadratic system of type I = -y + dx + lx 2 + mxy + ny2,
x
if
(2.3.1)
= x.
If d = 0, m(l + n) =1= 0, then 0 is a first order fine focus of (2.3.1). For definiteness, let m(l + n) < 0, then 0 is stable. Choose d > 0 sufficiently small, then 0 changes its stability and a unique limit cycle bifurcated from O. It expands monotonically as d increases, and becomes a homoclinic cycle passing through saddle N(O,~) as d = do, after which there is no limit cycle as d > do. It is shown that (2.3.1) undergoes a homo clinic bifurcation as d passes through do, and has no or one unique limit cycle for values of d on different sides of do. The bifurcation diagram is shown in Fig. 2.3.2. The uniqueness of limit cycles in the global has been proved for all parameters, see [188J, Sec. 12. Thus, Fig. 2.3.2 is complete in the whole parameter space. After these results were obtained for nondegenerate homo clinic and heteroclinic bifurcations, there are a number of papers considering the degenerate cases of homoclinic loop with D(N) = 0 and heteroclinic loop with D(Nl)D(N2) < 0, see [76], [113], [117], [118], [133], and related references in [115], [188J.
2.3.
Homoc1inic and Heteroc1inic Bifurcations
65
d
noLC
noLC
@
HoeB
Fig. 2.3.2 We now introduce the more general results obtained by the authors in [76], [113] in the next two paragraphs. 2.3.2.
Degenerate case
Suppose that C 3 system (2.0.1) has 0 as a saddle and a homo clinic loop Lo passing through 0 for)' = O. We denote the right-hand sides of (2.0.1) by Po(x,y), Qo(x,y) as). = O. Take a point Ao E Lo and a normal line segment l through Ao. Consider the orbit arc AIA2 with Al sufficiently near Ao and A2 being the next intersection point of the orbit and l. Let ~
A2 = f3(u)n
Lemma 2.3.4.
f3 '( u )
+ ao,
u < 0,
f3(u) < O.
(2.3.2)
We have
1 = --exp 1 + Eo
J~
AJA2
(8P -8o X
8 Qo ) + -8 Y
dt,
(2.3.3)
where Eo ---+ 0 as u ---+ O. Furthermore, if O. is a fine saddle, z. e., Do = D(O) = 0, then the infinite integral
o 8 Qo ) dt _;; (8P -+-Lo 8x 8y
G'-
converges.
(2.3.4)
Chapter 2.
66
Bifurcation of 2-Dimensional Systems
For the proof see [24]. Suppose that the system (Po, Qo) has 0 as a saddle. Then there exists an E neighborhood U of 0 and a C 3 transformation such that the system can be changed into the following normal form in U,
x = f..LIX + x2yRI(X, y),
(2.3.5)
where f..LI > 0, -f..L2 < 0 are the eigenvalues of the linearization at 0, and R I , R2 are continuous functions. Let (2.3.6) Then we have Lemma 2.3.5. The following conclusions hold: i) if Do < 0 (> 0), i5' --t -00 (+00) as Al --t Ao; ii) if Do = 0, i5' --t a as Al --t Ao.
Proof. The system can be taken in the form (2.3.5) in an E neighborhood of 0 as above, and Lo is shown in Fig. 2.3.3. Let 0 < 8 < E. Take two points BI = (0, -8), B2 = (8,0), and a ray l' starting from o towards the interior o! Lo. Then the lines y = -8, x = 8 and l' intersect the orbit arc AIA2 at C I , C 2 and Co, respectively. i5' can be represented by (2.3.7) It is easy to see that AIA2--t Lo if Al --t Ao. If Do i= 0, then the values of I~ + ~I have a positive lower bound for points on C;C2 ~
when C I C 2 is sufficiently near 0 iAI sufficiently near Ao). But the time interval for the orbit arc C I C 2 can be arbitrarily large as Al is sufficiently close to Ao. Thus, J ~ + J ~ is the main part of i5'. GIGo GoG 2 Then conclusion i) can be easily obtained.
2.3.
Homociinic and Heterociinic Bifurcations
67
Fig. 2.3.3 We now prove ii). Suppose that Do = O. By (2.3.5) in U, we have
oPo oQo () ax + oy = xyR x, y . CICo, CoC 2 can be represented by x = tp(y), y = 7/J(x), respectively. Then
r~ (OPo + oQo)dt=jYCO
ie1 e o
ax
oy
-6
tpR(tp(y),y)dy ~O -P,2+tpyR2(tp(y),y)
asAl~Ao.
Similarly,
Then, we obtain
- (1B1 Ao + 1) (OP oQO) -axO+ -dt = B2 A o oy
(T
~
~
~
.
o
The lemma is proved. Corollary 2.3.6. Let Do (T0).
(T
= o.
Then Lo is stable (or unstable) if
Consider a C 3 system in the following form,
x = Po(x,y) +af(x,y,a,b), if = Qo(x, y) + ag(x, y, a, b),
(2.3.8)
Chapter 2.
68
Bifurcation of 2-Dimensional Systems
where a E JR, b E JR n , n 2: 1. Suppose that (2.3.8)a=o has a homoclinic loop Lo passing through the saddle O. By local transformation (2.3.8) can be converted into
x = J.Ll(a,b)x + x 2yR1 (x,y,a,b), iJ
= -J.L2(a, b)y
+ xy2 R 2(x, y, a, b)
(2.3.9)
in the small neighborhood of O. Take Ao on x-axis near 0 (on the unstable manifold of 0), and a section line l through Ao. Then for lal small enough, the length of the vector with end points which are the intersection points of l and the stable and unstable manifolds of 0, represented by W~b and W:b respectively, can be represented by (see
[24])
where M(b) is the Melnikov function
(2.3.10) If Uab = 0, then W~b and W:b coincide; if Uab < 0 or Uab > 0, then their relative positions are as shown in Figs. 2.3.4( a) and (b) respectively.
o
(a)
(b) Fig. 2.3.4
2.3.
69
Homoc1inic and Heteroc1inic Bifurcations
We now consider the succession function with a spiral AlA2 inside L o , where AI, A2 E l, denoted by
Fo( u, a, b) = f3( u, a, b) - u,
u
< 0,
(2.3.11)
in which AI, A2 are written as (2.3.2) with f3 = f3( u, a, b). It is easy to see that limu-to Fo( u, a, b) = Fo( 0, a, b) exists. We prove the following lemma.
Lemma 2.3.7. Suppose that Do = i) if M(b) "/: 0, then Fo(O, a, b) = constant; ii) ~(u,a,b) = r.p(u, a, b) exp (/1 2 where p = p(u, a, b) > is the length tiple.
°
0. Then for every fixed b E IR n ,
aM(b)N
+ o(a),
where N is a
;:t In p), of IAoA21 with a constant mul-
Proof. For Uab > 0, we see that Fo(O, a, b) is the length of AoA~ from Fig. 2.3.4(b) and Fo(O, a, b) = Uab. i) is obtained. For Uab ::; 0, we have Fo(O, a, b) ::; 0, the absolute value of which is the length of AoA~ in Fig. 2.3.5. Let Ao = (f, 0) with f > small and consider points D l , El with Dl = (0, -f) and El = (Xo, -f) E W:b · We have aM(b) + o(a) X----;====~~=~= 0JP;(Dt) + Q~(Dt)
°
It suffices to prove that
Fo(O, a, b) = f3(0, a, b) = -xo + o(a). Let,
= ,(a, b) = J.Ld J.Ll
with ,(0, b)
dy y -d = -,-(I x x
= 1.
(2.3.12)
From system (2.3.9) we get
+ xyR(x,y,a,b)),
in which R is a continuous function. Let p = yx'Y, then
(2.3.13)
Chapter 2.
70
Bifurcation of 2-Dimensional Systems
o A'.
Fig. 2.3.5 The solution of (2.3.9) through
Since
A~ =
E1
can be represented by
(f,{3(O,a,b)), we get {3(0, a, b) =
_f 1-1'
xJe-
J: l'yRdx.
(2.3.14)
o
Similar to the proof of Lemma 2.3.5, it is easy to prove that
lim (f
a-O lxo
If M(b) =J 0, from ,(a,b)
= 1 + O(a)
lim ( _ 1) In (_ a-O '
and then
f1-l'xr1 ---t
1 if a
{3(0, a, b)
yRdx = 0.
---t
we obtain
°
aM (b) + o( a) ) = Jp;(Dd + Q~(Dd ' 0. By (2.3.14), we get
= -x o (1 + o(a)) =
Since P;(D 1) + Q~(D1) = f2JL~(0,b) conclusion i) is proved. Now prove ii). By Lemma 2.3.4,
-Xo
+ o(a).
= f 2JLi(0, b) = P;(Ao) + Q~(Ao),
a{3 = _1_ exp ( {~ (aPo+ aQo + a of + a a9)dt) . au 1 + fo lAoA~ ax ay ax ay
(2.3.15)
2.3.
Homoclinic and Heteroclinic Bifurcations
71
Similar to Lemma 2.3.5, we can prove (2.3.16) Suppose that the orbit through Al intersects the line y = Then
-/0
at
E~.
f af nag ) dt= ~ ;' (a ag ) dt+i.pI(u,a,b), jA\A2 ~ (a!:j+a a!:j+a nuy uX uy E\A2 uX where
i.p~is
along
E~A2
continuous and bounded with i.p1(U, 0, b) = 0. By (2.3.9), we have
with Ro continuous, and then
in which the second term of the right-hand side can be shown to tend to zero as a ---t 0. Then we have
af + an ag ) dt = - J-LI - J-L2 In IYA21 jA\A2 ~ (a!:j + i.p2(u,a,b), uX UY J-L2 E
(2.3.17)
with i.p2 continuous and bounded and i.p2( u, 0, b) = 0. From (2.3.16) and (2.3.17), ii) is proved. 0 Now we can prove the following theorem.
Chapter 2.
72
Bifurcation of 2-Dimensional Systems
Theorem 2.3.8. If Do = 0, (J = fLo (~+ ~) dt f::. 0 and M(bo) f::. for some bo E IR n , then for lal, Ib - bol f::. sufficiently small, system (2.3.8) has a unique limit cycle Lab near Lo if and only if a(J M (b o) > Moreover, Lab is stable (or unstable) as (J < 0 (or> 0) if it exists.
°
°
°.
Proof. From Corollary 2.3.6 and Fig. 2.3.4, we see that there are an odd (resp. even) number of limit cycles in a small neighborhood of Lo if a(JM(b o) > 0 (resp. < 0). It suffies to prove that there is at most one limit cycle near Lo for M(b o) f::. 0, (J f::. (small). Let Lab be a limit cycle of (2.3.8) and Lab --t Lo as a --t 0, b --t boo Let € > be small and Lab meet the lines Y = -€, X = € at points E l , E2 respectively. Then similar to Lemmas 2.3.5 and 2.3.7, we can prove that as a --t 0, b --t bo ,
°
°
;; Lab
OPo OQO)d (+- t--t(J aX oy
and
;; Lab
of ( a£l uX
(9) dt = - /-Ll-/-L2IYE11 ( + a£l UY /-L2 In - € + i.p2 YE
with lima ..... O,b ..... bo i.p2(YEp a, b) o(a)l, we have
2,
a, b) ,
= 0. From IYE 1 > IUabl = laM(bo)N + 1
In IU;b I < In IY:1 I < 0,
°< I(/-Ll - /-L2) Thus as a
In IY:111
0), Lab is stable (unstable), and then unique.
0
2.3.
73
Homoc1inic and Heteroc1inic Bifurcations
Corollary 2.3.9. Suppose (J i: 0, f.L1 (a, b) == f.L2( a, b). Then there is at most one limit cycle bifurcated from Lo if Ibl is bounded and lal is small enough.
For the case that Lo consists of two homo clinic cycles L1 and L 2, let
(J Mi(b) =
=;;
(GPo !;}
Lo
vX
GQO)dt,
(2.3.18)
+!;}
vy
hi e- J:(¥:-+~)dt(f Po - gQo)a=odt,
i = 1,2.
(2.3.19)
Then we may prove (see [76]) Theorem 2.3.10. Suppose that Mi(b o) i: 0, i = 1,2, for some bo E IR n , then for lal, Ib - bol i: 0 small enough, there is exactly one limit cycle Lab (resp. no limit cycle) in the outer neighborhood of Lo if and only if a(JMi(b o) < 0 (resp. > 0), i = 1,2.
Consider the heteroclinic case. Suppose that a heteroclinic cycle Lo consists of two orbits L 1, L2 connecting the saddle points Sl, S2. Let f.L1i(a, b) > 0, -f.L2i(a, b) < 0 be the eigenvalues of Si(a, b) of (2.3.8) near Si, and for lal small A
L.l.i
= f.L1i - f.L2i'
Ii
f.L2i, 2. = 1, 2 . =f.Lli
Suppose (J and Mi(b o) have the forms (2.3.18) and (2.3.19) respectively. Note that Ii = 1 if and only if D(Si) = O. Then proceeding along the same lines as above but with more careful estimations (see [76]) we can prove the following main conclusions. Theorem 2.3.11. Suppose that 1112 i: 1 and Mi(b o) i: 0 for some bo E IR n , i = 1,2, then for lal, Ib - bol i: 0 sufficiently small, (2.3.8) has a unique limit cycle near Lo if a(l - 11I2)Mi(bo) > 0, i = 1,2, and has at most two limit cycles if a(l - 11I2)Mi(bo) < 0, i = 1,2. Moreover, in the latter case, if (1 - 11)(1 - 12) ~ 0 is added then no limit cycle can be bifurcated from Lo.
74
Chapter 2.
Bifurcation of 2-Dimensional Systems
Example 2. For the quadratic system
x= if
-y
= x
+ dx + lx 2 + ny2,
+ x2,
(2.3.20)
it is possible to create two limit cycles in the neighborhood of a heteroclinic cycle. There are two saddle points: S1 = (O,~) and S2 = (-1, Y2) with Y2 = 2~ [d - l - J(d _l)2 + 4n] for (d - l)2 + 4n > O. For l > 0, there appears a limit cycle created from 0 if d increases from 0 to positive. It is easy to get the bifurcation diagram as shown in Fig. 2.3.6, in which we see a heteroclinic cycle bifurcation point H and a region with two limit cycles of (2.3.20). If we fix l and change the parameters d, n from H to this region, then there are two limit cycles created from the heteroclinic cycle passing through S1, S2'
J5
d
/
H
o Fig. 2.3.6 Theorem 2.3.12. If /1 = /2 = 1 (i.e., S1, S2 are both fine saddles), then letting M(b o) = M 1(b o) + M 2(b o) we have i) Lo is stable (or unstable) if cr < 0 (or> 0); ii) (2.3.8) has a unique limit cycle (or no limit cycle) near Lo for lal, Ib - bol i= 0 sufficiently small if acrM(bo) > 0 (or < 0); iii) there is at most one limit cycle near Lo if cr i= 0 and ~i == 0, i = 1,2 (i.e., S1(a, b), S2(a, b) are both fine saddles).
2.3.
Homoc1inic and Heteroc1inic Bifurcations
2.3.3.
75
Highly degenerate case
We now consider the case that (2.3.8)a=0 has a singular closed orbit Lo with a family of closed orbits in its internal (or external, in the cases as in Figs. 2.3.1(a), (b)) neighborhood, and the problem about the limit cycles bifurcated from Lo. Let L>.. be a family of closed orbits of (2.3.8)a=0 with
L>.. : x
= x(t, A),
y
= y(t, A),
0 ~ t ~ T>.., 0
.. ~ Lo as A ~ +0, where Lo is a homoclinic loop through saddle O. (2.3.8) may be considered as (2.3.9) in the small neighborhood of o as above. We may construct curvilinear coordinates near each closed orbit L>.. and consider the corresponding succession function (see Sec. 2.4 below) a
F(u,A,a,b) = K(A) [ (A, b)
+ l(u,A,a,b)],
(2.3.21)
where
K(A) = P;(x(O, A), y(O, A))
(A, b) =
+ Q~(x(O, A), y(O, A)),
J e- J:(P~x+Q~y)dt(Pog LA
1 (0, A, 0, b) =
QoJ)a=odt,
(2.3.22)
o.
Let M(b) be the Melnikov function represented by (2.3.10). Then similar to Lemma 2.3.5, we can prove that
M(b) = lim (A,b). >"-+0
Let
~(a,b) =
(2.3.23)
f..L1(a, b) - f..L2(a,b).
Theorem 2.3.13. The following conclusions hold for lal, Ib-bol =1=
o small enough,
i) if M(b o) =1= 0 for some bo E mn , then (2.3.8) has no closed orbit near Lo; ii) if M(b o) = 0, ~~(O, bo) =1= 0, then (2.3.8) has at most one limit cycle bifurcated from Lo.
Chapter 2.
76
Bifurcation of 2-Dimensional Systems
Proof. From Lemma 2.3.7:
+ FI(u, a, b)]
Fo(u, a, b) = aN[M(b)
with FI(O, 0, b)
= o.
Thus, if M(b o) =/: 0, then for lal, Ib - bol =/: 0 small enough, Fo =/: 0, and i) holds. To prove ii), suppose there are two limit cycles L I , L2 with Li ~ L o, i = 1,2, as a ~ 0, b ~ boo First, if Uab ::; 0, let Bi be the intersection point of Li and I; then the y-coordinate Yi of Bi is the same as the u-coordinate of Bi on I, and Fo(Yi, a, b) = 0, for i = 1,2. We may assume Y2 < YI < o. By using the Rolle theorem for Fo on [Y2, YI], there is a U o E (Y2, YI) such that ~(uo, a, b) = O. Take Al E I with U o as its u-coordinate, and the orbit through Al meets I again at a point A 2. Then Y2 < YA 2 < YI < Uab· From Lemma 2.3.7 we obtain
cp( u o,a, b) exp ( -
,6. In f..L2
Po) -
1
= 0,
(2.3.24)
where Po = p(u o, a, b), from which we see that .
hm
,6.
a->O,b->b o
Then cp(u, 0, b) = 1 by Fo limit then exists,
=
lim
a->O,b->b o
--In Po = f..L2
0, ~
=
0 as a
o.
(2.3.25)
=
O. The following finite
~(cp(uo,a,b) -1) = a
C;
(2.3.26)
and by (2.3.24), (2.3.25) we have C
+
lim a->O,b->bo
exp ( -
.0. Ji2
In p ) - 1 a
On the other hand, by (2.3.25), lima->o ~ as a ~ 0, b ~ bo , we get
0
=
= O.
~~(O, bo)
(2.3.27)
=/: 0
and Po ~ 0
2.3.
Homoc1inic and Heteroc1inic Bifurcations
lim
a-O,b-bo
exp ( -
II i-L2
In p ) - 1 a
0
=
77
1( ~ - --lnpo)(l+o(l))=±oo,
lim
a-O,b-bo a
/12
which contradicts (2.3.27). Next, if Uab > 0, then by Fig. 2.3.4(b) we see that Ui = Yi + Uab < 0, and, just as in the above case, it comes to a contradiction. The proof is thus completed. 0 For the case b E IR, we may see the bifurcation curves more clearly.
°
Theorem 2.3.14. Suppose that M(b o) = 0, M'(b o) =1= for some bo E JR, then there exists a unique continuous function b = b* (u, a) with b*(O, 0) = bo, such that for small enough lal, Ib - bol =1= 0, i) (2.3.8) has a homoclinic cycle near Lo if and only if b = b*(O, a) = bo + O(a); ii) (2.3.8) has closed orbit near Lo if and only if b = b*(u,a) for sufficiently small u < 0. Now consider the relationship between the succession functions Fo and F. Let AA be the intersection point of LA and l, and U o = -d(A o , AA)' Then the u-coordinate of AA is
u('\', a, b) = {u o ('\'), u o (,\,)
+ Uab,
Uab:::; Uab >
0, 0,
which is differentiable and monotonic with respect to '\'. It is easy to see that, for u('\', a, b) :::; 0,
Fo(u('\', a, b), a, b) =
°
~
F(O,'\', a, b) = 0.
(2.3.28)
If M'(b o) =1= 0, then there is a unique continuous function b = b~('\') with b~(O) = bo, such that ('\', b~) = 0, and there is an E('\') > 0, and a unique continuous function b = bt(,\" a) = b~('\') + O(a), such that,
°
for < lal < E('\'),
Ib - b~('\')1 < E('\'),
F(O,'\', a, bi)
= p~) [ (,\, , bi) + 1(0,'\', a, bi)] = 0.
Chapter 2.
78
Bifurcation of 2-Dimensional Systems
Since A( a, b) ~ 0 satisfying u( A, a, b) = 0 implies that A ~ A( a, b) is equivalent to U(A, a, b) :::; 0, (2.3.28) holds for A ~ A(a, b) and F(O,A,a,b*(u(A,a,b),a)) = O. Thus, there exists an Co > 0 such that for A ~ A(a, b), 0 < lal < co, Ib - bol < co,
b*(U(A, a, b), a) = brp, a).
(2.3.29)
This shows that the domain of bi is A ~ A(a, b), 0 < lal < co, Ib-bol < co. From (2.3.29) and Theorem 2.3.14, we can obtain the following. Corollary 2.3.15. If M(b o) = 0, M'(b o) # 0, then there is an Co > 0, such that for 0 < lal < co, 0 < Ib - bol < co, the number of limit cycles of (2.3.8) near La is the same as the number of the roots
of b = bi(A,a) with A > A(a,b). Moreover, (2.3.8) has a homoclinic cycle if and only if b = bi(A(a, b), a). Theorem 2.3.16. Suppose that M(b o) = 0, M'(b o) # 0 and ~~(O,bo) # 0 for some bo E JR. Let 8 = sgn(-M'(bo)~~(A,bo)) for
small enough A > O. Then for small enough lal, Ib - bol # 0 and near La, (2.3.8) has a unique limit cycle if 8(b-b*(0,a)) > 0, has a homoclinic cycle if b = b*(O,a), and no limit cycle if 8(b - b*(O,a)) < o. Example 3. Consider the system (1.3.6) in Bogdanov-Takens bifurcation:
x = y, if
= x - x 2 + c(8y + ,xy),
where, > 0, which has a family of closed orbits 1
0< A if c = o. For the intersection points of H as abscissas, we obtain {X2
= A and x-axis with Xl
.
Ao(h*)
= ... = A~k-l)(h*) = 0,
(2.4.8)
then (2.0.1) cannot have more than k limit cycles near becomes sufficiently small.
rho
as 1>'1
For the proof see [188]. Corollary 2.4.2. If An(h*) = 0, A~(h*) =1= 0, then rho can bifurcate a unique limit cycle of (2.0.1) if 1>'1 =1= 0 is small enough. Example 1. Consider the van der Pol system x =y,
(2.4.9)
iJ = -x + J.Ly(1 - x 2 ), which has a family of closed orbits verify
=
l
0
k
rh
:
x 2 + y2 = h 2. It is easy to
h 2 cos 2 t(1 - h 2 sin 2 t)dt
~) = o. = 7rh 2( 1 -"4
Al(2) = 0, and A~(2) =1= O. Hence we know that (2.4.9) has a unique limit cycle if IJ.LI =1= 0 is small, which approaches the circle x 2 + y2 = 4 as J.L ~ o. It is useful to consider a perturbed Hamiltonian system in the form
.
8H
x = - 8y
.
8H
y = 8x
+ p(x, y, >'), (2.4.10)
+ q(x, y, >'),
in which we assume that p(x, y, 0) == q(x, y, 0) == 0, and the general integral H(x,y) = h
Bifurcation of 2-Dimensional Systems
Chapter 2.
82
of the Hamiltonian system (2.4.10),\=0 has a family of closed orbits rho Now
p~o
= p~(x, y, A),
Q~o
=
q~(x, y, A),
and from (2.4.4) we get
Al(h) =
kh [~~p~(x,y,O) + ~~ q~(x,y,O)] dt
= irh r p~(x, y, O)dy - q~(x, y, O)dx = J JG)p~,\(x, y, 0)
(2.4.11)
+ q~,\(x, y, O)]dxdy
== (h), where G h is the interior region of obtain
rho
Then from Theorem 2.4.1, we
Theorem 2.4.3. If (h*) = 0, '(h*) =J 0, then as
IAI
=J 0 and small enough, (2.4.10) has a unique limit cycle near rho. Moreover, if
.(h*) = '(h*) = ... = (k-l)(h*) = 0,
(k)(h*) =J 0,
then (2.4.10) cannot have more than k limit cycles in the neighborhood of r h*· 2.4.2.
Weakened Hilbert's 16th problem
Originally, the Poincare method was used to study the problem of limit cycles created from a system with the origin 0 as a center. For this system, there is a family of closed orbits r h covering a simple connected region including 0, and the boundary of this region might be a singular closed orbit with several critical points (including those at infinity). Thus each limit cycle of the perturbed system may take a r h as its limit position (that is the rigorous Poincare bifurcation
2.4.
Poincare Bifurcation
83
mentioned above), or an 0 as its limit position (it is often called center bifurcation and may be considered as a fine focus bifurcation of infinite order), or a singular closed orbit as its limit position, which corresponds to the highly degenerate homoclinic or heteroclinic bifurcation stated in Sec. 2.3.3. Taking all these cases into account, it is often considered as the generalized Poincare bifurcation. To study such a bifurcation we must consider the zero point of the function as in (2.4.11). Related to such highly degerate bifurcation and the limit cycle problem of polynomial systems, V. Arnold posed the so-called weakened Hilbert '8 16th probem (see [9]). That is, how many real zeros does the function
I(h) =
J{
JH C I ; they then move separately, until they coincide to become a new multiple-two limit cycle when C = C 2 ; it eventually disappears when C > C2 while the rotation number p > ~.
"/...
- - - - - --:-i -
--, I
I I I I I
I
__4-______~~------c C1 C2
Fig. 2.4.2
Chapter 2.
86
Bifurcation of 2-Dimensional Systems
We sketch the main idea of the proof, which is related to the Poincare bifurcation of toral systems. For (2.4.14) we may assume that B = 1 (by scaling of t), and consider the limit cycle bifurcations in the (A, C, D) parameter space. We shall prove that for each rotation number p = ~, the Poincare bifurcation creates exactly two limit and there are two bifurcation surfaces of multiplecycles with p = Be m two limit cycles with p = ~ which end in two points on the C and D axes, see Fig. 2.4.4. For each of these points, the system is of the central type. First let A = 0, (2.4.14) becomes sin x dy _ D + C -.-. dx sm y
(2.4.15)
Consider the case p = ~ and express (2.4.15) in the equivalent form dy = dx
(~
_
2
a) + K a s~n x, sm y
(2.4.16)
where K, a are new parameters. For the degenerate system (2.4.16)a=o, there are integral lines y = ~x+h on the plane and these are all closed orbits with rotation number p = ~ on torus. Consider the rectangle 8 + 81 : [0,21l'] X [0,1l']. For (x, y) E 81 = [1l',21l'] X [0,1l'] the toral equation has the form dy = (~ _ dx 2
a) _ K as~n x. sm y
(2.4.17)
We may expand the solution of (2.4.15) with respect to a for sufficiently small lal:
y(x)
=
yo(x)
+ aY1(x) + o(a),
(2.4.18)
where Yo(x) = ~x is the solution of (2.4.15)a=o through O. Substitute (2.4.18) into (2.4.17) and compare with the terms of a. After a complicated estimation for Y1 (x) we can prove that:
2.4.
Poincare Bifurcation
87
1) if K < %, a > 0 is sufficiently small, then (2.4.14) has the structure of limit cycle type, and there are exactly two limit cycles with p = ~ near y = yo(x) if lal is small; 2) if K = %, a > 0 is sufficiently small, then there is a unique limit cycle (multiple two) which passes through points 0, B, O2 , see Fig. 2.4.3. All other orbits on the torus approach this semi-stable limit cycle as t --+ ±oo. Hence for C near 0 and D near ~, the corresponding system has the structure of limit cycle type. On the (C, D)-plane, there exist two multiple-two limit cycle bifurcation curves through the points G,O) and (O,~) such that the system corresponding to (C, D) located between these curves has two limit cycles with p = ~.
Fig. 2.4.3 G
~
Fig. 2.4.4
D
88
Chapter 2.
Bifurcation of 2-Dimensional Systems
Let Do = ~ - ao, Co = ~ao and laol small enough, (2.4.17) has a multiple-two limit cycle with p = ~ through points 0, B, O2 • Denote Ro = (Co, Do). Fix Do and let C decrease from Co, this limit cycle splits into two simple limit cycles and move separately. Thus they must coincide to become a new multiple-two limit cycle through the points AI, Bl and A~ (see Fig. 2.4.3) when C = C*, then disappear when C < C*. Denote Qo = (C*, Do). Starting from Qo, by fixing C and increasing D we can get R-l' again by fixing D and decreasing C we get Q-I' and so on. Starting again from R o , and fixing C = Co and letting D decrease we get QI, and so on. We obtain the sequences ~, Qi, i = 0, ±1, ±2,' .. in the (C, D)-plane, see Fig. 2.4.4, for which the corresponding toral system has multiple-two limit cycles passing through 0, B, O 2 and AI, B l , A~ respectively. For the interior points of each ~Qj with Ii - jl ::; 1, (2.4.14) has exactly two simple limit cycles with p = ~. In view of the arbitrariness of R o , such end points ~ and Qi form two continuous curves, which are the graphs of multiple-two limit cycle bifurcations. It is not too difficult to show that the method above may also be applied to the limit cycles for any p = ~. Thus, for each rational number p, there are exactly two multiple-two limit cycle bifurcation curves on the (C, D)-plane. G
Fig. 2.4.5. M 2 LCB surfaces
71"
and
71"1
with p = 1
2.4.
Poincare Bifurcation
89
For A # 0, the continuation method of multiple-two limit cycles, similar to that following the proof of Theorem 2.2.4, can be applied to obtain the multiple-two limit cycle bifurcation surfaces in (A, C, D)space, see Fig. 2.4.5. The study of toral systems is also valuable to some practical models. In general, one may consider the equations 01 = WI, O2 = W2 as two circular motions, and couple them to obtain a toral system
01 = WI + hI (fh'(h), O2 = where
hI, h2
w2
(2.4.19)
+ h 2 (8 1 , 82 ),
are periodic in 81 , 82 . For example, the system
01 = WI
-
a sin (8 1
-
82 )
-
(1 - a) sin (8 1
-
282 ),
O2 = w2
-
asin(8 1
-
82 )
-
(1- a)sin(8 1
-
282 )
(2.4.20)
appears in biomathematics, and its topological structure on the torus was studied by some authors, who obtained pictures similar to Fig. 2.4.2 by numerical calculations, see A. Cohen etc, J. Maih. Bio. 13(1982).
Chapter 3 Bifurcation in Polynomial Systems
Lh~nard
It is well known that the second-order nonlinear differential equation (3.0.1) x + f(x)x + g(x) = 0,
referred to as a Lienard equation, can be changed into the system of equations x = y, (3.0.2)
iJ
=
-g(x) - f(x)y,
or the system
x = y - F(x), iJ
= -g(x),
(3.0.3)
by a Lienard transformation, where F(x) = fox f(s)ds. Many mathematical models appeared in the fields of science and technology have the form (3.0.1) or its equivalent forms (3.0.2), (3.0.3). For examples, the motive behavior of a class of nonlinear oscillators is described by the equation ([87])
x + (a + ')'x 2 )x + (3x + bx3 = o. In the analysis of axial-flow compressor stability it apears as a system of the following type ([121]) x
=
y, 91
Chapter 3.
92
Bifurcation in Polynomial Lienard Systems
The system
appears in the study of three-dimensional viscous flow structures near a plane wall (see [95]). In some cases the models are not originally in such forms, but we may use suitable change of variables to reduce them to Lienard types, such as the FitzHugh nerve conduction system ([47]) .
X
1 3
3
= Y - -x + x + j..l,
if = p( a - x - by),
which will be analysed in Sec. 3.4.1 by changing variables. In addition, quadratic systems or the more general nonlinear systems of the type
can also be reduced to the form of (3.0.3) by changing variables (cf. [188], Sec. 15). Therefore, Lienard systems have very important significance in both in theoretical and practical points of view. In the past sixty years, many mathematicians have carried out extensive and deep research on them. In particular the books [188], (Secs. 5-7), and [193], (Ch. 4, Secs. 1-5) contain excellent summaries of the main results obtained in this area before 1984. In this chapter, we discuss the case that F(x), g(x) in (3.0.3) are polynomials, for which (3.0.3) is called a polynomial Lienard system, and focus on the problem of limit cycle bifurcations of these systems. Most of the material is concerned with new results obtained by mathematicians in China and other countries in recent years. These results are all global in both the phase space and parameter space.
3.1.
Boundedness of Solutions and Existence of Limit Cycles
3.1.
93
Boundedness of Solutions and Existence of Limit Cycles
When F(x) and g(x) are polynomials of x, the system (3.0.3) often has many critical points. It is well known that: the interior of a limit cycle of a quadratic differential system can only have a unique critical point, which must be a focus (see [188], Sec. 11). But, as we can see later, even for a very simple cubic Lienard system there may exist limit cycle or sepratrix loop surrounding several critical points simultaneously. So first let us generalize the concept of critical points and some classical theorems about limit cycles. 3.1.1.
Concepts of critical point system
Consider the planar autonomous system
x = P(x, y), i;=Q(x,y),
(3.1.1)
where P and Q are continuous functions of x, y, the uniqueness of whose solutions is guaranteed. Definition 3.1.1. A set of critical points of (3.1.1) is called an unstable (or astable) critical point system if the sum of the indices of all the critical points is +1 and there exists a bounded region D, containing this set of critical points, such that from every point on the boundary aD only positive (or negative) semitrajectories of (3.1.1) leave D. Definition 3.1.2. A set of critical points of (3.1.1) is called an unstable (or astable) critical point-cycle sytem if the sum of the indices of all the critical points is +1 and there exists at least one limit cycle or separatrix loop containing one or several of these critical points in its interior, and there exists a bounded region D containing this set, such that from every point on the boundary aD only positive (or negative) semitrajectories of (3.1.1) leave D.
Chapter 3.
94
Bifurcation in Polynomial Lienard Systems
Definition 3.1.3 A limit cycle is called small if it surrounds only one critical point, and is called large if it surrounds a critical point system with more than one critical points, or a critical point-cycle system. 3.1.2.
Boundedness of solutions
Consider the Lienard system
x = y - F(x), iJ
=
(3.1.2)
-g(x),
and the generalized Lienard system
x=
h(y) - F(x), iJ = -g(x),
(3.1.3)
where (x,y) E R2, F(x) = fox f(s)ds, and the following conditions are satisfied: (HI) f(x), g(x) and h(y) are functions of class c1, and thus the existence and uniqueness of the solutions of (3.1.3) are guaranteedj (H2) there exists CI ::; 0 ::; C2 such that xg( x) > 0 for x fj. [CI. C2], and G(±oo) = +00, where G(x) = fox g(s) dSj (H3) there exists dl ::; 0 ::; d2 such that yh(y) > 0 for y fj. [d l , d2 ], and h(±oo) = ±oo. The following notations will be used in this paragraph:
N
= {(x,y):
x
E [XI,X2],jyj2:
N}, where -00 < Xl < 0
0, let a(f3) = b(f3) = (f3 + 1) / d\ a being a real number, with 0 < a < b. When h(y) = F(x) and there exists a single-value branch for x 2: C2 or one for x::; CI, we let C 2: max{jclj,C2},
v+ = {(x, y) : h(y) = F(x), x 2: c}, I = {(x, y) : h(y) > F(x), x 2: c}, II = {(x,y) : h(y) < F(x),x 2: c}, III
=
{(x, y) : h(y) < F(x), x::; -c},
3.1.
Boundedness of Solutions and Existence of Limit Cycles
~+
= {(r,y)
:y
~-
> Yr},
= {(r,y)
:y
95
< Yr}, (h(Yr) = F(r)).
It is easy to see that the trajectories of (3.1.3) have the following properties. Lemma 3.1.1. Any trajectory of (3.1.3) has no vertical asymptote. In particular, the slope of the trajectory at any point pEN is bounded. Thus for any fixed Yo : Iyol » 1, if the trajectory passing through Po( 0, Yo) does not go to any finite critical point, then it must intersect the lines x = Xl and X = X2. Lemma 3.1.2. System (3.1.3) has no positive semitrajectory ' : for which x --t +00, y --t +00 in region I, and has no "(: for which x --t -00, Y --t -00 in region III.
In what follows, we shall prove some lemmas describing the behavior of the trajectories of (3.1.3) under some additional conditions, which are mostly from [68] and related references. Lemma 3.1.3. Suppose that 1) there exists c 2:: C2 (or -c :::; cd, such that F( x) 2:: b[ G( x)]Q for x 2:: c (orx:::; -c); 2) there exists d 2:: d 2 such that h(y) :::; yf3 for y 2:: d. Then (3.1.3) has a family of negative (or positive) semitrajectories which remain in II (or III) and go to infinity.
Proof. We only prove the conclusion for II, the case of III being similar. Under the conditions above, h(y) = F(x) has a unique single valued branch V+ in the region x 2:: c, y 2:: d. Consider the equations dy dx
g(x) F("x)-h(y)'
(3.1.4)
dy g(x) . dx - b[G(x)]Q - yf3
(3.1.4')
and
Chapter 3.
96
Bifurcation in Polynomial Lienard Systems
in the region U = ((x,y) : d::; h(y) < b[G(xW\x 2: c}. From 1) we have 0< dYI < g(x) < dYI . dx (3.1.4) - b[G(xW' - h(y) - dx (3.1.4') It is not difficult to verify that I~ : Y = [G~x) lli~l is the integral curve
of (3.1.4') passing through the point (c, b[~lli~l). Note that x < 0 in U, and the directions of the trajectories of (3.1.3) crossing I~ are shown in Fig. 3.1.1. Thus the negative semitrajectory I; of (3.1.3) 0 passing through p can only remain below I~ and go to infinity.
dl---+--
o
Fig. 3.1.1 Lemma 3.1.4. The right shift solution of the equation
dy du
({3 + 1)u/3
(3.1.5)
au/3 - y/3 '
originating from Po( u o, Yo) (au~ > yg) must intersect the curve y/3 = au/3. Proof. If y is bounded and u .......... +00 along an integral curve of (3.1.5), then we have ~ . . . . . ~ > O. So, without loss of generality, we may assume U o > 0, Yo > O. Let v = ~, then (3.1.5) becomes dv du
({3 + 1 - av + v/3+ 1 ) u(a - v/3)
cp( v) - u(a - v/3)"
3.1.
Boundedness of Solutions and Existence of Limit Cycles
97 1
It is easy to see that cp'(v) = 0 has a unique root v* = (t3~I):B, and from 0 < a < b we can get cp(v*) > 0, and then cp(v) > 0 for all v> o. Hence we have (V
a - w t3
u
J"Vo cp () dw = In-, W Uo
Yo
= UoVo·
These equations define a unique function u = u( v) which satisfies 1 Uo = u(vo) and takes on minimum value at VI = all. Let UI = U(VI)' then it is easy to know that point (UI, UI VI) is the intersection point of the right shift solution of (3.1.5) from Po and the curve yt3 = au t3 .
o In the following we consider the equation
dz dy = F(z) - h(y),
z ~ Zo ~ 0
(3.1.6)
~ zo, h(y) ~ yt3 for y ~ d, then the right shift solution [p of (3.1.6) starting from p(zo, yp) on yz~ must return to intersect yz~.
Lemma 3.1.5. Suppose that F(z)
:s; az for z Q
Proof. Note that ~ < 0 above V+ : h(y) = F(z), so we need only to prove that [p must intersect V+. From F( z) :s; az and the comparison theorem of differential equations, it is enough to prove that the right shift solution ')'p of the equation Q
dy
1
dy du
(,8 + 1 )u t3
comes
au t3 - h(y)'
(3.1.7)
(3.1.7')
By using the condition h(y) ~ yt3 for y ~ d, and Lemma 3.1.4 as well as the comparison theorem, we know immediately that the right shift solution ')" of (3.1.7') starting from p must intersect h(y) = au t3 . 0
Chapter 3.
98
Bifurcation in Polynomial Lienard Systems
In the same way we can prove Lemma 3.1.6. Suppose that F(z) ~ -az Ci for z ~ zo, h(y) ::; -lyli3 for y ::; -d, then the right shift solution IP of (3.1.6) from P(ZOl yp) E ~~ must return to intersect ~~. Now we turn to discuss the criteria for the bounded ness of solutions of (3.1.3) with several critical points. Theorem 3.1.7. Suppose that system (3.1.3) satisfies (Ht}-(H3), and there exist i31 > 0, i32 > 0, C ~ C2, Xo ::; Cl, and d ~ d2 , such that F(x) ~ b2 [G(x)]Ci2 for x ~ c, F(x) ::; al[G(x)]Cil for x ::; xo, and yi31 ::; h(y) ::; yi32 for y ~ d, then all solutions of (3.1.3) are bounded in the positive sense. Proof. From Lemma 3.1.3 we know that system (3.1.3) has a family of negative semitrajectories which remain in II and go to infinity. Note that iJ < in II, so if the positive semitrajectories do not go to a finite critical point, then they must intersect the lines x = C and x = Xo by Lemma 3.1.1. Let z = G(x), x ::; x o, and Zo = G(zo). Notice that G(x) is strictly monotonically decreasing in (-00, x o), so it has an inverse function x = z*, there exist a unique x~ < x* and a unique x~ > x* satisfying G(x~) = G(x~) = Zoo In the region N = {(x,y) : x~ ~ x ~ x~, iyi ~ 1}, if ' : does not go
3.1.
Boundedness of Solutions and Existence of Limit Cycles
101
to any finite critial point, then it must intersect the lines x = x~ and x = x~ by Lemma 3.1.1. From the condition F a) in Theorem 3.1.8, and Lemmas 3.1.1, 3.1.5, and 3.1.6, ' : can remain neither in the part of x :2: x~, nor in that of x :s; x~. Thus it must turn around at the origin if it does not go to any finite critical point. By the transformation (3.1.9), the image of the connected part of containing p(O, Yo) and lying in the strip region x~ :s; x :s; x~, IYI < +00 can be presented piecewisely by
'P
Y=Yi(Z),
zEJi,
z:S;zo,
i=m,m+1,"',n,
where Yi(Z) satisfies (3.1.10)i and the initial conditions
Yo(O)
= Yo = Yl(O),
Yi(Zi)
= Yi+l (Zi), i = m, m + 1"
.. ,n.
That is, dz
(Z
Yi+l(Z)-Yi(Zi)=J>F( z,
Yi(Z)-Yi+l(Zi)
=l
z
Zi
)
'Pi+l dz
h(
Yi+l
),i=0,1, ... ,n,
F() h( )' i=m,m+1,'" ,0. 'Pi Yi
We now prove three lemmas under the conditions of Theorem 3.1.8. Lemma 3.1.9. As IYol
--t
+00,
uniformly in [0, zo], and thus
h(Yi(Z)) - h(Yj(z)) h(Yi(Z)) h(yo)
--t
1,
--t
ZE J
0,
i,j i,
Z E Ji n Jj , =
m,'" ,n + 1
uniformly in [0, zo].
Proof. Equivalently, we prove
h(y(x)) - h(yo)
--t
°
as IYol
--t
+00
Chapter 3.
102
Bifurcation in Polynomial Lienard Systems
uniformly in [x~, x~], where y(x) satisfies the equation
dy dx
g(x) F(x)-h(y)'
y(O) = Yo.
In fact, from the equation. above it is easy to see that y( x) - Yo as IYol -+ +00. Moreover,
-+
0
=
0
h'(y(~))g(~)
h(y(x)) - h(yo) = F(~) _ h(y(~))x, where
~ lies between 0 and x.
Thus from condition
lim hh'((y)) iyi-++oo y
in 2°, the lemma is proved.
0
Let (x*, y*) and (x*, y*) denote the intersection points of x = x* and x = x* respectively. Lemma 3.1.10. h2 (yo)(y* - y*)
-+
(x*) - (x) as
[p
with
IYol -+ 00.
Proof. Let S be the set of intersection points of the lines z = Zi, i = m,m + 1,··· ,n and the curve Z = G(x) in the region x* ~ x ~ x*, Izi < 00. Since G(x*) = G(x*) = z*, it is easy to know that the curve z = G(x), x E [x*,x*], is divided into an even number of small monotonical segments by S and these small segments occur in pairs, each pair of which corresponds to the same interval of z but with opposite monotonicity. Let L 1 , L 2 ,'" ,L K (obviously, each is a subinterval of Ji n J j ) denote such intervals of z, where K is the total number of such intervals (if in some Lk there exist k pairs of small segments of opposite monotonicity, then they are counted for k times). Thus we have
y* _ y* =
rx * i:h
g(x) dx F(x) - h(y)
_L ( [ 1 _ 1 ] dz - kEK iLk F( I.PkJ - h(YkJ F( I.PkJ - h(Yk2) _ L ( F(I.Pk 2 ) - F(I.PkJ + h(YkJ - h(YkJ dz - kEK iLk [F(I.Pk t ) - h(YkJ][F(I.PkJ - h(YkJJ .
3.1.
Let
Boundedness of Solutions and Existence of Limit Cycles
IYol -
00.
103
We get from Lemma 3.1.9 that
h2 (yo)(Y* - y*) - -
L /
[F( rpkJ - F( rpk2 )] dz
kEKLk
= -
fx:' F(x)g(x) dx. o
The proof is completed. Lemma 3.1.11. When is, y(x~) < y(x~).
IYol »
1, we have Yn+1(zo) < Ym(zo), that
Proof. We have
Let IYol - 00. From Lemmas 3.1.9-3.1.10 and the condition 10 b) in Theorem 3.1.8, we get
h 2(Yo)(Yn+l(Zo) - Y(Zo))
- - {L;O[F(rpn+l) - F(rpm)] dz
+ ~(x*) - ~(x*)} < O.
Proof of Theorem 3.1.8. Let "IPo be a trajectory of system (3.1.3) passing through Po(O, Yo) (Yo> 0). When Yo » 1, "I:' must intersect successively the lines x = x~, x = 0, and x = -x~ at the points PI, P2, P3, P4, and P5, and "I;' must intersect the line x = x~ at point P- 1 as shown in Fig. 3.1.3. Let Yi denote the ordinate of points Pi, i = -1,0, .. · ,5. There is no harm in supposing that when Yo is sufficiently large, IYi I, i = 2,3,4, can also be sufficiently large. Thus from Lemma 3.1.11 if Yo is sufficiently large, then Yl < Y-I, Y2 < Y4. Under the transformation (3.1.9), system (3.1.3) in the half-planes x ~ x~ and x ~ x~ are equivalent to the equations (3.1.10)m and
Chapter 3.
104
Bifurcation in Polynomial Lienard Systems
(3.1.10)n+1 in the half-plane Z ~ Zo respectively. From F(CPn+d ~ F( CPm) and Y2 < Y4, for the equations (3.1.10)m and (3.1.10)n+1, Z ~ Zo, using the com~arison theorem we can see that the image of trajectory segment P4P5 must lie on the left side of the image of that of P1 P2 ; therefore the image point of P5 is located below the image of H. But as Y1 < Y-1, the image of P5 is located below the image of P -1' Turning to the (x, Y)- plane, it is immediately seen that the point P5 is located below P- 1 (see Fig. 3.1.3). The theorem is proved. 0
Fig. 3.1.3 Remark. If
or
i~[F(CPn+1(Z)) - F(CPm(z))] dz
=
00,
then it is easy to see that the condition 10 b) in Theorem 3.1.8 holds. In the general case, the condition F in Theorem 3.1.8 can be verified by calculating the limits X-+'FOO lim
.
[~(~j = J-ti X Q,
i
= 1,2.
Corollary 3.1.12. For system (3.1.2), suppose that g(x) satisfies the condition in Theorem 3.1.8 and
lim x-+-oo
F(x) In ~ = J-t1 > -v8, yG(x)
.
hm X-++OO
F(x) In ~ = J-t2 < v 8. yG(x)
3.1.
Boundedness of Solutions and Existence of Limit Cycles
If J.L2
> J.Ll
or J.L2
= J.Ll,
105
and
then all solutions of system (3.1.2) are bounded in the positive sense.
In fact, since hey) = y, i31 = i32 condition 20 in Theorem 3.1.8 holds.
=
1, it is easy to see that the
Corollary 3.1.13. For system (3.1.3), suppose that h(y) and g(x) satisfy the conditions in Theorem 3.1.8 and that there exist c 2: max{lxml, x n } and k > k, such that F(x) ~ k for x ~ -c, F(x) 2: k for x 2: c, then all solutions of (3.1.3) are bounded in the positive sense. In fact, it is easy to see that the condition 10 in Theorem 3.1.8 holds. 3.1.3.
Existence of limit cycles
By using the annular region theorem and the theorem of boundedness obtained above we can immediately get the existence theorem of limit cycles as follows. Theorem 3.1.14. Suppose that for (3.1.3), the following conditions hold: 1) the hypotheses (Hr)-(H3); 2) g( x) has at most a finite number of zero points, among which is a generallized odd multiple zero point if it changes the sign of g( x); 3) (3.1.3) has at most a finite number of critical points in the closed region D = {(x,y): Cl ~ x ~ c2,d 1 ~ Y ~ d 2}, which form an unstable critical point system or a critical point-cycle system S; 4) the hypotheses in one of the following: Theorem 3.1.7, Theorem 3.1.8, Corollary 3.1.12, and Corollary 3.1.13, hold. Then (3.1.3) has at least one stable limit cycle surrounding S.
106
Chapter 3.
3.1.4.
Bifurcation in Polynomial Lienard Systems
Non-existence criteria of closed orbits
We now turn to establishing several non-existence criteria of closed orbits, which are very useful for determining the number and position of limit cycles. Theorem 3.1.15. If system (3.1.2) satisfies (Hr)-(H2)' and F(x) 2:: .jSG(x), g(C2) = 0, F(C2) = 0, that is, C2(C2, F(C2)) is a critical point of (3.1.2), then (3.1.2) has no closed orbits containing C 2 in its interior. Proof. From Lemma 3.1.3 we know that system (3.1.2) has a negative semitrajectory passing through C 2 and going to infinity. Thus the conclusion is obtained. 0 Now consider the system (3.1.3) under the suppositions as follows: h(y) E C1, yh(y) > O(y i= 0), h'(y) > 0, h(±oo) = ±oo; f(x), g(x) E C(lrl,r2), -00 ::; rl < 0 < r2 ::; +00, and xg(x) > O(x i= 0). Let D = {(x,y): rl::; x::; r2, Iyl < oo},
Z = G(x),
Zi = G(ri),
= 1,2. z* = max{zl, Z2}' Denote the inverse functions of Z = G(x) by Xl = Xl(Z), 0 < Z < Zl, and X2 = X2(Z), 0 < Z < Z2, respectively, then system (3.1.3) is i
equivalent to
dz dy = F(Xi(Z)) - h(y),O::; Z < Zi·
(3.1.11)i
Theorem 3.1.16. ([27]) If for (3.1.3) the above conditions hold and F(X2(Z)) - F(Xl(Z)) 2:: 0 (or::; 0) and to for Z E (O,z*), then (3.1.3) has no closed orbit or singular closed orbit in D. Proof. We only prove the case F(X2(Z)) - F(Xl(Z)) 2:: 0 and to. Consider the solutions Zi = Zi(Y) of (3.1.11)i with the same initial conditions Zi(YO) = 0 (Yo < 0), i = 1,2. Suppose that their common existence interval is [Yo, Yl). From
dYI 1 1 dYI dz (3.1.11h = F(X2(Z)) - h(y) ::; F(Xl(Z)) - h(y) = dz (3.1.11h'
3.1.
Boundedness of Solutions and Existence of Limit Cycles
107
and the comparison theorem we know that Zl(Y) < Z2(Y), Y E (Yo, Yd. Thus, whether they return to meet y;t or not, we know that, turning to the (x, y)-plane, the trajectory arcs ' : and ,; cannot intersect Yo+ at the same point. Therefore they cannot form a closed orbit or a singular closed orbit. D Theorem 3.1.17. ([27]) If for (3.1.3), the suppositions above are satisfied and the simultaneous equations
F(u)
= F(v),
G(u) = G(v),
rl
< u < 0,
0< v < r2
(3.1.12)
have no solution, then (3.1.3) has no closed orbit or singular closed orbit in D. Proof. Let Z = G(u) = G(v), which have inverse functions u = u(z), v = v(z), 0 < z < z*, respectively. We obtain that
F(v(z)) - F(u(z))
> 0« 0) for 0 < Z < z*.
Since there is no solution for (3.1.12), Theorem 3.1.16 can be applied. D
Corollary 3.1.18. Let F(x) = Fo(x) + Fe(x), where Fo(x) and Fe(x) are the odd and even parts of F(x) respectively, and Fe(O) = 0, xFo(x) 2:: 0(::; 0) and -¥=- 0, g(-x) = g(x). Then (3.1.3) has no closed orbit or singular closed orbit in D. Proof. Note that G( -x) = G(x), so from z = G(u) = G(v), rl u < 0 < v < r2, we have u(z) = -v(z), 0 < z < z*. It follows that
F(v(z)) - F(u(z))
= 2Fo(v(z)) 2:: 0(::; 0)
and
-¥=-
0 (y i= 0), h'(y) > 0, h(±oo) = ±oo; fo(x), g(x) E C(lTI,T2), -00 ~ rl < 0 < r2 ~ +00, fo(-x) = -fo(x), and xfo(x) > O(x i= 0); go(x) and ge(x) are the odd and even parts of g( x) respectively. If ge( x) is of a fixed sign and there exist rl < Cl ~ Co ~ C2 < r2 (Cl ~ 0 ~ C2) such that (x - co)g(x) < 0 for x E (Cl' C2), x i= co; (x - co)g(x) > 0 for x tt [Cl,C2], then (3.1.14) has no closed orbits intersecting x = Cl and x = C2 simultaneously.
3.2.
Criteria for Deciding the Number of Limit Cycles
It is a far more important and delicate question to determine the exact number of limit cycles in the global bifurcation of polynomial Lienard systems. In this section we give certain conditions under which the generalized Lienard system has at most one or at most two limit cycles. We consider the generalized form of Lienard system:
x = h(y) if and
=
x= if =
F(x, J.L),
-g(x), h(y), -g(x) - f(x,J.L)Y,
(3.2.1)
(3.2.2)
where (x,y) E D = {(x,y) : rl < x < r2,lyl < +oo}, -00 ~ rl < o < r2 ~ +00, the parameter J.L takes values in a certain interval I, F(x, J.L) = J; f(s, J.L) ds. It is always assumed that f, 9 and hare C l functions of their variables. We use the following notations: Ve : h(y) = F(x), the vertical isocline of (3.2.1), Le:
y
=-
i-
~~:~,
the level isocline of (3.2.2).
).(L) = f(x)dt, where L is an arc of the orbit of (3.2.1) (0£ (3.2.2)). For simplicity, the parameter J.L is omitted in f(x,J.L) and F( x, J.L) here and in the following if the meaning is clear.
3.2.
Criteria for Deciding the Number of Limit Cycles
3.2.1.
111
Preliminary lemmas
In order to decide on the number of limit cycles, we need to estimate the integrals of divergence along some arcs of the trajectories. We first give the following lemmas. Lemma 3.2.1. Suppose that yh(y)
o for bl
> 0 (y =F 0), h'(y) > 0; f(x) :S
:S x :S b2, and
are two arcs of the trajectories above Ve (or x-axis) of (3.2.1) (or (3.2.2)) withY2(x) > YI(X), orbelowVe (or x-axis) withY2(x) < YI(X). Then we have
The proof is omitted. Lemma 3.2.2. L~t L : y = y(x), a :S x :S (3, be an arc of the
trajectories of (3.2.1), then we have ).(L) = ±
=± where Ve·
+
[In IF({3) -
h(y( a)) F(a) - h(y(a))
1+ 1(3
[1 IF({3) - h(y({3)) I n
F(a) _ h(y({3))
a
r(3
+ Ja
[F({3) - F(x )]h'(y)g(x) dX] [F({3) - h(y)][F(x) - h(y)J2 [F(a) - F(x)]h'(y)g(x) d] [F(a) - h(y)][F(x) _ h(y)J2 x,
(or -) corresponds to the case that L lies above (or below)
Proof. We only prove the case for which L lies above Ve.
)'(L) =
1(3 a
f(x) dx = F(x) - h(y)
1(3 F'(x) a
h~(y) + h~(y) dx F(x) - h(y)
_lnI F ({3)-h(y({3))1+1(3 h~(y) dx F(a) - h(y(a)) a F(x) - h(y)
Chapter 3.
112
IF(f3) - h(y(a)) I
I
= n F(a) _ h(y(a)) -
+
Bifurcation in Polynomial Lienard Systems
1(3 a
h~(y) d F(f3) _ h(y) X
r(3
h'(y)g(x) 10 [F(x) _ h(y)]2 dx,
and the result is obtained by combining the last two integrals.
0
Suppose that there exist rl < a < Xo < b < r2, such that (x - xo)f( x) for Xo =1= x E [a, b]. Let Z = F(x), Zo = F(xo), Za = F(a), Zb = F(b), and x = Xl(Z), Zo ::; Z ::; Za, X = X2(Z)' Zo ::; Z ::; Zb, be the inverse functions of Z = F(x) on the interval [a,x o], and [xo,b] respectively. After changing x to Z and eliminating t, system (3.2.1) becomes [Z - h(y)]dy = ki(z)dz, Z 2: 0, (3.2.3)i
>0
where ki(z) Z
= g(Xi(Z))/ f(xi(z)) i = 1,2.
The curve
Ve
becomes
Ve :
= h(y).
< c ::; b (or a ::; c < x o) such thatF(a) = F(c) = z (orF(c) = F(b) = z), (x-xo)f(x) > 0 (or < 0) for Xo =1= x E [a,b], kl(Z) < k2(Z) for Zo < Z < z (or z < Z < zo), Lemma 3.2.3. Suppose that there exist Xo
and L : y = y(x), a::; x ::; b, is a trajectory arc of (3.2.1) above or below Ve, then we have A(L) < O.
Ve
Proof. Divide L: y = y( x), a ::; x ::; b, into two segments: L1
:
y
= Yl(X),
a::; x ::; Xo;
L2 : y
= Y2(X),
xo::; x ::; b.
Let £1:
y = Yl(Xl(Z)), Zo::; Z ::; Za,
£2: y = Y2(X2(Z)), Zo::; Z ::; Zb,
denote their images under the transformation Z = F(x), respectively. Suppose, for definiteness, L lies above Ve, then Ii lies above Ve, so that Z - h(y) < O. From
dYI _ dYI _ kl(Z) - k2(Z) > 0 dz (3.2.3h dz (3.2.3h Z - h(y)
3.2.
Criteria for Deciding the Number of Limit Cycles
113
and we know that
Thus
A(L)
= =
f(x) d l f(x) dx+ Ib f(x) dx i Xo F(x)-h(y) x+ xoF(x)-h(y) F(x)-h(y) 1] dz + hZb z - 1h(Y2) dz < O. 1 [1 o z - h(Y2) z - h(Y1) c
a
c
2
-
~
2
Lemma 3.2.4. Suppose that k(z)/z is monotonically decreasing for z > Tt 2: 0, and L 1, L2 are two integral curve arcs of (z - y)dy = k(z)dz with end points (Tt, Yit) (Y21 < Yl1 < 0) and (Tt, yd (Y22 > Y12 > 0). Then along the counterclockwise direction of Li we have
- h
h_ -1-Y dz > O.
J = _ -1- dz L2 Z -
Y
LI Z -
Proof. For i = 1,2, let ~(ai, ai) be the intersection points of Li and y = z, and Yi1(Z), ydz) denote the arcs of Li below y = z and above y = z respectively. Then we have
- [la z - Y211 (z ) dz - lal z - Yl11 (z ) dz] al 1 dz - la 1 dz] == J- + J-2. + [l z - Y12(Z) z - Y22(Z) 2
J =
T}
T}
2
1
T}
T}
In the first integral of J1, let u = pz( 0 < p = ad a2 < 1). Then J- 1 = JcfT}
fY11 U -
1fh(u) du + lal [1 1] du, - u - Yl1(u) u - fh(u) T}
Chapter 3.
114
Bifurcation in Polynomial Lienard Systems
where ih(u) = PY21(p- 1U). It is easy to see that the first integral in J1 is positive, and it is not difficult to verify that Y = fh( u) is a solution of the differential equation
dy du
pk(p- 1u) u- y
(3.2.4)
From
-dYI du
Yll(U),
rJ ~ u
< al·
It follows that
lal [1 U - ih(u) -
1]
u - Yll(U) du = lim fa [ 1 _ 1 ] du > O. a ...... allTJ u-fh(u) u-Yn(u) TJ
Therefore,
J 1 > O.
Similarly we can prove
J 2 > O.
o
Corollary 3.2.5. For the system (3.2.1) with h(y) = y, suppose that in the interval (r1'~], ~ ~ 0 (resp. [~,r2)' ~ ~ 0), F(~) = rJ ~ 0, f(x) < 0, (resp. > 0), the function )'(Ld (resp. )'(L 2) < )'(Ld)·
Similar to Lemma 3.2.4 and Corollary 3.2.5, we obtain Lemma 3.2.6. Suppose that k(z)/z is monotonically increasing for z > 0, and L 1, L2 are two integral curve arcs of (z-y)dy = k(z)dz
with end points (0, Yil) (Y21 < Yll < 0) and (0, Yi2) (Y22 > Y12 > 0).
3.2.
Criteria for Deciding the Number of Limit Cycles
115
Then along the counterclockwise direction of Li we have J-
= ;;_ -1- dz L2 Z -
Y
- ;;_ -1- dz L1 Z - Y
< O.
Corollary 3.2.7. For the system (3.2.1) with h(y) = y, suppose that in the interval (rl,a], a ~ 0 (resp. [b,r2), b ~ 0), F(a) = 0, f(x) < (resp. F(b) = 0, f(x) > 0), thefunctioncp(x) is monotonically decreasing (resp. increasing) and L 1 , L2 (resp. L 2, L 1 ) are two trajectory arcs of (3.2.1) with end points (a,Yil) (Y21 < Yl1 < 0) and (a, Yi2) (Y22 > Y12 > 0). Then we have >'(L2) < >.(Ll) (resp. >'(L2) > >'(Ld)·
°
Lemma 3.2.8. Suppose that system (3.2.2) with h(y) = Y satisfies: 1) there exists 0 ~ C2 < r2 such that g(C2) = 0, g(x) > 0 for C2 < x < 1'2; 2) the functions f(x), g2(X) = g(x)/(X-C2), and (x-c2)f(x)/g(x) are monotonically increasing for C2 < x < r2. Then along any two trajectory arcs L 1 , L2 of (3.2.2) with end points (C2,Yi2) (Y22 > Y12 > 0) and (C2,Yid (Y21 < Yu < 0), we have >'(L2) < >.(L 1 ). Proof. Let x = C2 + r sin 0, Y = r cos 0, 0 ~ 0 ~ from ~+). System (3.2.2) is transformed into
r=
r
sin 0 cos 0 - cos Og( C2
+r
sin 0) -
r
cos 2 0f(C2
7r
(0 starting
+ r sin 0),
iJ = !(xcosO - ysinO) r
= cos 2 0 + sin 2 Og2( C2 + r
sin 0)
+ sin Ocos 0 f( C2 + r
sin 0)
= S( 0, r).
First we show that any trajectory arc L of (3.2.2) with end points (C2' Y2) (Y2 > 0) and (C2' yd (Yl < 0) is convex with respect to the point C 2 (C2, 0). From the monotonicity of f(x), we know that in the half-plane x > C2, Le has only one branch or two branches L~ and L~, as shown in Fig. 3.2.l. In either case, we have xcos 0 > 0 (0 =1= ~) for x ~ C2.
Chapter 3.
116
Bifurcation in Polynomial Lienard Systems
y = f(~)
(c)
(b)
(a)
Fig. 3.2.1
°
For case (a), from iJ < we know that iJ Le. Note that below Le we have
>
°for any point above
1 + _y_. f(x) < 1 _ g(x) . _1_. (x - c2)f(x) = 0.
x-
C2
g2(X)
f(x)
x-
g(x)
C2
For fixed (),
.
() = cos 2 ()
[
+ sin 2 ()g2( C2 + r sin ()) 1 +
+ r sin ()) cot ()] ( . ()) 9 C2 + rsm
f(C2
is monotonically decreasing with r increasing. Bu~ x < 0, iJ > 0, r is monotonically decreasing as t inceases. Thus () is monotonically increasing with t increasing. For case (b), above Le we have < () < ~ and x > 0, iJ > 0, and it follows that r increases with t. Thus for fixed (); S((), r) is monotonically increasing with r increasing. Therefor~ () is monotonically increasing with t increasing. Below Le we have () > since iJ < 0. For case (c), in the region between L~ and L~, we have iJ > since iJ < 0. For the arc above L~, the situation is similar to that above Le in (b); and for the arc below L~ the situation is similar to that below Le in (a). . Summarizing the above discussions we obtain that () > along L. Thus Li can be represented by
°
°
°
°
L i : ri
= ri(()),
i
= 1,2,
0:::; ():::;
7f.
3.2.
Criteria for Deciding the Number of Limit Cycles
Let Xi(O)
= C2 + ri(O)sinO.
117
Then we have
Lemma 3.2.9. Suppose that system (3.2.2) with h(y) = y satisfies: 1) there exists rl < CI ~ 0 such that g( CI) = 0, g( x) < 0 for rl < x < CI; 2) the functions f(x), gl(X) = g(x)/(X-CI), and (x-cI)f(x)/g(x) are monotonically decreasing for rl < x < CI. Then along any two trajectory arcs Ll, L2 of (3.2.2) with end points (CI,Yil) (Y21 < Yll < 0) and (cI,yd (Y22 > Yl2 > 0), we have >'(L2) < >'(LIo). On using the transformation x the result follows.
3.2.2.
--t
-x, y
--t
-y and Lemma 3.2.8
Generalization of some classical theorems
Consider the generalized Lienard system (3.2.1). We assume that yh(y) > 0 (y 0), h'(y) > 0 except in the Theorems 3.2.12-3.2.13. We give the generalization of Sansone's theorem (see [188], Sec. 6).
t=
Theorem 3.2.10. Suppose that for (3.2.1), there exist rl < al < o < a2 < r2 such that F(ad = F(O) = F(a2) = 0, g(x)F(x) ~ 0 for x E (aI, a2), f(x) 2:: 0 for x ~ (aI, a2), xg(x) 2:: 0 for x t= 0, and G(al) = G(a2)' Then (3.2.1) has at most one limit cycle in D, which is simple and stable, if exists. Proof. From the hypotheses we know that
V(x, y) = H(y)
+ G(x)
- G(ad = O(H(y)=g h(u)du)
Chapter 3.
118
Bifurcation in Polynomial
Lif~nard
Systems
is a closed curve passing through the points A 1(a1,0) and A2(a2'0), and that
dVI dt
= -g(x)F(x)
~ 0, x E (a1,a2).
(3.2.1)
Thus any closed orbits r of (3.2.1) must intersect the lines x = a1 and x = a2 simultaneously, as shown in Fig. 3.2.2. It is obvious that
From Lemma 3.2.2 we have
Thus we obtain
).(r)
= 1r - f(x) dt < O. o
The limit cycle, if exists, must be stable. 11
z
Fig. 3.2.2 In 1958 Zhang Zhifen proved the following result (see [193]). Theorem 3.2.11. If system (3.2.1) with h(y) = y satisfies that xg(x) > 0 for 0 =1= x E (r1,r2), f(x)jg(x) is monotonically increasing in (r1, 0) U (0, r2) and is not a constant in any neighborhood of x = 0, then (3.2.1) has at most one limit cycle in D and it is stable if exists.
3.2.
Criteria for Deciding the Number of Limit Cycles
119
In the past thirty years it has been applied successfully to prove the uniqueness of limit cycles in various problems. The following two theorems generalize Theorem 3.2.11 to cover systems with several critical points. Theorem 3.2.12. ([172]) Suppose that system (3.2.1) satisfies: 1) there are 1'1 < Cl ::; 0 ::; C2 < r2 and d l ::; 0 ::; d 2 such that xg(x) > 0 for x ~ [Cl, C2], and yh(y) > 0, h'(y) > 0 for y ~ [d l , d 2]; 2) there exist real numbers a and (3 such that the function fl (x) = f(x)+g(x)[a+{3F(x)] has simple zero points bl , b2 with bl ::; Cl, C2 ::; b2 ; 3) system (3.2.1) has only a finite number of critical points in the region R = {(x, y) : bl ::; x ::; b2, dl ::; Y ::; d 2}, which form an unstable (resp. a stable) critical point system or critical point-cycle system S, and any closed orbits surrounding S of (3.2.1) contain R. Then we have: a) if h(x) ::; 0 for x E [b l ,b2] and h(x)/g(x) is monotonically increasing for x ~ [bl, b2], then (3.2.1) has at most one limit cycle (resp. two limit cycles) surrounding S in D. b) if fl (x) ~ 0 for x E [bl, b2] and h(x)/ g(x) is monotonically decreasing for x ~ [bl, b2], then (3.2.1) has at most two limit cycles (resp. one limit cycle) surrounding S in D. Proof. We only prove a). Note that along any closed orbit f of
(3.2.1) we have irg(x)dt
= 0,
irh(y)g(x)dt
= 0,
irg(x)[h(y) - F(x)]dt = 0,
and thus
ir - f(x) dt ir - fl(x) dt. =
Let f l , f2 be two limit cycles of (3.2.1) with S : S eRe int.fl int.f 2, as shown in Fig. 3.2.3.
c
120
Chapter 3.
Bifurcation in Polynomial Lienard Systems
Fig. 3.2.3 Consider the characteristic exponent A(fi) of fi. From Fig. 3.2.3 and Green's formula we have
If S is unstable, then fl is inner stable, so A(fd :S O. Now we show that f 1 cannot be a semistable limit cycle. Suppose this is not so, i.e., fl is an inner stable and outer unstable limit cycle of (3.2.1)JL as IL = ILl. Then from the bifurcation theoryl we know that for a suitable IL =1= ILl, (3.2.1)JL has at least one stable limit cycle fll and one unstable cycle f12' with fll c fl C f 12 , and A(fll) < 0, A(f12) > 0, which contradicts A(f12) < A(fll) proved above. Therefore in this case (3.2.1)JL has only one stable limit cycle fl. If S is stable, then f1 is inner unstable, so A(f 1) 2: O. It is not difficut to see that if f 1 is outer stable, then in the exterior of f 1 there cannot be any limit cycle, and if f 1 is an unstable limit cycle, then in the exterior of f 1 there exists at most one stable limit cycle. Therefore in this case (3.2.1)JL has at most two limit cycles. 0 IIf system {3.2.1} does not contain any parameter, the proof can be proceeded as in Theorem 6.4. In what follows this will not be mentioned again.
[188]
Sec. 6,
3.2.
121
Criteria for Deciding the Number of Limit Cycles
Theorem 3.2.13. ([70]) Suppose that system (3.2.1) satisfies: 1) xg(x) > 0 for 0 =1= x E (rl' r2); 2) there exist rl < Cl ::; bl < a < b2 ::; C2 < r2 such that f( x) < 0 for x E [bI, b2], f(x) ~ 0 for x fj. (b l , b2), xF(x) < 0 for 0 =1= x E (Cl' C2), and G(cd = G(C2); 3) there exist d l < 0 < d2 such that yh(y) < 0 for 0 =1= y E (d l , d2), yh(y) > 0 for y fj. [dl, d2], and H(dd = H(d 2); 4) function f (x) / g( x) is monotonically increasing for x E (rl' 0) U (0,r2)' Then (3.2.1) has at most two limit cycles surrounding three critical points simultaneously in D. Proof. It is easy to see that (3.2.1) has three finite critical points:
Dl(O, d l ), 0(0, 0), and D 2 (0, d2 ). First we prove that any closed orbit of (3.2.1) surrounding D l , 0, and D2 must contain the points P 2(Cl, d 2), Q2(C2, d2), H(Cl, dd and Ql( C2, d l ). In fact, we define V(x, y) = H(y) +G(x). Consider a closed set defined by V (x, y) = V (C2' d2). It is either a simple closed curve if V( C2, d 2) ~ 0, or with two simple closed components if V( C2, d 2) < O. The set includes the points P2, Q2, PI and Ql in its interior for either of the case since H(dd = H(d 2) and G(cd = G(C2)' Therefore, from conditions 1) and 2) we have
dVI d
t
= -g(x)F(x) >
0 for V(x, y) < V(C2' d 2) and x
=1=
O.
(3.2.l)
It follows that any closed orbit of (3.2.1) surrounding the three critical points D l , 0, and D2 must surround the set V(x, y) = V(C2' d2). Suppose (3.2.1) has two closed orbits r l C r 2 both surrounding the points D l , 0, and D2 as shown in Fig. 3.2.4. Then from the proof above the points AI, A2 on the line x = bl , and B l , B2 on the line x = b2 lie above the line y = d2, and E l , E 2, Fl , and F2 lie below the line y = dl , see Fig. 3.2.4.
122
Chapter 3.
Bifurcation in Polynomial Lienard Systems 11
Fig. 3.2.4 The remaining part of the proof can be proceeded exactly as for Theorem 3.1.12. 0 3.2.3.
Several new results
Now suppose the following condition is satisfied for (3.2.1). (A) There exist 0 < b < a < r2, 0 < e < r2, such that F(O) F(a) = 0, (x - b)f(x) > 0 for b =J x E (r1,r2), and g(O) = g(e) = 0, xg(x) > 0 for x E (r1,r2), x =J O,e. Let Z = F(x), Zo = F(b), Zl = F(r1 + 0), Z2 = F(r2 - 0), z* = min{zl' Z2}, and x = X1(Z), Zo ::; Z ::; Zl; x = X2(Z), Zo ::; Z ::; Z2 be the inverse functions of Z = F(x) on the intervals (r1,b) and (b,r2) respectively. After changing x to Z we get (3.2.3k From the supposition above, it is easy to see that
k1(z) < 0 ::; k2(Z) for Zo < Z < 0, k1 (z) 2: 0 for 0 ::; Z < Zl, k2(Z) 2: 0 for Zo ::; Z < Z2·
(3.2.5)
under the transformation Z = F(x), for any closed orbit r of (3.2.1), the part r 1 in x < b is changed into the integral arc f\ of (3.2.3)1 from point (zo, yd (Y1 < 0) to (zo, Y2) (Y2 > 0), and the part r 2 in x > b is changed into the integral arc f'2 of (3.2.3h from point (zo, Y2) to (zo, yd, see Fig. 3.2.5.
3.2.
Criteria for Deciding the Number of Limit Cycles
123
z
(a)
(c)
(b) Fig. 3.2.5
In addition, V(x, y) closed curve. By
dVI dt
= H(y) + G(x) = C, for each C> 0, is still a =
-g(x)F(x)
~ 0,
(3.2.1)
we know that any closed orbit of (3.2.1) must intersect x = a. We now prove a special case of Theorem 3.1.19, that is, Lemma 3.2.14. ([171]) If the condition (A) holds fOT system (3.2.1) and the simultaneous equations
F(u) = F(v),
g(u) g(v) f(u) = f(v)'
T1
< u < 0, a < v < r2
(3.2.6)
have no solution, then (3.2.1) has no closed orbits in D.
Proof. Based on (3.2.5), (3.2.6) has no solution implies that k1(Z) < k2(Z) for Zo < Z < z*, and it follows that
dYI dz
_ dYI {3.2.3h
dz
= k1(Z) - k2(Z) > 0 « 0) (3.2.3h
z - h(y)
for z - h(y) < 0(> 0). If (3.2.1) has a closed orbit r, then its images f\ and t2 are the integral arcs of (3.2.3h and (3.2.3h passing through
124
Chapter 3.
Bifurcation in Polynomial Lienard Systems
the points (zo, yd and (zo, Y2) respectively. By using the comparison theorem we know that the configurations of 1 and 2 in the halfplane z ~ Zo must be as shown in Fig. 3.2.5(c). Thus we have
r
0=
r
JAnt.r - f(x) dxdy = JAnt.(rlUr2) dzdy i= O.
The contradition shows that
r
o
cannot exist.
Theorem 3.2.15. ([171]) If for system (3.2.1), condition (A) and the following assumptions are satisfied: 1) there exists 0 ::; z < z*, such that k1(Z) < k2(Z), Zo < z < z; k1(Z) ~ k2(Z), Z < z < z*; 2-1) the function Il> (x) = f (x) / g( x) is monotonically increasing for i) < z < 1'2 and 0 < c < a, or 2-2) the function Il> (x) is monotonically decreasing for 1'1 < z < u, and Zl ~ Z2, where u = X1(Z), i) = X2(Z), then (3.2.1) has at most one limit cycle in D, which is simple and stable, if exists. Proof. Let r be any closed orbit of (3.2.1), L(x£, Y£) and R(XR' YR) are its right-most point and left-most point, A and B be the intersection points of r and x = b, see Fig. 3.2.5(a). We now prove that ).(r) < O. Let
LB: y = Yn(x), BR: y = Y21(X),
LA: Y = Y12(X), AR: Y = Y22(X),
and Y = Yij (Xi (z )) (i, j = 1, 2) denote the corresponding images in (z, Y )- plane respectively. The proof will be divided into five steps as follows. (1) To prove YR > Y£. Suppose this is not so, i.e., YR ::; YL, then h(YR) ::; h(YL), ZR = F(XR) ::; F(XL) = ZL. The image L(ZL' YL) of L lies in the upper right side of the image R(ZR' YR) of R as shown in Fig. 3.2.5(c). Note that in the region of Z - h(y) > 0,
dYI _ dYI = k1(Z) - k2(Z) dz (3.2.3h dz (3.2.3h Z - h(y)
< 0 (> 0), -
3.2.
125
Criteria for Deciding the Number of Limit Cycles
for Zo < z.-:5 z, (for z~ Z < z*). By using the comparison theorem we see that EL crosses ER at most once. But in fact, EL does not meet ER. Similarly, AL does not meet AR in the region of Z - h(y) < O. Consequently, the image of r in the (x, Y )- plane must be as the graph shown in Fig. 3.2.5(c). Similar to Lemma 3.2.14, a contradiction is deduced. (2) r must intersect the lines x = ii and x = v simultaneously. From the step (1) we know that R lies in the upper right side of L, and f\ must intersect f' 2, as shown in Fig. 3.2.5(b). Let P( Zp, yp) and Q( zQ, YQ) be the intersection points of f'l and f' 2 respectively. Note that in the interval (zo,z), we have k1(z) < k2(Z). Thus from the proof of Lemma 3.2.14 we obtain
Yl1(Xl(Z)) < Y21(X2(Z)), Y12(Xl(Z)) > Y22(X2(Z)), ZO < Z < z. These show that f'l does not intersect f'2 in the interval (zo, z). Therefore, Zp > z, ZQ > z. Moreover, we get ZR > ZL > z and YR > YL > y, where z = h(y). (3) Let E, G (H, J) be the two intersection points of r and x = ii (x = v). Then we have
>.(r)
=
(lEAH + flBa + faTE + lHlfJ)( - f(x) dt).
From condition 1) and by using Lemma 3.2.3, we obtain
f_
leAH
- f (x) dt < 0,
f_ -f(x)dt < O.
ilEa
(4) Under the condition 2-1), we take a point C(xe, Ye) E Ve with Ye = YL and Xc > v, and an orbit 'Y of system (3.2.1) passing through C. Let M and N be the intersection points of 'Y and x = v. The images of MC and eN on the (z, y)-plane can be represented by
Ne : Y = Y31(X2(Z)),
Me: Y = Y32(X2(Z)), z:::; Z < ZL
respectively. Note that in the interval (z, ze), we have
dYI dz
= k1(z) - k2(Z) < 0(> 0)
_ dYI (3.2.3h
dz
(3.2.3h
Z -
h(y)
-
-
126
Chapter 3.
for z - h(y)
< 0 (> 0),
Bifurcation in Polynomial Lienard Systems
and
Y31(X2(Zc))
= Yc = YL = Yn(Xl(Zc)).
By the comparison theorem we obtain
Thus
r_ + JHRJ r_ )(- f (x) dt) = ( Jcn L - JNC r_) (- f (x) dt) (JCLE + ( JMC r_ - JEL L) (- f (x) dt) + ( JHRJ r_ - JMCN r_ )(- f (x) dt) == II + h + 13 , It is easy to see that
II = lim a->zc
12
=
lim
rb [
1 _ 1 ] dz z - Yn (z ) z - Y31 (z)
< 0,
1 ] dz < 0, Z - YI2(Z) d (f(x)) f(v) f(v) Ja dx g(x) dxdy + JNJ g(v) dy + JBM g(v) dy
b->zc
h =-
ra [
lz
1
lz z - Y32(Z)
Jr
_
r
r
< 0,
where (J" is the interior of the closed curve MCNJRHM. Thus ).(r) < 0 is proved for this case. (5) Under the condition 2-2), we take a point K(xk, Yk) E Ve with Yk = YR, Xk < XL, and an orbit, of (3.2.1) passing through K. In the same way as for step (4), we can show that ).(r) < O. 0 Corollary 3.2.16. Suppose that condition (A) is satisfied for (3.2.1) with 0 ::; c ::; a, and the function g( x) / f (x) is monotonically decreasing for x E (rl' 0) U (a, r2)' Then (3.2.1) has at most one limit cycle in D, which is simple and stable, if exists. Proof. Let
o ::; z < z*.
3.2.
Criteria for Deciding the Number of Limit Cycles
127
By supposition we have
K(O)
= g(O) _ g(a) = _ g(a) < 0
f(O) f(a) f(a) - , K'(z) = f(x)g'(x)3- f'(x)g(x) I _ f(x)g'(x)3- f'(x)g(x) I > o. f (x) xa So K(z) has at most one zero point in [0, z*). The stated conclusion is thus obtained from Lemma 3.2.14 and Theorem 3.2.15.
0
Remark 1. For the system (3.2.1) with h(y) = y, the conditions 2-1) and 2-2) may be weakened into the following: 2-1') the function 7jJ(x) = f(x)F(x)/g(x) is monotonically increasing for v < x < r2 and 0 < c ~ a; 2-2') the function 7jJ( x) is monotonically decreasing for rl < x < ii, and Zl > Z2. For example, if 2-1') holds, then from Corollary 3.2.5 we know at once that h < 0 in the step (4) above. Remark 2. For a = b = c = 0, Theorem 3.2.15 and Corollary 3.2.16 are also valid. Remark 3. If the simultaneous equations (3.2.6) has only one solution (ii, v), then the condition 1) in Theorem 3.2.15 holds. For the system (3.2.1) with h(y) = y, by the change of variables x ---7 -x, t ---7 -t, and using Theorem 3.2.15 and Remarks 1 and 3, we can get Theorem 3.2.17. ([28-29]) If there exist rl < Xl < Xo < 0, rl < such that F(XI) = F(O) = 0, (x - xo)f(x) > 0 for Xo =1= x E (rl' r2), and g(~) = g(O) = 0, xg(x) > 0 for x E (rl' r2) and
~ ~ 0,
x
=1= O,~,
then: a) when the simultaneous equations g(u) g(v) F(u) = F(v), f(u) - f(v)'
have no solution, (3.2.1) has no closed orbits in D;
(3.2.7)
128
Chapter 3.
Bifurcation in Polynomial Lienard Systems
b) when (3.2.7) have exactly one solution, and the function 'IjJ( x) is either monotonically decreasing in x E (r1' Xl) and Xl :S ~ :S 0 or monotonically increasing in x E (0,r2), (3.2.1) has at most one limit cycle in D, which is simple and unstable, if exists. Similarly, we can prove the following two theorems. Theorem 3.2.18. ([41]) Suppose the following conditions hold for (3.2.1) with h(y) = y: 1) there exist r1 < Xl < Xo < 0 such that F(xd F(O) = 0, (x - xo)f(x) > 0 for Xo =1= x E (r1' r2); 2) there exist r1 < C1 < C2 < 0, such that g(x) > 0 for x E (C1' C2) U (0, r2), g(x) < 0 for x E (r1' cd U (C2' 0) and g'(C2) < 0; 3) the simultaneous equations (3.2.7) have at most one solution; 4) 'IjJ(x) is monotonically increasing in x E (0, r2) and Zl :S Z2. Then: a) (3.2.1) has no small limit cycle surrounding the left critical point. b) If C2 < Xl, then (3.2.1) has at most one limit cycle or homoclinic loop. It can be either a small limit cycle around the origin or a large one surrounding all three critical points simultaneously. c) If Xl < C1 < C2 < Xo and the only solution (u,v) of (3.2.7), if exists, satisfies u < Xl < 0 < v, then (3.2.1) has at most one large limit cycle surrounding three critical points or a homoclinic loop. The limit cycle (homoclinic loop) is attracting and hyperbolic. Moreover, there is no small limit cycle around the origin. d) If C1 < Xl < C2 < x o, then (3.2.1) has no small limit cycle around the origin. Theorem 3.2.19. ([174]) Suppose the following conditions hold for (3.2.1): 1) there exist r1 < a1 < b1 < 0 < b2 < a2 < r2 such that F(a1) = F(O) = F(a2) = 0 and f(x) < 0 for x E (b 1, b2), f(x) > 0 for x tf. [b 1 , b2 ];
3.2.
Criteria for Deciding the Number of Limit Cycles
129
2) there exists 0:::; c :::; a2 such that g(O) = g(c) = 0, xg(x) > 0 for x E (rl, r2), x -=J 0, c, and G(aI) 2: G(a2); 3) the simultaneous equations (3.2.6) with bl < u < 0, a2 < v < r2 have at most one solution; 4) the function f (x) / g( x) is monotonically increasing in x E (a2' r2). Then (3.2.1) has at most one limit cycle in D, it, if exists, has a negative characteristic exponent. Remark 1. For (3.2.1) with h(y) = y, the condition 4) may be weakened to that 1fJ(x) is monotonically increasing for a2 < x < r2. Remark 2. For a2 = b2 =
C
= 0, Theorem 3.2.19 is still valid.
Theorem 3.2.20. ([175]) Suppose the following conditions are satisfied for (3.2.1) with h(y) = y: 1) there exist rl < CI :::; 0 :::; C2 < r2 such that g(CI) = g(O) = g(C2) = 0, and xg(x) < for -=J x E (CI,C2), xg(x) > for x ~ [CI' C2]; 2) there are b :::; C2 :::; a < r2 such that F(O) = F(a) = 0, and (x - b)f(x) > for b -=J E (rl,r2);
° °
°
°: :; ° x 3) the function 't/(x) = g(x)/f(x)F(x) is monotonically decreasing
for x E (rl,O,) and the function 1fJ2(X) = f(x)[F(x) - F(C2)]/g(X) is monotonically increasing for x E (C2' r2). Then (3.2.1) has at most two large limit cycles surrounding the three critical points simultaneously in D. Proof. Let r l C r 2 be any two large limit cycles of (3.2.1) surrounding the three critical points as shown in Fig. 3.2.6. Our aim is to prove A(r 2 ) < A(r l ). By using Lemmas 3.2.1, 3.2.2 and Corollary 3.2.7, we can get
Now let into
~
= x - C2,
'T/
= Y - F(C2), and system (3.2.1) is transformed
~ = 'T/ - F(~),
i} = -g(~), ~
2: 0,
130
where F(~) Note that
Chapter 3.
= F(x) - F(C2), !(OF(~) g(~)
Bifurcation in Polynomial Lienard Systems
!(~)
= f(x), and
g(~)
= g(x), x
~
C2.
f(x )[F(x) - F(C2)] g(x)
and from the second part of condition 3), we know that the function !(~)F(~)/g(~) is monotonically increasing for ~ ~ 0, and F(O) 0, !(~) > 0 for ~ > O. By using Corollary 3.2.5 we obtain
Combining the above results we get A(r 2 ) < A(rd. The remainning part of the proof follows in the same way as in Theorem 3.2.12. 0
Fig. 3.2.6 Similarly, we can prove Theorem 3.2.21. Suppose the following conditions are satisfied for (3.2.1) with h(y) = y : 1) the same as in Theorem 3.2.20;
3.2.
Criteria for Deciding the Number of Limit Cycles
131
2) there are rl < al :S cI :S bl :S 0 :S b2 :S c2 :S a2 < r2 such that F(al) = F(O) = F(a2) = 0, f(x) < 0 for x E (b l ,b2) and f(x) > 0 for x rt [bl, b2 l; 3) the function '¢I(X) = f(x)[F(x) - F(cdljg(x) is monotonically decreasing for x E (rl, CI), and the function '¢2(X) is monotonically increasing for x E (C2, r2). Then (3.2.1) has at most two large limit cycles surrounding the three critical points simultaneously in D.
Theorem 3.2.22. ([198]) Suppose the following conditions are satisfied for (3.2.2) with h(y) = y: 1) there exist rl < CI :S 0 :S C2 < r2 such that g( cd = g( C2) = 0, xg(x) > 0 for x rt (CI,C2) and f(x):S 0 for x E (CI,C2); 2) functions f(x), g(x)j(x - CI) and (x - cdf(x)jg(x) are monotonically decreasing for x E (rl,cI), and functions f(x), g(x)j(X-C2) and (x - c2)f(x)jg(x) are monotonically increasing for x E (c2,r2). Then (3.2.2) has at most two large limit cycles surrounding all the critical points in D. Proof. Let r l C r 2 be any two large limit cycles of (3.2.2) surrounding all critical points, and Ai, B i , E i , and D i ( i = 1,2) be the intersection points of r l , r 2 with x = C2 and x = CI respectively, as shown in Fig. 3.2.7. We shall prove A(r 2 ) < A(rl)' 11
z
Fig. 3.2.7
132
Chapter 3.
Bifurcation in Polynomial Lienard Systems
From Lemma 3.2.1 we have
By using Lemmas 3.2.8-3.2.9 we get
Combining the above results we get at once >.(r2) < >.(rl). The remaining argument is similar to that for Theorem 3.2.12. D Similarly, we can prove Theorem 3.2.23. ([198]) Suppose that for (3.2.2) with h(y) = Y there exist rl < CI :::; bl :::; c~ < 0 < c~ :::; b2 :::; C2 < r2 such that 1) xg(x) < 0 for x E (CI' cD u (c~, C2), xg(x) > 0 for x E (rl' CI) U (C2' r2); 2) f(x) :::; 0 for x E (b l ,b2), f(x) ~ 0 for x ~ (b l ,b2); 3) F(cd = F(cD, F(c~) = F(C2); 4) same as 2) in Theorem 3.2.22. Then (3.2.2) has at most two large limit cycles surrounding all the critical points in D. It follows that the proof of Theorem 3.2.23 is still applicable if bl = c~ = Oor C2 = b2 = c~ = o.
CI =
3.3.
Global Bifurcation of Cubic Lienard Systems
As mentioned at the beginning of this chapter, the mathematical models in many practical problems are often described by cubic Lienard systems. Theoretically, it is also the case of most classic nonlinear systems. In this section, we present some complete results which are obtained in [103,168-173,41,68].
3.3.
Global Bifurcation of Cubic Lienard Systems
3.3.1.
133
General remarks
If in (3.1.2) F(x), g(x) are polynomials of a degree not higher than n, then it is called polynomial Lienard system of degreen, which we can write in the form
x= iJ
+ a2x2 + ... + amxm) == y + b2x2 + ... bkx k ) == -g(x),
y - (alx
= -(/-LX
F(x)
(3.3.1)
where max{ m, k} = n. The Hopf bifurcation for the system
x= y
=
y - (alx
+ a2x2 + ... + anx n ),
-X,
(3.3.2)
has been mentioned in Sec. 2.1. [103] proved that (3.3.2) has at most one limit cycle if n = 3 and conjectured that (3.3.2) has at most k limit cycles if n = 2k + lor n = 2k + 2. Unfortunately, up to now the question that (3.3.2) with n = 4 has at most one limit cycle is still not solved completely. [104] and others studied the problem of the number of the small-amplitude limit cycles of (3.3.1) with /-L = 1, and gave a formula for the focal values of some special form of (3.3.1). Here we cite their result as follows. Lemma 3.3.1. For system (3.3.1) with /-L values at 0(0,0) are
= 1, the first three focal
~(2a2b2 - 3a3), 'T/6 = co ( 6a2a4 + 20a4b2 - 15a3b3 - 15a5), 'T/2
where
Co
= -aI,
'T/4
=
is a positive constant.
For the cubic Lienard system
x= iJ =
y - (8x + nx 2 + mx 3), m> 0, -x(/-L + kx + ex 2 ), /-L > 0,
(3.3.3)
134
by the scaling x
Chapter 3.
--t
xVii, t
.
--t
Bifurcation in Polynomial Lienard Systems
tVii, (3.3.3) may be written in the form
{;
x = Y - (-x Vii
n 2 + -x + -m- x3 ), JL
JLVii
Y. = -x ( 1 + - k- x + -E: x 2) ,
JLVii JL2 then from Lemma 3.3.1 we can get the first three focal values of the critical point 0(0,0) of (3.3.3), namely {; 2nk - 3mJL 15comE: T}6 = - JL 3Vii . T}2 = - Vii' T}4 = 8JL2Vii From this we can obtain at once
Lemma 3.3.2. When {; > 0 « 0),0 is a stable (an unstable) elementary critical point; when {; = 0, 2nk - 3mJL < 0 (> 0), 0 is a stable (an unstable) fine focus of order 1; when {; = 0, 2nk - 3mJL = 0 and E: > 0 « 0), 0 is a stable (an unstable) fine focus of order 2. Specifically, when {; = m = 0, 0 is a stable (an unstable) fine focus of order 1 if nk < 0(> 0); when {; = k = 0, or {; = n = 0, 0 is a stable fine focus of order 1.
In what follows, without loss of generality we can always assume am > 0 in (3.3.1), the case am < 0 can be treated by a change of variables y --t -y, t --t -to Note that any real root of g(x) = 0 with g(x) alternating in sign is certainly an odd multiple, and so from Corollary 3.1.13 and Theorem 3.1.14 we can get the following results.
Theorem 3.3.3. Suppose that m, k are odd numbers and bk > 0 in (3.3.1). Then all solutions of (3.3.1) are bounded in the positive sense. Moreover, if all finite critical points of (3.3.1) form an unstable critical point system or critical point-cycle system S, then (3.3.1) has at least one stable large limit cycle surrounding S. By translating and scaling, any general cubic Lienard system
x= iJ =
y - (a o + alx + a2x2 + a3x3), -(f3o + f31X + f32x2 + f33 x3 )
3.3.
Global Bifurcation of Cubic Lienard Systems
135
can be changed into
x= y if
+ a2x2 + a3x3), + bx ± x 2).
(alx
= -x(J.1-
(3.3.4)
Let a3 > 0 as stated above. Furthermore, we assume a2 > 0 (otherwise, let x --t -x, y --t -y). But, in general, (3.3.4) does not form a family of generalized rotated vector fields, hence we turn to considering the equivalent system of (3.3.4) on the phase plane x
if
=
y, = -x(J.1- + bx ± x 2) - (al
+ 2a2x + 3a3x2)y.
(3.3.5)
System (3.3.5) forms a family of generalized rotated vector fields with respect to the parameter al (or a3), and the vectors rotate clockwise as al (or a3) increases. Moreover, the positions of critical points do not change as the paramaters aI, a2, a3 are varied, which is quite convenient for discussing the evolution of the phase portaits following the parameters. But it is well known that theorems of existence and uniqueness of closed orbits for systems of type (3.3.5) are quite few. Therefore we sometimes use the form of (3.3.4) again, in order to overcome one's shortcomings by learning from the strengths of others. 3.3.2.
Integrable cases
We list the results of some special classes of integrable cubic Lienard systems in the following, which will be used later. The system x = y, if = ±ax + x 3 - 2a2xy with a > 0, a2 > 0 has a general integral 2 2 2 2 a2 1 2Y + (a2 - c) (x 2 ± a) I Inl2y +2a2Y(x ±a)-(x ±a) I--In 2 ( )( 2± ) =CI c y + a2 + c x a
and two special integrals 1 it : Y = -,2(a 2 + c)(x 2 ± a), where c
=
';a~
+ 2. The phase portraits are shown in Figs. 3.3.1-3.3.2.
136
Chapter 3.
Bifurcation in Polynomial Lienard Systems
Fig. 3.3.1
(a) 0 < a2
0, a2
> 0,
(3.3.6)
by using the above results and the property of rotated vector fields with respect to al and a3 we can get Theorem 3.3.4. The system (3.3.6) has no closed orbit or singular closed orbit if al a3 2: 0 (ai + a~ 0).
t=
138
3.3.3.
Chapter 3.
Bifurcation in Polynomial Lienard Systems
One-critical point case
Consider the system
x = y,
iJ =
-x - (al
+ 2a2x + 3a3x2)y,
(3.3.7)
or its equivalent form ·23
X = Y - (alx iJ = -x.
+ a2x + a3x
),
(3.3.7')
It has only one critical point 0(0,0) with index +1.
Theorem 3.3.5. ([103]) a) If al and (3.3.7) has the general integral 2a~x2
=
a3
= 0,
+ 2a2Y -In 12a2Y + 11
then 0 is a center
= C;
b) if al ~ 0 and ar + a5 #- 0, then (3.3.7) has no closed orbit or singular closed orbit; c) if al < 0, then (3.3.7) has exactly one single stable limit cycle. The phase portraits are shown in Fig. 3.3.5.
(a) al > 0
(b) al
= a3 = 0
(c) al < 0
Fig. 3.3.5 Proof. a) By using symmetry principle (cf. [188], Sec. 15) the result follows. b) The conclusion can be obtained from Corollary 3.1.18.
3.3.
Global Bifurcation of Cubic Lienard Systems
139
c) Consider the system (3.1.7'), and the existence of limit cycles is obtained at once from Theorem 3.3.3. Let
F(x)
= a3x3 + a2x2 + alx,
f(x) = 3a3x2 + 2a2x + aI,
g(x)
= x.
The roots of F(x) = 0 and f(x) = 0 are
and
a2 - .ja~ - 3ala3 3a3 respectively, and x2-lxll = -a2/a3:::; 0, so G(xI) = G(-XI) ~ G(X2). After simplifying and putting z = u+v, w = UV, the simultaneous equations
F(u)
g(u)
g(v)
= F(v), feu) = f(v)' bl < u < 0, X2 < V < +00
(3.3.8)
can be reduced to
We have ZI,2
= -
6~3 (3a2 ± .j9a~ -
24ala3 ) .
It is not difficult to verify Zl < bl + X2 < Z2 < X2, that is, (3.3.8) has only one solution. Again, from f(O) = al < 0, f"(x) = 6a3 > 0, we know that f(x)/x is monotonically increasing with x in 0 < x < +00. Consequently, from Theorem 3.2.19 we get that (3.3.7') has at most one single stable limit cycle. 0
In the same way we can show that (cf. [174]) Theorem 3.3.6. For the systems :i; =
y,
iJ = _x3 - (al + 2a2x + 3a3x2)y,
Chapter 3.
140
Bifurcation in Polynomial Lienard Systems
and
= y, iJ = -x(l + x 2) - (al + 2a2x + 3a3x2)y, = a3 = 0, then 0 is a center;
x
a) if al b) if al ~ 0 and ai + a5 =1= 0, then they have no closed orbit or singular closed orbit; c) if al < 0, then they have exactly one single stable limit cycle. Their phase portraits are similar to those shown in Fig. 3.3.5.
Remark. The system
x = y,
iJ = -x(J.L + bx + x 2) - (al
+ 2a2x + 3a3x2)y
with J.L > 0 has only one critical point 0(0,0) as b2 - 4J.L < O. But this case is different from the above, and the system may have two limit cycles (cf. Sec. 3.4.1 later). 3.3.4.
Two-critical point case
In this paragragh the three classes of systems having two critical points are considered.
(A) Consider the system
= y, iJ = ±elx ± x
e2 x2 - (d l
(3.3.9)
+ 2d2x + 3d3x2)y,
By the scaling x --t ~x, Y --t elVeI y, t --t b t, (3.3.9) may be changed e2 e2 vel into x = y, iJ = ±x ± x 2 - (al + 2a2x + 3a3x2)y, Ve I , a3 where al = 4, a2 = d 2 e2 Vel only discuss the system
x = y == P(x,y),
iJ = -x + x 2 - (al
=
d 3 e l Ve,. e2
In what follows, we shall
+ 2a2x + 3a3x2)y == Q(x, y).
(3.3.10)
Other cases can be reduced to (3.3.10) by changing the variables: x --t x-I; x --t -x, t --t -t, or x --t 1 - x, t --t -to
3.3.
141
Global Bifurcation of Cubic Lienard Systems
The two finite critical points of (3.3.10) are 0(0,0) and A(l, 0). A is a saddle. Consider the equivalent form of (3.3.10): (3.3.11) From Lemma 3.3.2, 0 is a stable (an unstable) elementery critical point if a1 > 0 « 0) and a stable (an unstable) fine focus of order 1 if a1 = 0, 2a2 + 3a3 > « O)j if a1 = 0, 2a2 + 3a3 = 0, (3.3.10) becomes integrable with a general integral
°
1 1 - y + -2ln 13a3Y 3a3 9a3 and a special integral Y
=
1 , -3 a3
11 -
1 2 -x
2
1 3 + -x =
3
C,
while 0 is a center.
Theorem 3.3.7. For any fixed a3 > 0, 1) if 2a2 + 3a3 = 0, then when a1 = 0, (3.3.10) is integrable; and when a1 0, (3.3.10) has no closed orbit or singular closed orbit; 2) if 2a2+3a3 > 0, then there exists an all = all(a2,a3): -H2a2+ 3a3) < all < 0, such that when all < al < 0, (3.3.10) has a unique stable limit cycle; when a1 = all, (3.3.10) has an inner stable separatrix cycle passing through A and surrounding OJ when a1 < all or a1 ~ 0, (3.3.10) has no closed orbit or singular closed orbit; 3) if 2a2 + 3a3 < 0, then there exist a12 = a12(a2, a3) : < a12 < - ~ (2a2 + 3a3), such that when < a1 < a12, (3.3.10) has a unique unstable limit cycle; when a1 = a12, (3.3.10) has an inner unstable separatrix cycle passing through A and surrounding OJ when a1 > a12 or a1 ::; 0, (3.3.10) no longer has closed orbits or singular closed orbits. The phase portraits of the cases 1) and 2) are shown in Figs. 3.3.63.3.7, and those of case 3) can be obtained by rotating an angle 7r and changing the time t to -t in the pictures of Fig. 3.3.7.
t=
°
°
Chapter 3.
142
Bifurcation in Polynomial Lienard Systems
1/
(a)
al
>0
(b)
al
=0
(c)
al
3a2, ~
= 36(a2 - 2a3b)2 - 4Sa3b(3a3b - 2a2) = 12(3a2 - 4a3b) < 0,
so (3.3.21') has only one solution z = 2b > b, that is, (3.3.21) has no solution. The second statement of the lemma follows from Theorem 3.1.17. 0 Lemma 3.3.13. For a3 > 2a2/3b, system (3.3.19) has at most one small limit cycle, and if it exists, it has a positive characteristic exponent. Proof. Let t =
-T,
dx dT
Y
---t
-y, (3.3.19) is converted into
= Y - Fll(X),
dy dT
= -gl(X),
where Fll(X) = -Fl(x), fll(x) = - ft(x). It is easy to see that 0 < Xl < Xl < b = X2 < X2, Fn(O) = Fn(Xl) = 0, (x - x)fn(x) > 0 for Xl =F x E (-oo,b), xg(x) > 0 for 0 =F x E (-00, b).
3.3.
Global Bifurcation of Cubic Lienard Systems
151
An elementary calculation shows that
~(9l(X)) = _ (x - b)2 H(x) dx fl1(x)
J'A(x)
,
with H(x) = 3a3x2 + 2(2a2 - 3a3b)X - b(2a2 - 3a3b). The discriminant .6. of H(x) satisfies .6. = 8a2(2a2 - 3a3b) < 0. Therefore, 1x(9l(X)/f1l(x)) < holds for x E (-oo,b)U(Xl,b). The result follows from Corollary 3.2.16. 0
°
° < a3 ::; 2a2/3b, system has no small limit cycle and has exactly one simple stable large limit cycle. Lemma 3.3.14. For
(3.3.19)
Proof. Let P(x, y) = y - Fl(X) and Q(x, y) = -gl(X). Then for B(y) = exp(~y) we have div(BP,BQ) =
-(X-b)C~3x2+2a2-3a3b)B > 0,
for-oo
< x < b.
By Dulac criterion, (3.3.19) has no closed orbits that do not intersect x = b. It follows that 0 and Al(b, Fl(b)) form an unstable critical point system, so (3.3.19) has at least one stable large limit cycle by Theorem 3.3.3. We now prove the uniqueness of this large limit cycle by using Theorem 3.2.19. It is easy to see that Xl < Xl < < b = X2 < X2 and the condition 1) of Theorem 3.2.19 is satisfied. An elementary calculation shows that the inequality G l (Xl)-G l (X2) ~ (or::; 0) holds for 3a3b ::; (JI3 - 2)a2 (or (JI3 - 2)a2 < 3a3b < 2a2), therefore the condition 2) of Theorem 3.2.19 is satisfied for
°
°
3a3b ::; (JI3 - 2)a2. The simultaneous equations
gl(U) ft(u)
gl( v)
ft(v) , Xl
dx
gl(X)
1 __ 1_ > 0 x-x2 x-b '
°,
that is, the conditions of'Remark 1 of Theorem 3.2.19 hold. Consequently, for 3a3b :S (VI3 - 2)a2 all conditions of Theorem 3.2.19 are satisfied and the Lemma is proved. For the case (VI3 2)a2 < 3a3b :S 2a2, on changing variables X - t -X, Y - t -y, (3.3.19) becomes x = y - F I2 (X), (3.3.19') iJ = -gI2(X) with F I2 (X) = -FI ( -x), g12(X) = -gl( -x). It is easy to see that for (3.3.19'), G I2 (Xl) 2:: G I2 (X2) if (J13 - 2)a2 < 3a3b :S 2a2' The other conditions of Theorem 3.3.19 and Remark 1 can be checked in a similar way. The lemma is completely proved. 0 Lemma 3.3.15. For 2a2/3b < a3 < a2/b, system (3.3.17) has at most two large limit cycles.
3.3.
Global Bifurcation of Cubic Lienard Systems
Proof. ~or a3b
153
< a2 < 3a3b/2, we have a2 a3
- - < -b =
Cl
2a2 = b1 < 0, 3a3
< --
and the first three conditions of Theorem 3.2.23 with rl +00, C2 = b2 = c~ = are satisfied. In the interval (0, +00), f'(x) = 6a3x + 2a2 > 0,
°
= -00, r2 =
°
( Xf(x))' = 3a3b - 2a2 > g(x) (x+b)2 '
and thus the functions f(x), g(x)/x, xf(x)/g(x) are increasing for increasing x in < x < +00. In the interval (-00, -b), f'(x) < -6a3b + 2a2 < 0,
°
°
( g( x) )' = 2x < x+b '
( (x
+ b)f(X))' __ 2a2 ( ) gx
-
x
2
< 0,
and thus the functions f (x), g(x)/(x + b), (x + b)f(x)/g(x) are decreasing for increasing x in -00 < x < -b. Consequently, from Theorem 3.2.23 and the related remark, we know that (3.3.17) has at most two large limit cycles surrounding the two critical points A and 0 simultaneously. 0 Proof of Theorem 3.3.10. Based on the five lemmas above and on using the theory of rotated vector fields and the bifurcation theory, Theorem 3.3.10 follows. 0 Remark. For a more general system with two critical points:
x = y, iJ = -x 2(x + b) - (al + 2a2x + 3a3x2)y,
(3.3.23)
the Hopf bifurcation and homoclinic bifurcation can be shown in a way similar to the above, but at this time (3.3.23) may have two small limit cycles or two large limit cycles. The question of whether (3.3.23) has at most two small limit cycles or two large limit cycles is still open.
154
3.3.5.
Chapter 3.
Bifurcation in Polynomial Lienard Systems
Three-critical point case
In this paragragh three classes of special systems are discussed. (A) If a cubic Lienard system with linear damping has two saddles and one antisaddle, then it is easy to see that the antisaddle must lie between the two saddles. By translating the antisaddle to the origin, the system can be written in the form
x =y, iJ = -x(J..L + bx - x 2) - (al + 2a2x)y,
(3.3.24)
where a2 > 0 and J..L > O. The three finite critical points of (3.3.24) are 0(0,0), A(CI, 0), and B(C2' 0) with CI
=
~ (b -
Jb 2 + 4J..L) < 0
0 « Consider the system
~ (b + Jb 2 + 4J..L) = C2,
0) for b > 0 «
x= y iJ
0), A and B being saddles.
+ a2x2), + bx - x 2),
(alx
= -x(J..L
(3.3.25)
for which 0 is a stable (an unstable) elementary critical point if al > o « 0), a stable (an unstable) fine focus of order 1 if al = 0 and b < 0 (> 0). Note that the critical points of the system x = y, iJ = -x(J..L - x 2) - bx 2 - 2a2xy
(3.3.26)
have the same behavior as those of (3.2.24) with alb < 0, and then (3.3.26) has no closed orbits around 0 by Theorem 3.1.22. From the theory of rotated vector fields, (3.3.24) has no closed orbits around 0 if alb < O. In the following, we discuss only the case b < 0 and al < 0, other cases can be treated by the variable change x ---t -x, t ---t -to Theorem 3.3.16. For any fixed a2 > 0 and b < 0, there exists ai = ai(a2, b) with -~(b + v'b 2 + 4J..L) < ai < 0, such that when ai
0 for any fixed Yo> 0, and if al::S (or~) (a2-v'a~ - 2)b, then the separatrix Ul (or sd go to infinity as t --t +00 (or -(0) at the lower part of V(x, y) = o. Consequently, (3.3.27) has no closed orbits surrounding AI, 0 and A2 simultaneously. 0 (B 2 ) The subcase where 0 < a2 < J2 and b < o. First we discuss the behavior of the trajectories of system (3.3.28). Lemma 3.3.18. ([68]) For 3al > 2a2b (3al < 2a2b), all solutions of (3.3.28) are bounded in the positive sense (in the negative sense).
Bifurcation in Polynomial Lienard Systems
Chapter 3.
158
Proof. Suppose that U o < 0 < Vo are two roots of G(x) = 0, then it is easy to see that xg(x) > 0 for x ~ [u o, vol. Let z = G(x), and x = u(z) < uo, x = v(z) > vo, z > 0, be the inverse functions. Denote cp(z) = v(z) + u(z). We check the conditions of Corollary 3.1.12. First it is easy to see that
.
hm
x-doc
F(x) In friT:::\ = 2a2 < v 8. yG(x)
Next, we prove that lim cp(z) =
z..... oc
Since b < 0, G(v(z)) = G(u(z)) for x » 1. It follows that v(z) Moreover, from
-~b. 3
> G( -u(z)), and G'(x) = g(x) > 0 > -u(z), i.e., cp(z) > 0 for z » 1.
= G(u) -
G(v) - G( -u)
G - u)
2 3 = -bu 3
and
G(v) - G( -u) =
-u(z) < ~ < v(z),
g(~)cp(z),
we have
u3
2b
cp(z) = Note that when z» 1, g(-u(z))
cp(z) < 2b
u3
2b
3· g(~). < g(~) < g(v(z)). From this we get
u3
3· g( -u)' 2b
_
li~cp(z) :::;
cp(z) > - . = ---[v - cp(z)] 3 g(v) 3g(v) and thus lim cp(z) =
z ..... oc
3 --t
2
-"3 b,
2 3
--b
as z
--t
-~b. 3
Also, we have
F(v(z)) - F(u(z)) = (v - u)(a2cp(z)
+ al) > 0
« 0),
00,
3.3.
Global Bifurcation of Cubic Lienard Systems
fz~OO[F(v(z))
- F(u(z))] dz =
+00 (-00)
159
for 3al - 2a2b
> 0 « 0).
Hence the lemma can be proved by using Corollary 3.1.12.
D
Similarly, we have (see [68]) Lemma 3.3.19. For 3al = 2a2b, all solutions of (3.3.28) are bounded in the positive sense. Lemma 3.3.20. For 3al 2: 2a2b, system (3.3.27) has no closed and A2 simultaneously. orbits surrounding AI,
°
Proof. From the rotatedness of vectors of (3.3.27) with respect to aI, we need only to prove the lemma under 3al = 2a2b. By translating the origin to the point (-~, 0), (3.3.27) can be reduced to
x = y, iJ = -x (JL -
~b2 + X2)
-
~b (~b2 -
JL) - 2a2xy.
(3.3.27')
Note that ge(x) = 1b(~b2 - JL) < o. It follows that (3.3.27) has no closed orbits surrounding AI, 0, A2 simultaneously by Theorem 3.1.22. Theorem 3.3.21. 1) For al 2: 0, system (3.3.27) has no closed orbits around A2 alone and surrounding AI, 0, A2 simultaneously; 2) for any fixed JL < 0, b < 0, and 0 < a2 < ..;2, there exists al3 = aI3(a2, b, JL) with ~(b+Vb2 - 4JL)a2 < a13 < (b+Jb 2 - 4JL)a2, such that when al3 < al < (b+Jb 2 - 4JL)a2 (al = aI3), (3.3.27) has exactly one simple stable small limit cycle (or an inner stable separatrix cycle) around Al alone, and when 0 ~ al < al3 or al 2: (b + Jb 2 - 4JL)a2, (3.3.27) has no closed orbit around Al alone. Proof. First, from f(x) > 0 for x > 0 and Lemma 3.3.20, 1) holds. Next consider system (3.3.28), when al = ~(b + Jb 2 - 4JL)a2, we have Xl = Cl, g(x)F(x) < 0 for -00 < x < 0, x ::/= Xl, and thus from
Chapter 3.
160
Bifurcation in Polynomial Lienard Systems
Theorem 3.1.20, (3.3.28) has no closed orbits around Al alone and Al is unstable. It follows that (3.3.27) has no closed orbits around Al alone as a1 < ~(b + v'b 2 - 4J.L)a2, by Al still being unstable and the theory of rotated vector fields. Now consider (3.3.29)1' Let Do = {(~, 1']) : -00 < ~ < -C1, 11']1 < +oo}. We can prove as above that (3.3.29)1 has no closed orbits in the region Do when a1 2: (b + v'b 2 - 4J.L)a2. We claim that for -a2c1 < a1 < -2a2c1, (3.3.29)1 has at most one simple stable limit cycle in Do. The roots of F1(~) = 0 and f1(~) = 0 satisfy
It is not difficult to see that the simultaneous equations
gl(U)
JI(u)
gl(V) f1(V)
-
have a unique solution. In the interval
00
< u < 0, 6 < v
0 ~ + C1 ~ - v'b 2 - 4J.L .
Then all conditions for Theorem 3.2.15 are satisfied and the claim is reached. By using the theory of rotated vector fields the conclusion 2) follows. 0 In what follows, we discuss the case a1 < O. First we can prove the following lemma as for Theorem 3.3.21. Lemma 3.3.22. 1) For any a1 < 0 system (3.3.27) has no closed orbit or singular closed orbit around Al alone;
3.3.
Global Bifurcation of Cubic Lienard Systems
161
2) For al < (b - ylb 2 - 4J.L)a2, (3.3.27) has no small limit cycle around A2 alone and also no large limit cycle surrounding AI, O,A2 simultaneously;
°
3) For any fixed J.L < 0, b < 0, and < a2 < V2, there exists all = all(a2, b, J.L) : -2a2c2 < all < -a2C2, such that when -2a2c2 < al < all(al = all), (3.3.27) has exactly one single unstable small limit cycle (a inner unstable separatrix cycle) around A2 alone, whereas for all < al < 0, (3.3.27) has no closed orbit around A2 alone. Lemma 3.3.23. For -2a2c2 < al :S -a2C2, or a2b < al < ~a2b, (3.3.29h has at most one limit cycle surrounding three critical points simultaneously, and if it exists, it is simple and unstable. Proof. The roots of F 2(0 respectively
=
0, h(~)
=
0, and g2(0
°
are
and d l = -ylb 2 - 4J.L < -C2 = d 2 < 0. In the same way as for Theorem 3.3.21, we can verify that the conditions for Theorem 3.2.18 are all satisfied. The lemma then follows. 0 We conjecture for -a2C2 < al < a2b, system (3.3.29)2 has at most one limit cyle surrounding three critical points simultaneously, and if it exists, it is simple and unstable. This needs to be proved. Summarizing the conclusions above for system (3.3.27), we get
°
Theorem 3.3.24. For any fixed J.L < 0, b < 0, and < a2 < V2, there exist all = al1(a2,b,J.L), a12 = a12(a2,b,J.L) anda13 = aI3(a2,b,J.L) with -2a2c2 < all < -a2C2, all < a12 < ~a2b, -a2Cl < a13 < - 2a2cl, such that (3.3.27) has the different phase portraits as shown in Fig. 3.3.13.
Chapter 3.
162
Bifurcation in Polynomial Lienard Systems
rckJ 00
,/
./
/'
f
(a)
(S)G)
al :::; -2a2c2
~
~ /" (d) all < al < a12
/'
(b) -2a2c2 < al < all
(c)
.--/
-'
al = all
0. The proof of Theorem 3.3.25 is completed by the following lemmas. Lemma 3.3.26. For -00 < al ::; -3a3, system (3.3.31) has no closed orbit or singular closed orbit around B alone. Proof. From 1), we need only to prove the conclusion for al = 3a3. At this time, (3.3.31) becomes
x = y, iJ = (1 - x2)(x
(3.3.32)
+ 3a3Y)'
If (3.3.32) has a closed orbit r in x > 0, then it can only lie above l : x + 3a3Y = 0, i.e., x + 3a3Y > 0, as shown in Fig. 3.3.15. Hence, along r we have h(r)
= 1r (~: + ~~) dt = 3a3 loT (1 = 3a3
l C
Xr - Xl
d
+ 3a3Y)(Xr + 3a3Y)
(Xl
X2) dt
dy
> 0.
This shows that r is simple and unstable with the same stability of the critical point B inside. Thus r cannot exist. 0 11
:c
Fig. 3.3.15 Lemma 3.3.27. For al > -~a3' (3.3.31) has no closed orbit or singular closed orbit around B alone.
166
Chapter 3.
Bifurcation in Polynomial Lienard Systems
Proof. By translating B(l,O) to the origin, the equivalent form of (3.3.31) can be written as
~
= y - FI(~)'
iJ =
-gl(~)'
~ E 1= (-1,
(3.3.33)
+(0),
With FI(~) = a3e + 3a3e + (3a3 + ar)~, fr(~) = 3a3e gl(~) = ~(~2 + 3~ + 2), G1(~) = i~4 + +
e e.
Obviously, fr(~), gl(~) Eel, and ~gl(~) > 0 for 0 The simultaneous equations
+ 6a3~ + 3a3 + aI, =1=
~ E I.
can be reduced to a3(u 2 + uv + v 2 ) + 3a3(U + v) + al + 3a3 = 0, u 3 + (u 2 + UV + v 2 )v + 4( u 2 + UV + v 2 ) + 4( u + v)
= O.
It follows that
When al 2: -ia3, we have 3a3u + lla3 + al > 0, -a3u3 + 8a3u + 4al + 12a3 2: 0, so (3.3.34) has no solution. The conclusion of the lemma follows from Theorem 3.1.17. 0 Lemma 3.3.28. For -3a3 < al < -ia3, (3.3.31) has at most one small limit cycle around B alone, which is simple and unstable, if exists.
Proof. Consider (3.3.33). In the interval I, FI(~) and fr(~) each has one zero, -1 < 6 = ~( -3a3 + J9a§ - 4(al + 3a3)a3) < 0 and 1 J-3ala3 < 0 respectively. It is easy to see that -1 < ~o = -1 + -3 a3 (~ - ~o)fr(O > 0 for ~o =1= ~ E I.
3.3.
Global Bifurcation of Cubic Lienard Systems
167
As stated before, the simultaneous equations
can be reduced to
h(z)
= [3a3(Z + 1)2 + 9a3z + 15a3 + 2al](a3z2 + 3a3z + 3a3 + at}
+ a3(al + 3a3)(Z + 1)(z + 2) = 0 for -1 < Z < 6. It is easy to verify that h( -1) < 0, h(6) > 0, h"(z) > 0, and, furthermore, if 2al + 3a3 ~ 0 then h'(z) > 0 for z > -1, if 2al + 3a3 < 0 then there exists a unique Zo E (-1,6) such that
h' (z) < 0, h( z) < 0, 0 < z < Zo;
h' (z) > 0, Zo < z < 6.
Therefore, in either case, h(z) = 0 has a unique root in the interval (-1,6), which corresponds to the exactly one solution of (3.3.35). Consider the function
cp(O =f{(~)9l(~) - fr(~)9~(~) =-[3a3(~
+ 1)4 + 3(al + a3)(~ + 1)2 -
all.
Since we have
for ~ < 3.2.17.
-
~(fr(O) cp(~) < 0 d~ 91(0 - 9r(0
o.
The conclusion of the lemma follows on using Theorem 0
From the results proved above and 1), we know that for any fixed a3 > 0, there exists an all = all(a3) with -3a3 < all < -~a3 such that when -3a3 < al < an, (3.3.31) has only two small limit cycles around A and B alone, which are simple and unstable; when al ~ -3a3 or al > all, (3.3.31) has no closed orbits around A and B
168
Chapter 3.
Bifurcation in Polynomial Lienard Systems
ro
alone; whereas for al = all, (3.3.31) has a separatrix cycle passing through 0 and surrounding A and B simultaneously, and r 0 is both inner and outer unstable by Theorem 2.3.1. In addition, when al ::; all, A,O and B form an unstable critical point system (al ::; -3a3) or a critical point-cycle system (-3a3 < al ::; all), whereas for al > all, A, 0, B become a stable critical point system. Lemma 3.3.29. For al 2: -a3, system (3.3.31) has no closed orbits surrounding A, 0 and B simultaneously.
Proof. Since al 2: -a3 > all, A, 0, B form a stable critical point system, from 1), we need only to prove the conclusion for al = -a3. Consider the equivalent system of (3.3.31) with al = -a3, X = y - a3(x 3 - x), iJ = -x(x 2 - 1),
(3.3.36)
and a family of closed curves surrounding A,O and B simultanneously, V(x,y) = x4 - 2x2 + 2y2 = C, C 2: o. We have -dVI
= -4a3x 2 (x 2 - 1) 2 ::; O. dt {3.3.36} Therefore, the lemma is proved by the Poincare tangential curve method, see [188]. 0
Lemma 3.3.30. For -00 < al < -a3, system (3.3.31) has at most two large limit cycles surrounding A, and B simultaneously.
°
Proof. Let g(x)
= x 3 - x,
The roots of F(x)
=
f(x)
= 3a3x2 + all
0 and f(x)
= 0 are
F(x)
= a3x3 + alx.
3.4.
169
Global Bifurcation in Some Applied Models
and Xl
~
~
-
= -~-3a; < 0 < X2 = ~-~'
respectively. The roots of g(x) = 0 are -1, 0 and 1. Since al < -a3, we have Xl < -1 < 0 < 1 < X2, and, obviously, Xl ::; -1, X2 2: 1 for -00 < al ::; -3a3; -1 < Xl < 0 < X2 < 1 for -3a3 < al < -a3. It is easy to see that the functions
are monotonically decreasing for
-00
.
.~--~--~--~+~--------~--~
x2
".,~)
The conclusion of the lemma follows from Theorem 3.2.19 and its Remark 1. 0 Lemma 3.4.3. For (b, p) E A, system (3.4.2) has no closed orbits
if
""0 ::; "., < 2""0. Proof. Now the roots aI, a2 of F(x)
= 0 satisfy
-3"., < al < a2
z + 317 > 0, 4( 17 + 170)2 - 817(17 + 170) = 4(17 + 170)( 170 - 17) < 0, we obtain H(z) > 0 for -317 < al < z < a2 < O. That is, (3.4.3) has no solution. Then (3.4.2) has no closed orbits by Theorem 3.1.17. 0 Lemma 3.4.4. For (b, p) E B, the following conclusions hold: a) if 17 = 170' then system (3.4.2) has exactly one simple stable limit cycle; b) if 0 < 17 -170« 1, then (3.4.2) has at least two limit cycles.
i-
Proof. a) Since 17 2 = 17; = 1 - bp > 1, 0 is an unstable fine focus of order 1. By Theorem 3.3.3, (3.4.2) has at least one stable limit cycle. The uniqueness can be proved in a way similar to that for Lemma 3.4.2. b) At this time, 17 2 > 17; = 1 - bp > 1, and 0 becomes a stable critical point. Hence an unstable small amplitude limit cycle is bifurcated from 0 when 0 < 17 -170 « 1, while the original stable limit cycle still remains. Therefore, (3.4.2) has at least two limit cycles. 0
i-
If we can prove that there are at most two limit cycles, then we get the following complete result (as a conjecture).
3.4.5. 1) For (b,p) E A, system (3.4.2) has 17 = 170 as the Hopf bifurcation, and has exactly one limit cycle (simple and stable) if 0 ~ 17 < 170 and no closed orbits if 17 2:: 170' 2) For (b, p) E B, there exist an 17* E (170,2170) such that when 17 = 17*, (3.4.2) has a multiple-two limit cycle. It has exactly one limit cycle (simple and stable) if 0 ~ 17 ~ 170' and has two limit cyles as 170 < 17 < 17*, and no closed orbits if 17 > 17*· Theo~em
3.4.
Global Bifurcation in Some Applied Models
3.4.2.
175
A self-excited system
A self-excited system with three equilibria governed by the equation
ii - (/3 - 8il)if -
+ ,y3 = 0,
ay
a,/3",8> 0,
(3.4.4)
was studied in [142] using a numerical method. It is found that system (3.4.4) has one large limit cycle surrounding three equilibria simultaneously and two small limit cycles each around one of the equilibria with index +1. A more complete qualitative analysis for (3.4.4) was given in [172]. Later on, the problem about the number of large limit cycles was solved completely in [70]. We now introduce the results of [172] and [70]. Let if = x. By the scaling x - t :fix, Y - t -~y, t - t .)at, (3.4.4) is transformed into :i; =
if =
y3 - Y - (bx 3 - ax), -x.
(3.4.5)
It is easy to see that (3.4.5) has three critical points: 0(0,0), A( 0,1) and B(O, -1). 0 is a saddle, A and B are of index +1. The trajectories are symmetric with respect to the origin 0 since (3.4.5) remains unchanged under the trasformation x - t -x, y - t -yo System (3.4.5) is integrable if a = b = 0, then it has a general integral 2
Vc(x, y) = x - y
2
+ '12 y 4 = c,
1
--2 < C < +00 .
Vo( x, y) = 0 consists of double loops passing through 0 and lying in the strip region Ixl ~ For -~ < C < 0, Vc(x, y) = C consists of two closed orbits around A and B; and for C > 0, Vc(x, y) = C consists of one large closed orbit surrounding A, 0 and B simultaneously.
Yf.
Note that (3.4.6)
Chapter 3.
176
Bifurcation in Polynomial Lienard Systems
and hence (3.4.5) has no closed orbits if ab ~ 0(a 2 + b2 -=1= 0). We only need to discuss the case a > 0 and b > 0, since the case a < 0 and b < 0 can be treated by a change of variables x -+ -x, t -+ -to Let = x, TJ = y - 1, (3.4.5) becomes
e
e= ae + 2TJ + 3TJ2 -e·
iJ =
be
+ TJ3,
(3.4.7)
It is not difficult to see that A is an unstable focus when 0 < a < 2.j2, a node when a ~ 2.j2, or a stable fine focus when a = 0, b > O. The behavior of B is the same as A by symmetry. Let h(y)
= y3 -
y,
F(x)
= bx 3 -
ax,
g(x)
= X.
h(y), F(x) and g(x) satisfy the conditions of Corollary 3.1.13, and all solutions of (3.4.5) be bounded in positive sense. System (3.4.5) forms a family of generalized rotated vector fields with respect to a, and the vectors rotate counterclockwise as a increases. Based on the above fact, we now consider the limit cycles of (3.4.5).
Lemma 3.4.6. Fora ~ b/2, (3.4.5) has no closed orbit or singular closed orbit around A and B alone. Proof. Consider a family of closed curves Vc( x, y) = C, - ~ < C ~ O. As stated above they all lie in the strip region JxJ ~ Thus when a ~ b/2, we know from (3.4.6) that the trajectories of (3.4.5) always cross the curves Vi(x, y) = C in the same direction. The conclusion of the lemma follows by the method of tangential curves. 0
4.
Lemma 3.4.7. ([70]) For any a > 0 and b > 0, system (3.4.5) has at most two large limit cycles surrounding three critical points simultaneously. Proof. Let f(x) = 3bx 2 - a,
H(y)
=
1 4 1 2 -y - -y . 4 2
3.4.
Global Bifurcation in Some Applied Models
177
It is easy to see that: 1) xg(x) > 0 for 0 =1= x E (-00, +00); 2) f(x) < 0 for x E (-lfi, Ifi), f(x) > 0 for x E (-lfi, Ifi), and xF(x) < 0 for 0 =1= x E (-If, If), G( -.;1') = G( If); 3) yh(y) < 0 for 0 =1= y E (-1,1), yh(y) > 0 for y rt (-1,1), and
H( -1) = H(1); 4) the function f(x)/x is monotonically increasing for x E (-00, O)U (0,+00) from f(O) < 0,1"(0) > o. Consequently, the lemma follows from Theorem 3.2.13. 0 Theorem 3.4.8. For any fixed b > 0, there exist a o = ao(b), al = al(b), and a2 = a2(b) with 0 < al < a o < a2 < b/2, such that: (1) a = a o is a homoclinic bifurcation, for which system (3.4.5) has a sepamtrix cycle r 0, both inner and outer unstable, passing through o and surrounding A and B simultaneously, and at the same time, inside the two loops of r 0, there exist a small stable limit cycle around A and one around B alone. There exists exactly one large stable limit cycle Ll outside of r 0 with the picture shown in Fig. 3.4. 3( d); (2) when al < a < a o, the two small limit cycles still exist, and r 0 becomes another unstable large limit cycle L2 inside of L l , and they coincide into a multiple-two limit cycle L12 when a = ai, which corresponds to a (large) limit cycle bifurcation of multiple-two; (3) when 0 < a < ai, there are two limit cycles, one around A alone and one around B alone, but L12 disappears and there is no longer any large limit cycle; (4) when a increases from a o (opposite to (2)), two small limit cycles are bifurcated from r 0, one around A and one around B, which are outside of the two small stable limit cycles that exist before; when a = a2, they coincide to become two small multiple-two limit c;ycles around A and around B, i. e., a = a2 (b) corresponds to a (small) limit cycle bifurcation of multiple-two; (5) when a> a2, the limit cycles around A and B alone disappear, while the large limit cycle Ll still exists.
Chapter 3.
178
(a) 0 < a < al
Bifurcation in Polynomial Lienard Systems
(b) a = al
(d) a
(f) a = a2
(g) a> a2
= aa
Fig. 3.4.3 The pictures in Fig. 3.4.3 represent the evolutions of the phase portraits of (3.4.5) with the change of parameter a.
Proof. For any fixed b > 0, when a = 0, (3.4.5) has no closed orbit or singular closed orbit, as stated above, and all solutions are bounded in the positive sense by Corollary 3.1.13. But A and Bare stable, so we get the separatrix configuration as shown in Fig. 3.4.3(a). When a ~ b/2, (3.4.5) also has no closed orbit or singular closed orbit around A and B alone from Lemma 3.4.6 and A and B are unstable, hence the separatrix configuration must be as shown in Fig. 3.4.3(g). There must exist a unique aa = aa(b) with 0 < aa < b/2, such that the corresponding system (3.4.5) has a double separatrix loop ra passing through 0 and surrounding A and B, that is both inner and outer unstable by Theorem 2.3.1. Moreover, at this time, the critical points A and B are unstable, and all solutions are bounded in the positive sense. Therefore, in the interior of r a there must be two small stable limit cycles, one around A and one around B alone, and in the
3.4.
Global Bifurcation in Some Applied Models
179
exterior of r 0 there must be one stable large limit cycle surrounding this critical point-cycle system. That is the situations shown in Fig. 3.4.3( d). The remainding conclusions can be obtained by using the above results and the theory of rotated vector fields. 0 We conjecture that (3.4.5) has at most two small limit cycles around A and B alone, which needs to be proved. 3.4.3.
Bogdanov-Takens system (continued)
We already meet the important Bogdanov-Takens system in Chapters 1-2. We now consider the system in the form
x
= y,
iJ = Al + A2Y + x 2 - T/XY,
T/ = ±1,
(3.4.8)
which is equivalent to (1.3.4) as seen by changing (y, t, A2) -+ (-Y, -t, -A2). The bifurcation curves near (AI, A2) = (0,0) have been given in Fig. 1.3.4. It is obvious that the Hopf bifurcation curve (AI = -AD and saddle-node bifurcation curve (A2 axis) can be considered as global results. We now use the uniqueness theorem in Sec. 3.2 to prove that the limit cycles of (3.4.8) are at most one for (AI, A2) E IR? Thus, the homoclinic bifurcation curve in Fig. 1.3.4 can also be extended to the global, and considered as a complete result. We only need to consider the case Al < 0, in which translating the antisaddle to the origin changes (3.4.8) to
x = Y,
iJ = -X(2J-A1 - x) - (-J- A1 - A2 + x)y.
(3.4.9)
The two critical points are: 0(0,0) with index +1 and A(2J-A1'0) which is a saddle. Change (3.4.9) to the Lienard system
x=y-F(x), iJ = -g(x),
(3.4.10)
Chapter 3.
180
Bifurcation in Polynomial Lienard Systems
where 12
1\
F(x) = "2x -(y-Al+A2)X,
g(x) = X(2V-Al - x),
From Lemma 3.3.2, 0 is an unstable (a stable) elementary critical point if A2 > - ) - Al (A2 < -) - AI); and is a stable fine focus of order 1 as A2 = -)-Al' Let
D = {(x, y) : -00 < x < 2V-Al, Iyl < +oo}. For fixed Al < 0, system (3.4.9) forms a family of generalized rotated vector fields with respect to A2, and the vectors rotate counterclockwise as A2 increases. The roots of F(x) = 0 and f(x) = x - ()-Al + A2) = 0 are
respectively. If A2 ~ 0, then from Xl ~ 2)-Al' g(x)F(x) < 0 for x E (-00, 2)-Al), we know that system (3.4.10) has no closed orbit or singular closed orbit in D by Theorem 3.1.20. Furthermore, we can prove
o :f=.
Lemma 3.4.9. For Al < 0 system (3.4.9) has no closed orbits as A2 ::; - ) - AI, and at most one single stable limit cycle as - ) - Al < A2 < 0 in D.
Proof. Consider system (3.4.10). For -)-Al < A2 < 0, obviously, the condition (A) in Theorem 3.2.15 is satisfied. After simplifying and putting z = u + v, w = uv, the simultaneous equations
F(u) = F(v),
g(u)
g(u)
f(u)
f(u)"
which can be reduced to
z W
= 2( )-Al + A2), =
< 2)-Al ()-Al + A2)(Z - 2)-Ad, w < O. -00
o.
We consider the following cases in turn. Case 1. a20 = aol, that is, b = O. (3.4.14) becomes X
iJ
= y, = -/-LX
+ x3 -
(al
+ 2a2x)y,
(3.4.15)
which is integrable if al = O. The phase portaits can be seen in Figs. 3.3.1-3.3.2. By the theory of rotated vector fields, it is seen that there is no closed orbits if al i= o. Case 2. al o = 0, a20 i= aol, that is, /-L = 0, b i= O. (3.4.14) becomes
=y, iJ = -x 2(b - x) - (al
;i;
+ 2a2x)y,
(3.4.16)
which has two finite critical points: 0(0,0) a saddle-node and A(b, 0) a saddle. Then (3.4.16) has no closed orbits from the index theory. Case 3. al o i= 0, a20 i= aol, which can be divided into two subcases: (1) al o < 0, that is, /-L > O. (3.3.14) is a system of the type of (3.3.24). From the results in Sec. 3.3.5 (A) we have the following conclusions: for any au < -2 and al o < 0, if aol(a20 - aol) < 0 then
3.4.
Global Bifurcation in Some Applied Models
183
°
(3.4.14) has no closed orbits; if aol(a20 - aol) > then (3.4.14) has at most one simple limit cycle, and, if it exists, is unstable. (2) al o > 0, that is, J.L < 0. If b2 + 4J.L ~ 0, i.e., la20 - aoll ~ 2V-aloan, then (3.4.14) either has only one saddle 0(0,0) or has a saddle 0 and a saddle-node C(~, 0), there being no closed orbits in both cases; and if la2o-aoll > v-aloan then (3.4.14) has three critical points: 0(0,0) (a saddle) and C I (CI,0),C2(C2,0). By translating one of the CI, C2 with index +1 to the origin, we may obtain a system of the form of (3.3.24) again. It has at most one limit cycle by the results in Sec. 3.3.5 (A), see [168] for details. To sum up, we have proved that (3.4.14) has at most one limit cycle in any case.
Chapter 4 Periodic Perturbed Systems and Integral Manifolds Parallel to the study of autonomous systems, another active area of research concerns the theory of non-autonomous systems, among which the case where time dependence is periodic is very useful in many applied fields. So in the remaining chapters we shall deal partly with the non-autonomous systems which are periodic in t. In this chapter we present first the methods of local bifurcation of periodic solutions for periodic perturbed systems. Then we give a brief introduction to the theory of method of averaging and integral manifolds. Finally, as an application we consider the bifurcations of an invariant torus for time-periodic perturbed systems.
4.1.
Bifurcation of Periodic Solutions
4.1.1.
Poincare maps and uniqueness of periodic solutions
Consider the following system of differential equations
x=f(t,x),
( 4.1.1)
where n ~ 1, f: IR x U ----+ IRn is a CT function and U is an open set in IRn. Suppose that the time dependence of (4.1.1) is periodic with period T > 0, i.e., f(t,x) = f(t+T,x). (4.1.2) 185
Chapter 4.
186
Periodic Perturbed Systems and Integral Manifolds
Let x(t, x o) be a solution of (4.1.1) satisfying x(O, x o) = Xo and x(t,xo) E U for all t E IR. Then from (4.1.2) it is easy to see that the solution is T-periodic iff x(T, x o) = Xo' We call the function x(T, x o) of Xo a Poincare map of (4.1.1), denoted by P(x o), and P(x o) - Xo the succession function of (4.1.1). It is clear that the number of Tperiodic solution of (4.1.1) on U is equal to that of the fixed points of P or of the roots of P - id in a suitable open set V cU. If U = IRn, we can choose V = IRn, in which case P is defined on IRn. If f(t,O) = 0 and U is an arbitrary neighborhood of the origin, we may choose V = U. It should be clear that a k-periodic point of P corresponds to a periodic solution of (4.1.1) with period kT. A T-periodic solution x(t, x o) is said to be hyperbolic if Xo is a hyperbolic fixed point of P, i.e., the matrix DP(x o ) has no eigenvalue with unit norm. We shall discuss the existence of a unique periodic solution near a given T-periodic solution. Without loss of generality, we may suppose the solution is zero, and consider the following system
x = A(t)x + f(t, x, >'),
(4.1.3)
where A is a continuous T-periodic n x n matrix and f is T-periodic in t and is C r (r ~ 1) for all t and (x, >.) in a neighborhood of (0,0) E IRn x IRm with
f(t, 0, 0) = 0,
Dxf(t, 0, 0) =
o.
(4.1.4)
Let X (t) be a fundamental matrix of the corresponding linear homogeneous system x = A(t)x. (4.1.5) Then we have
x(t, x o, >.) = X(t)X-l(O)xo
+ i t X(t)X- 1 (s)f(s, x(s, Xo, >'), >.)ds, 0
(4.1.6)
where x(t,x o,>') is a solution of (4.1.3) with initial value Xo at t The Poincare map of (4.1.3) is given by
P(xo, >.)
=
X(T)X-l(O)xo
+ loT X(T)X-l(t)f(t, x o, >'), >')dt.
= O.
(4.1.7)
4.1.
Bifurcation of Periodic Solutions
187
Now we can prove the following: Theorem 4.1.1. Suppose that (4.1.5) has no nonzero T-periodic solution. Then there exists {; > 0 such that for IAI < {; (4.1.3) has a unique T-periodic solution x*(t, A), which is C r with respect to (t, A) and satisfies Ix*(t, A)I < {; and x*(t, 0) = o. Proof. Note that the general solution of (4.1.5) is
x(t, c)
=
X(t)X-l(O)C,
(4.1.5) has no nonzero periodic solutions iff the linear equation
has no nonzero solution, where In denotes a n x n identical matrix, or equivalently det(X(T)X-l(O) - In) i= O. Then from (4.1.4), (4.1.7) and x(t, 0, 0) = 0 we have
DxP(O,O)
=
X(T)X-l(O).
Under our assumption we have
Hence by the implicit function theorem, the equation P(xo, A)-Xo = 0 has a unique C r solution Xo = Xo(A) with xo(O) = O. The desired periodic solution is given by x(t, Xo(A), A) == x*(t, A). D Since
DxP(Xo(A), A) = X(T)X-l(O) + g(A), g(O) = 0 from (4.1.7), we have immediately Corollary 4.1.2. Suppose that the zero solution of (4.1.5) is hyperbolic, then the periodic solution x*(t, A) is also hyperbolic and has the same stability as the zero solution.
Chapter 4.
188
Periodic Perturbed Systems and Integral Manifolds
In a similar way we may discuss the existence of periodic solutions for the following (n + m) dimensional system:
x = AP[Ax + !l(t, x, y, A)],
(4.1.8)
iJ = By + h(t, x, y, A),
where x E IRn , y E IRm , A > 0, pEN (set of natural numbers), and A and Bare n x nand m x m real constant matrices respectively. Suppose that the functions h, 12 in (4.1.8) are C 2 , T-periodic in t, and satisfy h(t, x, y, A) = O(IAI + Iyl + IxI 2), (4.1.9)
h(t, x, y, A) = O(IAI + lx, YI2). Let z(t,zo,A) = (x(t,zo,A),y(t,zo,A)) be a solution of (4.1.8) with initial value Zo = (xo, Yo) at t = O. Then similar to (4.1.6) we have VAt xO+/\Joe 'P rt APA(t-s)!1( ( ),Ads, ) ( ) Xt.,Zo,A=e s,ZS,Zo,A (4.1.10)
y(t, ZO, A) = eBtyo + fot eB(t-s) h(s, z(s, ZO, A), A)ds. Note that eAPAT - In = AP(AT + O(A P)). If we denote by P(zo, A) the Poincare map of (4.1.8), then from (4.1.10),
P(zo, A) - Zo = (APP1(zo, A), P2(Zo, A)), where
P1(zo, A) = (AT + O(AP))xo + foT eAPA(T-t) h(t, z(t, ZO, A), A)dt, (4.1.11)
P2(zo, A) = (e BT - Im)yo
+ foT eB(T-t) h(t, z(t, ZO, A), A)dt.
It is obvious from (4.1.10) and (4.1.9) that
x(t, ZO, A) Then
=
O(lxol + IAP+11 + APlzol),
y(t, ZO, A)
=
O(IYol + IAI + IzoI2).
4.1.
Bifurcation of Periodic Solutions
189
where G(O, 0) = o. Thus by the implicit function theorem, we obtain the following theorem: Theorem 4.1.3. Suppose that det A
=J 0, det( eBT - 1m) =J o.
Then there exists 0 > 0, such that for 0 < .:\ < 0 (4.1.8) has a unique T-periodic solution z(t, ':\), which is C r with respect to (t,.:\) and satisfies Iz(t, .:\)1 < 0 and z(t,O) = o. Furthermore, if both A and B have eigenvalues with nonzero real parts, then z(t,.:\) has same stability as the zero solution of the linear system
if
x=.:\PAx,
4.1.2.
=
By,
The Liapunov-Schmidt method
In this section, we shall study the bifurcation of periodic solutions for system (4.1.3). From Theorem 4.1.1, we may suppose that the linear system (4.1.5) has a nonzero T-periodic solution. Let p ;::: 1 be the maximal number of linearly-independent T-periodic solutions, and cI>(t) an n X p matrix whose columns form a base of T-periodic solutions of (4.1.5). Note that the inverse X-l(t) of the fundamental matrix X(t) of (4.1.5) is a fundamental matrix of the system
if = -yA(t).
( 4.1.12)
There is a p X n matrix w(t) whose rows form a base of T-periodic solutions of (4.1.12). Let
c = foT (t)cI>(t)dt,
D
= foT w(t)~(t)dt,
where and ~ are transpose matrices of cI> and W respectively. C and Dare nonsigngular (see [62]). We now introduce the following two Banach spaces:
BT = {J : IR
---t
IRn 1f is continuous and T - periodic }
Chapter 4.
190
Periodic Perturbed Systems and Integral Manifolds
with norm If I = SUPt IIf(t)lI, and B~
= {f
with norm IfiI = SUPt(lIf(t) I By and Q on BT by
E
BTl!' E Br}
+ IIf'(t)II).
Define the projections P on
P f = if!(-)C- I loT 0.
2m
If A = Ai and 1/ "I 0, then we may choose H = (e B1T Pi = In in this case. If 1/ = 0, then
It is clear that
-
In)-i since
194
Chapter 4.
Periodic Perturbed Systems and Integral Manifolds
(H~-I ~ ) .
Hence we may take H = a
=1=
0 or a = 0 and
f3
H = PI(e B2T - In)-I. If a
For the case of A = B 2 , if
2~1r, kEN, then similar to the above = 0 and f3 = 2~1r for some kEN, then
=1=
Thus we can take
H
=
h)
0 ( HilO
n m-- 2·
'
o
This ends the proof. Remark 1. It is easy to see that if
A then Po
= diag(O,B),
= diag (1, 0) A
= diag
and H
det(e BT - In-d
= diag (1, (e BT -
=1=
0,
In_It l . If
) ( ( _ 02~n 2k1r) ~ ,B ,
then Po = diag(h,O), H = diag(h, (e BT - I n_2)-I). Now suppose x(t,x o,>') is a solution of (4.1.22) with x(O,x o,>') Xo. By the method of variation of constants, we have
x(t, x o, >.)
= eAtxo + lot eA(t-s) f(s, x(s, x o, >.), >.)ds.
=
(4.1.24)
The Poincare map is x(T, ., >.). x(t, x o, >.) is T-periodic iff
(eAT - In)x o + eAT loT e- At f(t, x(t, x o, >.), >')dt
= 0,
which is, by Lemma 4.1.7, equivalent to
PIx o + HeAT loT e- At f(t, x o, >.), >')dt
= O.
(4.1.25)
4.1.
Bifurcation of Periodic Solutions
Let Xo = Poxo
+ PI Xo == a + b,
F(a,b,).)
195
a E Eo, b E E I , and
= forT e-At f(t,x(t,a+b,).),)')dt.
( 4.1.26)
From (4.1.23), it is easy to see that (4.1.25) is equivalent to
PoHF(a,b,).) = 0, b + PIH eAT F(a, b,).)
(4.1.27)
= O.
Since ~f(O,O,O) = 0 from (4.1.4), the second equation of (4.1.27) has a unique CT function b = b*(a,).) = 0(1).1 + lal 2 ) for (a,).) near the origin. Let G(a,).) = PoH F(a, b*(a, ).), ).), (4.1.28) then (4.1.27) is equivalent to G( a, ).) = O. This result can be summarized in
Ixol < 8, 1).1 < Xo = a + b*( a, ).)
Theorem 4.1.8. There exists a 8> 0 such that for
8, x( t, x o, ).) is aT-periodic solution of (4.1.22) iff and G(a,).) = o.
The significant terms of the Taylor's expansion of F(a, b,).) can be computed by using f(t, x, ).). For instance, let
+ fl(t, ).). + A1(t)x(v·).) + 0(lxl k + 1 + 1).1 . IxI 2 ),
f(t, x,).) = fk(t, x)
(4.1.29)
where 0 =1= v E IRm, f k (t, x) is a homogeneous polynomial of order k in x, k = 2 or 3. From (4.1.24), (4.1.26), and (4.1.29) we have
x(t, x o,).) = eAtxo + lot eA(t-s) !I(s, )')'ds + O(lxo . ).1
+ Ix o l2 )
and
F(a,b,).)
=
loT e-Atfk(t,eAt(a+b))dt+ loT e-Atfl(t,O)'dt + O(la, bl k + 1 + 1).1 . la, bl
+ 1).1 2 ).
(4.1.30)
Chapter 4.
196
If h(t,)..)
= 0,
Periodic Perturbed Systems and Integral Manifolds
then
F(a, b,)..) =
faT e- At fk(t, eAt(a + b))dt + (v· )..) X
faT e-AtAl(t)eAt(a + b)dt
+ 0(1)..1 + Ixol)(I)..I· Ixol + Ixol k )).
(4.1.31)
A special case is that A = diag (0, B) with B a (n - 1) x (n - 1) matrix with nonzero real part eigenvalues. In this case, it follows from Remark 1 after Lemma 4.1.7 that
G(a,)..) = (Gl(al, )..), ol,
a = (ab ol, G l , al E JR.
Using (4.1.28), (4.1.30), Remark 1, and e- At
Gl(al,)..) = a·)"
= diag (1, e- Bt ),
+ f3ka~ + 0(1)..1 2 + I)..' all + lall k+1),
we get
(4.1.32)
where f3k is the first component of the vector II A(t, eAta)dt, and a E JRm is the first line of the n x m matrix II fl(t, O)dt. Generically, we have k = 2 and f321al i= 0. For example, if k = 2 and alf32 i= 0, then from the implicit function theorem the following system of equations,
has a unique solution al = ai()..2,· .. ,)..m), )..1 = )..i()..2,··· ,)..m) for (aI, )..1) near the origin. The Taylor's expansion yields
Gl(ab)..) = Gl(al,)..) - Gl(ai, )..i, )..2,'" ,)..m)
=
~~; (ai, )..i, )..2,' ..
,)..m)()..l -
{)2G2l (* * )..2,'" + -() aI' )..1'
al
,)..m
)..i)
)( al - a *)2 l
+ 0(1)..1 - )..i1 2 + lal - ail' 1)..1 - )..il + lal = al()..l - )..i)[l + 0(1)..1 + lall)] + 2f32(al - ai)2[1 + 0(1)..1 + lall)]·
- ail 3 )
4.1.
Bifurcation of Periodic Solutions
197
Using the implicit function theorem again we know that function G l has exactly two (resp. one multiple or no) zeros if al!J2( Al - An < 0 (resp. = 0 or > 0) for lall, IAI small. For this case, we say that the periodic solutions of (4.1.22) undergo a saddle-node bifurcation. If f(t, 0, A) = 0, then, similarly, we get from (4.1.31)
where 13k is as above and b is the first component of
When bf3k i= 0, there is a trans critical bifurcation (for k = 2) or a pitchfork bifurcation (for k = 3). In certain cases we need to consider a T-periodic equation of the form
x = AP(Ax + f(t, x, A, J.L)),
( 4.1.33)
where A, J.L E JR, x E JRn , p > 0, and
For simplicity, we suppose that A = diag (0, B) and any eigenvalue of the (n -1) x (n -1) matrix B has a nonzero real part. From (4.1.10), system (4.1.33) has a solution x( t, x O , A, J.L)
=
( 01 e)'.P0Bt ) (a) b
where we have assumed
Xo
=
+ AP iort
(e).P B(t-s) f(1) f(2) ) ds,
(a,bf, f = (J(1),J(2))T, a,f(1) E JR.
Chapter 4.
198
Periodic Perturbed Systems and Integral Manifolds
Hence
= AP (GI(a, b, A, f,L)) . G2(a, b, A, f,L) Obviously we may solve out b = b*(a, A, f,L) G 2 = O. By inserting it into G I , we get
GI(a, b*, A, f,L)
= O(a2 + IAIP + 1f,L1)
from
= a2loT h(t, 1, O)dt + f,L loT f~l)(t)dt + A loT fJI)(t)dt + ... == f32a 2 + f,Lal + Aa2 + ... ,
(4.1.34)
which has the form of (4.1.32) with k = 2. Thus, if alf32 01= 0, then we have a saddle-node bifurcation with the bifurcation curve
f,L
= f,L*(A) = - a2 A + O(A1+P ) al
on the (A, f,L) plane.
4.2.
Method of Averaging and Integral Manifolds
In this section we give a general theory of the method of averaging and then state a theorem concerned with the existence of integral manifold which was proved in [62J. 4.2.1.
Method of averaging
Consider a system of the form
x = ).'p f(t, x, y, >"), iJ = By + g(t, x, y, >"),
(4.2.1)
4.2.
Method of Averaging and Integral Manifolds
199
where ,\ E IR, p 2:: 1, x E IRn , y E IRm , B is a m x m matrix having eigenvalues with nonzero real parts. We assume that f and 9 are Tperiodic in t and are C r (r 2:: 2) in their variables with g(t, x, y, 0) =
O(IYI2). Define the averaged equation of (4.2.1) as u=,\P/(u),
where
f(u)
v=Bv,
(4.2.2)
1 (T
= Tio f(t,u,O,O)dt.
Then we have the following averaging theorem. Theorem 4.2.1. There exists a c r transformation of coordinates
+ ).Pw(t,u),
y = v,
(4.2.3)
u = ,\p/(u) + '\Ph(t,u,v,'\), v = Bv + gl(t, u, v, ,\),
(4.2.4)
x = u
under which (4.2.1) becomes
where hand gl are T -periodic in t, and h(t, u, v,'\)
= f(t, u, v, 0) - f(t, u, 0, 0) + h(t, u, v, ,\),
h(t, u, v,'\)
=
f. J.\fij)(t, u, v, O),\j + ,\P[!~(t, u, v, O)w
j=1
- w~(f(t, u, v, 0) -
(4.2.5)
wDl + o (,\p+l ),
Moreover, if (u o , 0) is a hyperbolic critical point of (4.2.2), then there exists a '\0 > such that for < ,\ :::; '\0' (4.2.1) possesses an isolated hyperbolic periodic solution of period T
°
°
(x(t, ,\), y(t, ,\))
= (u o , 0) + 0('\),
with the same stability property as (u o , 0).
Chapter 4.
200
Periodic Perturbed Systems and Integral Manifolds
Proof. We use the to-be-determined method to determine the transformation (4.2.3). Notice that
+ APw~tl = I
(I
- A.PW~
+ O(A 2p ).
Then from (4.2.1) and (4.2.3) U = AP(J(t, u, v, 0) - wD
v = Bv + 91(t, u, v, A).
+ APh(t, u, v, A),
(4.2.6)
Now we choose w(t, u) as a T-periodic solution of the equation w~ =
f(t, u, 0, 0) - /(u).
Then (4.2.4) follows from (4.2.6). The last part of the theorem can be obtained from Theorem 4.1.3 by setting it, = u - U o . 0 Remark 1. If p = 1 and y does not appear in (4.2.1), then
x=
Af(t, x, A).
(4.2.7)
In this case, there exists a C r periodic change of coordinates of the form X
= U + AW(t, u)
r-l
+ A 2: Aiwi(t, u), i=l
which carries (4.2.7) into r-l
U = A/(U)
+ A 2: Ai h(u) + Al+r fr(t, u, A), i=l
where, from (4.2.5),
fl(U) =
~ foT[f~(t, u, 0) + f~(t, u, O)w - w~(J(t, u, 0) - w~)]dt.
Next, we consider the following multiple periodic system
e=
W + AS(t, (), r, A), r = AR(t, (), r, A),
(4.2.8)
4.2.
Method of A veraging and Integral Manifolds
201
where A E JR, w, e E JRn(n ~ 1), r E JRm(m ~ 1), Sand Rare Coo functions and are 271'-periodic in t and in each component of () respectively. From [64], we have Lemma 4.2.2. Let
Po(t,e,r)
= T->oo lim T1 loT S(t+T,e+wT,r,O)dT, 0
Qo(t,e,r) = lim T1 T->oo
iT R(t+T,e+wT,r,O)dT. 0
Then there exist functions u(t, e, r, A) and vet, e, r, A) which are smooth enough and 271'-periodic in t and in each component of (), such that for bounded r
+ UBW Iv~ + VOW lu~
+ poet, e, r)1 < O'(A), R(t, e, r, 0) + Qo(t, e, r)1 < O'(A), Set, e, r, 0)
where 0' is a non-negative continuous function for A ~ 0 with 0'(0) = O. Moreover, AU, AV, AU~, AU~, AU~, AV~, AVO and AV~ approach zero uniformly for bounded r as A --t O. We are now in a position to prove the generalized averaging theorem. Theorem 4.2.3. Under the change of coordinates
e
= c/> + AU(t, c/>,p, A),
r
= p + AV(t, c/>,p, A),
(4.2.9)
the system (4.2.8) becomes
;p = W + APo(t, c/>,p) + AP1(t, c/>,p, A), P= AQo(t, c/>,p) + AQl(t, c/>,p, A),
(4.2.10)
in which W = (WI,'" ,Wn ), PI and Ql are continuous in A and 271'periodic in (t, c/», and satisfy H(t,c/>,p,O)
= 0,
Ql(t,rp,P,O)
= o.
Chapter 4.
202
Periodic Perturbed Systems and Integral Manifolds
Moreover, if the following nonresonance condition holds, ±ko ±
kIWI
± ... ± knwn i= 0 for all k i E N, i = 0" .. , n, (4.2.11)
then Po and Q 0 are independent of (t, c/J), and equal to
10
Po(p) = (27r1)n+ I
211" 1211" ... S(t, 0 0
Qo(p) = ( 27r1) n+ I
10211" .. . 10211" R(t, 0, p, O)dtdO. 0
O,p, O)dtdO, (4.2.12)
0
Proof. From (4.2.9),
iJ = AU~ + (In + AU',p)¢ + AU~P,
(4.2.13)
r = AV; + (In + AV',p)¢ + (Im + AV~)p.
By Lemma 4.2.2, (4.2.8), and (4.2.13), we know that there exists a continuous function 7](A) with 7](0) = 0, such that ( In
+ ;u',p AV¢
AU~
I) (~) = (w + AS(t, 0, r, A) - ~U~) P AR(t, 0, r, A) - AV = (w + AS(t, c/J,p, 0) - ~U~) + O(A7]). AR(t,c/J,p, 0) - AV
1m + AVp
t
t
Substituting (4.2.10) into the equality above we obtain
(In
+ AU',p)(W + APo + API) + AU~(AQo + AQI) = W + AS - AU~ + O(A7]),
AV',p(W + APo + API) + (Im = AR - AV~ + O(A7]),
+ AV~)(AQo + AQI)
or equivalently,
(In + AU',p)PI + AU~QI = S(t,c/J,p, 0) - Po(t,c/J,p) - u~ - u',p + 0(7]), AV',pPI + (Im + AV~)QI = R(t, c/J,p, 0) - Qo(t, c/J,p) - v~ - v',p + 0(7]). (4.2.14)
4.2.
Method of Averaging and Integral Manifolds
203
Notice that the right-hand side of (4.2.14) does not depend on PI and Ql. We can solve PI and Ql from (4.2.14). Obviously, from Lemma 4.2.2, PI, Ql = 0(0' + TJ). Hence, the system (4.2.10) is determined completely. From the theory of almost periodic functions, Po and Qo are independent of (t, e), and are given by (4.2.12) if the condition (4.2.11) 0 holds. The proof is finished. Remark 2. If S(t, e, r, 0) and R(t, e, r, 0) are vector polynomials in cos t, sin t, cos ej , and sin ej , j = 1"" , n, then one may prove that PI, Ql = O()') in (4.2.10). If S(t, e, r,).) and R(t, e, r,).) are such kind of polynomials for each fixed), and the nonresonance condition (4.2.11) holds, then, for any natural number n, the system (4.2.8) can be transformed into the form n-l
;p =W + L ).j+lpAp) + ).n+lPn(t,.n+lQn(t, .).
j=O
For the proof we refer the reader to [24], [78J. 4.2.2.
Integral manifolds
Consider the following multiple periodic system,
w(>.) + S(t, e, x, y, >'), x = A(>')x + Fl(t, e, x, y, >'), if = B(>')y + F2(t, e, x, y, >'),
iJ
=
(4.2.15)
where (>', e, x, y) E IR X IRk X IRn x IRm , A(>') and B(>.) are matrices, and 5, Fl and F2 are 21T-periodic in e = (e l , .. · ,ek)' We will give the fundamental results on integral manifolds for (4.2.15) based on the results of [24J and [62J. A set 5 c IR X IRk X IRn x IRm is said to be a local invariant set of (4.2.15) if any solution (e(t), x(t), y(t)) of (4.2.15) with (to, e(to), x(t o), y(to)) E 5 for some to E IR exists and
Chapter 4.
204
Periodic Perturbed Systems and Integral Manifolds
satisfies (t, O(t), x(t), y(t)) E S for It - tol small. If the property holds for all t E 1R., S is said to be an invariant set. If S has a manifold structure, then it is called an invariant manifold or an integral manifold. To discuss the existence of an integral manifold, we make the following assumptions on (4.2.15). (HI) The functions A, B, w, S, FI and F2 are all continuous and bounded on 1R. X 1R.k X 0(0', .\0)' where
0(0', .\0)
= {(x, y,.\) : Ixl < 0', Iyl < 0', 0< .\ ::;
.\o}
(H2) There exist continuous functions ",(0', '\), ,(0',.\) nondecreasing in 0' and .\, such that Sand Fi on 1R. X 1R.k X 0(0',.\) are uniformly Lipschitz continuous with respect to 0 with Lipschitz constants ",(0', .\) and ,(0', '\), respectively. (H3) There exist continuous functions JL(O',.\) and 8(0', '\), nondecreasing in 0' and .\, such that Sand Fi on 1R. X 1R.k X 0(0',.\) have Lipschitz constants JL(O',.\) and 8(0',.\) with respect to (x, y). (H4) There exists a continuous nondecreasing function N(.\) for o < .\ ::; .\0' such that for (t, 0) E 1R. X 1R.k, 0 < .\ ::; .\0
IFi(t, 0, 0, 0, .\)1 ::; N(.\),
i
=
1,2.
(H5) There exists a constant K > 0 and a continuous function a(.\)
> 0 defined for 0 < .\ ::; .\0' such that
I ::; K e-a(A)(t-T), leB(A)(t-T) I ::; K e-a(A)(t-T), leA(A)(t-T)
(H6) There exist continuous functions
o < .\ ::;
r
t 2: ~(.\)
T,
> 0, D(.\) > 0 for
.\0' such that
l-Ta sup
[K ,(D(.\),.\)
a(.\)~(.\) + .
",(D(.\),.\)
+ JL(D(.\), .\)~(.\)] a(.\)
KN('\)
l~ sup a('\)D('\) < 1,
< 1,
4.2.
Method of Averaging and Integral Manifolds
. [8(lJ(A),A) l~ a(A)
205
+ f.L(lJ(A), Ah(lJ(A), A) ] = O.
Then from Theorem 7.2.1 in [62], we have Theorem 4.2.4. If system (4.2.15) satisfies the above assumptions (Hl)-(H6), then it admits a unique integral manifold of the form
SA = {(t,O,x,y): (x,y) = f(t,O,A), (t,O) E IR x IRk}, where f is continuous for (t, 0) E IRx IRk and 0 in 0 and satisfies If(t, 0, A)I ::; lJ(A),
.(v) and 8(v) are any given C 1 functions for v 2:: 0 with >'(0) = 0, 8(0) = 80 • Then letting>. = >.(v), 8 = 8(v) and introducing the scaling r --t v 1/ 2 r, we obtain from (4.3.16)
() = 1 + v1/2Q1(O)r + O(v 1/ 2), r = v1/2[R1(O)r2 + R2(t, 0)>.'(0) + R3(0)8'(0)] + o(v 1/ 2).
(4.3.18)
Now we can prove
Let
Theorem 4.3.5.
and let -
27f T
. . be zrratwnal and suppose (4.3.15) hold,
loT ~(O)dO, i = 1,3, 0 1 -T loT dO 1211" R 2(t,0)dt.
= -T1
~
- = R2
27f
0
(4.3.19)
0
If
R1[R2>.'(0)
+ R 38'(0)] < 0
(resp. > 0),
then system (4.3.1) has precisely two (resp. no) invariant tori near L x Sl for>. = >.(v)' 8 = 8(v) and v > 0 small. Prqof. Since
~
is irrational, the application of the averaging
theorem (Theorem 4.2.3) to the system (4.3.18) yields
() = 1 + v 1/ 2Q1r + o(v 1/ 2), r = v 1/ 2[R1r2 + R2>.'(0) + R 38'(0)] + o(v 1/ 2). o
The conclusion follows from Theorem 4.2.5. Next, suppose that 27f
p
T
q
(4.3.20)
is rational with (p, q) = 1. Then setting ¢ obtain the 27fq-periodic system in (t, ¢)
;p = r=
v 1/2Q1(¢ + t)r + o(v 1/ 2), v 1/ 2[R1(¢ + t) + R 2(¢ + t)>.'(O)
=
0 - t in (4.3.18) we
+ R3(¢ + t)8'(0)] + O(v 1/ 2), (4.3.21)
4.3.
Periodic Perturbed Systems on a Plane
211
which is in the standard form of the method of averaging, with the averaged equation
¢ = v 1/ 2 {hr, r = v 1/ 2[R1r2 + R2(..'(0) + R 38'(0)],
( 4.3.22)
in which R 1, R3 are given by (4.3.19), and 1 {T Q1 = T 10 Q1(fJ)d{},
!o27rQR2(t, 0, Sand Rare 21T-periodic in (t,O) and satisfy S, R
= o(r 2 ).
( 4.3.42)
Also, obviously, ( 4.3.43) Then the following result can be obtained. Theorem 4.3.11. If (4.3.25) holds, and
Ibol =f. ~ p
for kEN and p = 1,2,3,4,
(4.3.44)
218
Chapter 4.
Periodic Perturbed Systems and Integral Manifolds
then there exists a function ,(A, 8) = a( 8) + O( A), such that (4.3.1) has a unique invariant torus near {O} x Sl for IAI, 18 - 80 1small if and only if ,(A,8)a1(80 ) < O. Moreover, the torus is asymptotically stable (unstable) if a1(80 ) < 0 (> 0).
Proof. From (4.3.42) and (4.3.43), we see that (4.3.41) (and therefore (4.3.1)) has no invariant torus for IAI, 18-80 1small if ,a1(80 ) 2:: O. Let ,a1(80 )
., 8)
= >.290(t) + >'91 (t)x + 92(t, x) + 0(18 -
80 1+ Ixl 3 + 1>'1 3 ),
(4.3.46) where 92(t,X) is homogeneous in x of degree 2. Then introducing the scaling
X - t vx, >. = V/1, 8 - 80
= v 2bo,
v> 0,
lbol = 1,
and noting (4.3.25), we obtain
x = Box + v 2[boB'(8o)x + B1(8o)xlxI 2 + X(t, x, /1)] + o(v 2), in which
X(t, x, /1)
=
+ /1 291(t)X + /192(t, x). variable x = eBoty in the above equation
/1 39o(t)
Furthermore, changing the we obtain the following 27l'p-periodic system,
(4.3.47) with Y(t, y, /1) = e-BotX(t, eBoty, /1). The application of the averaging method to (4.3.47) yields
iJ = v 2[boB'(8 o)Y + B1(8o)YIYI2 + Y(y, /1)] + o(v 2), where Y(y, /1)
(4.3.48)
1 !o27rP
= -2-
7l'P
0
Y(t, y, /1)dt.
The associated autonomous system is a cubic system: (4.3.49) It is clear that we can apply Theorems 4.2.1, 4.3.1, and 4.3.4 etc. to (4.3.48) and (4.3.49) to study the subharmonic solutions of order p and invariant tori of system (4.3.1). We omit the details here.
220
Chapter 4.
4.4.
Periodic Perturbed Systems and Integral Manifolds
Hopf Bifurcation of Invariant Torus
In this section we are concerned with the Hopf bifurcation of an invariant torus near a fine focus of higher order or near a center [74J. Consider the following Coo systems:
x=
f(x)
+ Ag(t,X,A,b),
(4.4.1)
and (4.4.2) where (A,b) E JR x JR, k> 0, x E JR2, and g is 27r-periodic in t. Suppose that the autonomous planar system
x=
f(x)
(4.4.3)
has a fine focus of order n 2: 1 at the origin. That is, we may assume that 2n-l
f(x)
=2:
Bjlxl 2j x
+ O(lxI4n),
(4.4.4)
)=0
= (~j ~bj), j = 0"" , 2n - 1, with bo ) ) for j = 0, ... , n - 1. If
where Bj
aj =
°
for any p, kEN and 1
~ p ~
=1=
0, an
2n,
=1=
0, and
(4.4.5)
then from Lemma 4.3.9, the system (4.4.1) can be transformed into an equation of the form (4.3.34) with q = n - 1. By introducing the polar coordinates u = (r cos 0, r sin 0) and noting (4.4.4), we have the following 27r-periodic system in (t,O): •
0= f3(A, b)
n-l
2' 2n + 2: bjr ) + bnr + S(t, 0, r, A, b), j=l
n-l
n-l
j=l
j=o
r = r[-y(A,b) + 2:ajr2j + 2:an+jr2n+2j + R(t,O,r,A,b)],
( 4.4.6)
4.4.
Hopf Bifurcation of Invariant Torus
221
where 5 and Rare 27T-periodic in (t, B) and have the following orders:
5 = O(lrI 2n+ 1 + IAllrI 2n - 1 ),
R = O(lrl 4n + IAlr 2n - 1 ).
(4.4.7)
Theorem 4.4.1. If the nonresonance condition (4.4.5) holds, and ,HO,8) i= 0 for 8 in a compact set V c JR, then for IAI small enough, (4.4.1) has a unique invariant torus in the neighborhood of {O} x 51
if and only if ,(A,8)an < O. Moreover, the stability of the torus is determined by the sign of an. Proof. From the second equation of (4.4.6), there is no invariant torus if ,an 2: o. Then we suppose ,an < o. Notice that any invariant torus of (4.4.6) must have the form
where ,0(8)
= ,>'(0,8). This allows us to make the change of variables r = cv(l
+ p),
so that (4.4.6) becomes n
0=, + L.:b j [cv(l + p)fj + O(IAlv2) + 5,
j=1 n-1 p = (1 + p)boA(l - (1 + p)2n) + L.:lij,\(O, 8)A[cv(1 + p)]2j j=1 n-1 + L.:an+j(cv(l + p))2n+2j + O(IAI2) + R}. j=1
From (4.4.7), this system can be rewritten as
0=, + v 2[5 1(p, v) + O(v2n-1)], p = v 2n [R1(P) + R 2(p, v) + O(v2n-1)],
(4.4.8)
Cbapter 4.
222
Periodic Perturbed Systems and Integral Manifolds
in which
Sl(P,I/) =
n
Lbj c2j (1 + p)2 j I/2 j -2,
j=l R1(P) = -,osgn-X(l + p)[(l + p)2n - 1], n-1 R 2(p, 1/) = (1 + p) L rajA (0, 8)sgn-X + a n+jC2n (1 j=l For n
+ p )2n]( cl/(l + p) )2j.
= 1, we have
In this case, the theorem follows from Theorem 4.2.4 directly. For n ~ 2, we have n-1 Sl(P,I/) = R1(P)
L
j=o
Pj(1/2)pi n-1
+ R 2(p, 1/) = L
j=o
+ O(jpjn) == P(p, 1/2) + O(jpjn), Qj(1/ 2)pi
+ O(jpjn) == Q(p, 1/ 2 ) + O(jpjn),
where Pj and Qj are polynomials of 1/ 2 with Qo(O) = 0, Q1(0) -2n'osgn-X. By letting p = 1/ 2 P1' we obtain from (4.4.8)
iJ = I + 1/2[p(1)(p1, 1/ 2 ) + O(1/ 2n - 1 )],
(4.4.9)
PI = 1/2n[Q(l)(P1, 1/ 2 ) + O(1/ 2n - 3)], where P(1)(Pl,1/ 2) We can write
= P(1/ 2Pl' 1/ 2 ), and Q(1)(pl,1/ 2) = Q(1/ 2Pl,1/ 2)/1/2.
in which each Q?) is a polynomial of 1/ 2 with Q;l}(O) n - 1). Denote
=0
(2 ::; j ::;
4.4.
Hopf Bifurcation of Invariant Torus
223
Then (4.4.9) becomes
iJ = f3 + 1/2[p(2l(P2, 1/ 2) + 0(1/ 2n - 1)], P2 = 1/2n[Ql(0)P2 + Q(2)(p2, 1/ 2) + 0(1/ 2n - 3)], where
p(2)(p2, 1/ 2)
(4.4.10)
= P(1)(p2 + pi, 1/ 2), n-l
Q(2)(p2, 1/ 2) = L Q)l) (1/ 2)(p2
+ p;y.
j=2
Obviously,
Q(2)(p2, 1/ 2)
=
n-l
LQ)2)(1/2)~, Q)2)
= 0(1/ 2),
j
= 0, ...
,n-l.
)=0
Then for n = 2, we can finish the proof by using Theorem 4.2.4. For n ~ 3, we furthermore make a series of changes of variables:
We can obtain from (4.4.10)
iJ
=
f3 + 1/2[p(2n-2)(P2n_2, 1/ 2) + 0(1/ 2n - 1)],
P2n-2 where
= 1/2n[Ql(0)P2n_2 + Q(2n-2)(P2n_2, 1/ 2) + 0(1/)],
(4.4.11)
8P(2n-2)
-=---
= 0(1/2n-2), Q(2n-2)(P2n_2,0) = O. 8P2n-2 Then the conclusion follows from Theorem 4.2.4.
o
For system (4.4.2), similar to Theorem 4.3.12 we have Theorem 4.4.2. Suppose that (4.4.4) holds. Then Theorem 4·4·1 holds for system (4.4.2) without the condition (4·4·5).
Chapter 4.
224
Periodic Perturbed Systems and Integral Manifolds
In the following, we suppose that (4.4.3) has a Coo first integral given by (4.4.12) Then
DH(x)f(x) ==
°
for Ixl small.
(4.4.13)
In this case, we may assume that (4.4.4) holds for any natural number n 2: 1 with aj = 0, j = 1" .. ,n. It follows from (4.4.4) and (4.4.13) that n b H(x) = L -._J-(xi + x~)j+1 + O(lxI 2n +3). (4.4.14) j=o 2J + 1 From (4.4.12), bo k
Ibol=J -
p
=J 0. By Lemmas 4.3.8 and 4.3.9, if for p, kEN and 1
:s; p :s; 2n + 2,
(4.4.15)
then the time-periodic system (4.4.1) can be transformed into the form .
Y=
n
L
Pj (>.., 8)YIYI
2'
J
-
_
+ Y(t, y, >.., 8) = Y(t, y, >",8),
(4.4.16)
j=o
where
Y = O(I>"IIyI2n+2 + lyI2n+3), Pj (>.., 8) = (Qj(>..,8) (3j(>.., 8)
-(3j(>..,8)) = B j
+ 0(>..)
Qj(>",8)
for j = 0,1" .. ,n. From the proofs of Lemmas 4.3.8 and 4.3.9, we know that Y(t, y, 0, 8) = f(y). Therefore, for the function H given by (4.4.14), H = C is a first integral of (4.4.16) for>.. = 0. Let r > be small and the closed orbit Lr defined by H(y) = r2 have the timeparameter representation y = y( t, r) with period Tr in t. Without loss of generality, we may suppose that y(O, r) lies on the positive Yl-axis. Then (4.4.17) H(y(t,r)) == r2, y(O,r) = (ly(O,r)I,O).
°
4.4.
Hopf Bifurcation of Invariant Torus
225
It is easy to see from (4.4.14) and (4.4.17) that
=
ly(t,rW
where U1 =
2
Ibol'
n+1 L,ujr2j j=l
+ O(r 2n+3),
(4.4.18)
2b 1 U2 = - b~ , ... , Un+1 are all constants. In order to
find an approximation of y(t, r), we introduce the polar coordinates by y = (p cos 0 such that for 0 < IAI < Ao the system (4.4.1) has no invariant torus in the neighborhood of {O} x 5\ (ii) if a o(0,8 0 ) = 0 and al(0,80 ) =1= 0, then there exist Ao > 0, Al > 0 independent of n, such that for 0 < IAI < Ao, la o(A,8)1 < AIIAI 2/ 2n-l, system (4.4.1) has a unique invariant torus in the neighborhood of {O} X 51 if and only if ao(A, 8)al(0, 80 ) < 0, for which the torus is stable (resp. unstable) if nl(A,8) < 0 (resp. > 0).
-i
Proof. If a o(0,8 0 ) 0 or a o(0,8 0 ) = 0, al(0,80 ) =1= 0 and ao(A, 8)al(0, 80 ) ~ 0, then from (4.4.39), for IAI =1= 0, 18 - 80 1 small, (4.4.38) has no invariant torus near {O} x 51. Suppose that ao(O, 80 ) = o and ao(A, 8)al (0,8 0 ) < O. From (4.4.22) and (4.4.23), we have
2b l r 2 + O( r4). -27T -_ b0+ -b Tr
0
ao Therefore, by letting r = ( - N2
)1/2 (1 + p),
N2 = a oNI
2al
+ ha'
and
from (4.4.38), we obtain
e= bo + a
+ A[51 (p, ao) + O(laol n+!)], p = A[-2pao + aoP(p, ao) + O(laol n+!)], 0
5 0 (p, ao)
in which
+ la olp52(p, ao), = -p2(3 + p) + O(lnol).
5 1 (p, ao) = 5 1 (0, ao) P(p, ao)
(4.4.40)
Chapter 4.
230
Periodic Perturbed Systems and Integral Manifolds
Suppose p = p*(ao) = O(laol is a unique solution of the equation 2p = pep, a o). Under the change 17 = P - p*( a o), (4.4.40) becomes
iJ = (3 + ao[Fo((j, a o) + AF1((j, ao)J + AO(laoln+~), C5 = Aa ol7[-2 + G((j, ao)J
(4.4.41)
+ AO(laoln+~)
with (3
= bo + AS1(P*,ao), F1 = 0(1171),
Fo((j,a o) = So((j
G((j, ao) = 0(1171
+ p*,a o),
+ laol).
Then by choosing ~ = D = Nlaoln-~ with N large enough, the conclusion follows from the application of Theorem 4.2.4 to (4.4.41).
o
°
°
For the case of Tr == 27l'/lbol, we have So = in (4.4.40) and Fo = in (4.4.41). So, we can choose ~ = D = Nla ol1j2 , and use Theorem 4.2.4 to obtain the following result.
°
Corollary 4.4.5. Suppose that a o(0,80 ) = 0, a1(0,80 ) i= for some 80 E R. If Tr == 27l'/lbol, Ibol i= ~ for p, kEN, and 1::; p::; 4, then there exists Ao > 0, such that for < IAI < Ao, 18 - 80 < Ao, (4.4.1) has a unique invariant torus near {O} x Sl if and only if ao(A, 8)a1(0, 80 ) < 0.
°
1
Notice that the constants Ao and Al in Theorem 4.4.4 are independent of n. Letting n --t 00 we obtain immediately
°
Corollary 4.4.6. Suppose a o(0,8 0 ) = 0, a1(0,80 ) i= for some 80 E JR. If bo is irrational, then the conclusion of Corollary 4.4.5 holds.
For system (4.4.2), we can prove the following theorem.
°
Theorem 4.4.7. If a o(0,8 0 ) i= for 80 E JR, then the system (4.4.2) has no invariant torus near {O} x Sl for IAI small. If a o(0,8 0 ) = 0, a1(0,80 ) i= 0, then for IAI > 0, 18 - 80 1 small, (4.4.2) has a unique invariant torus near {O} x Sl if and only if
4.4.
Hopf Bifurcation of Invariant Torus
231
ao(A,b)al(O,bo) < O. The torus is stable (unstable) if al(A,b) < 0
(> 0). Proof. For (4.4.2), the condition (4.4.15) becomes IAlklbol
i= p2 for p,
q E Nand 1 ::; p ::; 2n
+ 2,
which is satisfied if (4.4.42) Thus, from Theorem 4.4.4 we know that there exist no E N and Al > 2 0, such that for n ~ no, 0 < IAI < lin and laol < AIIAI2n-l, (4.4.2) has a unique invariant torus near {O} x 8 1 if and only if ao(A, 8)a(0, 80 ) < O. Hence, in the region
of (A, ao )- plane, the bifurcation set is given by
Bn = {(A, a o) : a o = 0, 0
'o'(h)DhG(e, h) A get, G(e, h), >',8), where o'(h)
(4.5.7)
271"
= T(h)' a A b = a1b2 - a2bl. Then we have
Lemma 4.5.1. (4.5.1) has an invariant torus x
=
Set, c/J, >., 8)
satisfying Set + 271"m, c/J, >., 8) = Set, c/J + T(h), >., 8) = Set, c/J, >., 8), Set, c/J, 0, 8) E Lh, if and only if (4.5.7) has an invariant torus h = R(t, e, >.,8) satisfying R(t + 271"m, e, >., 8) = R(t, e + 271", >.,8) = R(t, e, >., 8) and R(t, e, 0, 8) = h. In this case we say that (4.5.1) possesses an invariant torus of order m, which is generated by the periodic orbit L h . Let
M(e,h,8) M(h,8)
=
r f(x) Ag(e,x,0,8)dt, iLk
1 1211" = -271" M(e, h, 8)de. 0
Chapter 4.
234
Periodic Perturbed Systems and Integral Manifolds
First we prove Theorem 4.5.2. ([72]) If for 1>'1 > 0, 18 - 80 1 small, (4.5.1) has an invariant torus which has Lh X Sl as its limit position as (>.,8) ---+ (0,8 0 ), then M(h, 80 ) = 0. Proof. Fix a E I. Suppose that La generates an invariant torus = a. Then for any solution (h(t),
h = R(t, 0, >., 8) with R(t, 0, 0, 8) O(t)) of (4.5.7) on the torus
h(t) = R(t, O(t), >., 8). By differentiating the above equality with respect to t and using (4.5.7), we obtain
>'f(G(O, a)) 1\ g(t, G(O, a), 0, 80 )
+ 0(>.) =
R~
+ R~n(R) + 0(>.),
since Ro(t, 0, 0, 8) = 0. Notice that this equality holds for all (t,O). Integration over (t,O) on [0, 2m7r] X [0, 27r] yields 27r (27rm
M(t, a, 80 )dt + 0(>')
>'T(a) Jo
= fo27r dO fo27rm R~dt + fo27rm dt t7r n(R)R~dO + o( >.)
= o( >'),
since R is 2m7r-periodic in t and 27r-periodic in O. Thus -
1 !o27rm M(t,a,8 0 )dt 2m7r 0 and the proof is completed.
M(a, 80 ) = -
=
° o
In order to investigate the existence of an invariant torus and subharmonic solutions of (4.5.1), we consider the function 9 in the following three cases: Case A:
g(t, x, >., 8) = h(x, 8)
+ >'h(x, 8) + >.2 h(t, X, >., 8).
(4.5.8)
Case B: g(t, x, >., 8) = f1(x, 8)
+ >'h(t, x, 8).
(4.5.9)
Case C (general Case): g(t, x, >., 8) = h(t, x, 8)
+ >'h(t, x, >., 8)
(4.5.10)
4.5.
Poincare Bifurcation of Invariant Torus
4.5.1.
235
Case A
Consider first the following planar system,
x=
f(x)
+ )..h(x, 8) + )..2 fz(x, 8).
(4.5.11)
Since (4.5.11) is autonomous, from (4.5.7), it can be transformed into dh d() = )"R((), h,).., 8),
(4.5.12)
1 where R((), h, 0, 8) = O(h/(G((), h)) 1\ h(G((), h), 8). Let
M 1 (h,8)
= hh 1 f(x)
1\ h(x, 8)dt.
(4.5.13)
Then the application of the averaging method to (4.5.12) yields the following result. Lemma 4.5.3. Suppose that
(4.5.14) for some (ho,8 0 ) E I x m. Then for 1)..1 =1= 0 and 18 - 80 1small enough, Lh o generates a unique limit cycle of (4.5.11). Furthermore, the following holds. Lemma 4.5.4. Suppose Lh is oriented clockwise. Then
M{h(h,8)
= ±1 div h(x,8)dt == ±O"(h, 8), hh
( 4.5.15)
where "+" (resp. "-") is taken when Lh expands (resp. shrinks) with h increasing.
Proof. Fix ho E I. For definiteness suppose that Lh expands with h increasing. Then for h > ho, h E I, applying the Green's formula we get ( 4.5.16)
236
Chapter 4.
Periodic Perturbed Systems and Integral Manifolds
where D(h) denotes the annulus bounded by Lh and Lh o ' Changing variables by x = q(t,r), 0 ~ t ~ T(r), ho < r < h, and noting (4.5.5) that
8q(t,r) DH(q)· Drq(t,r) = det 8(t,r) = 1,
we obtain from (4.5.16),
Ml (h, 8) - Ml (h o, 8)
= iho rh dr rT(r) div fl (q( t, r), 8)dt. io
Then differentiating the above with respect to h yields I
M 1h (h,8)
rT(h) .
= io
dlV
h(q(t, h), 8)dt.
o
The lemma is proved.
Theorem 4.5.5. If (4.5.8) and (4.5.14) hold, then for IAI "I 0, 18 - 80 1 small enough, Lh o generates a unique invariant torus of
(4.5.1) of the form M).. = ((t,x): x = u(t,O,A,8), t E [0,271"],
°E [O,T(ho)]},
satisfying u(t, 0, 0, 80 ) = q(O, h o). Moreover, M).. is asymptotically stable (unstable) if Acr(ho,80 ) < 0 (> 0). Proof. From Lemma 4.5.3, the system (4.5.11) has a unique limit cycle r)..,8 : x = u(t, A, 8), 0 ~ t ~ T(A, 8),
where T denotes the period of to (4.5.1) along r)..,8, we have
r)..,8'
Applying Lemmas 4.3.2 and 4.3.3
iJ = 1 + h(O,p,A,8) + A3 F1 (t,O,p,A,8), p = A(O, A, 8)p + fz(O,p, A, 8) + A3 F2(t, O,p, A, 8), where
(4.5.17)
4.5.
Poincare Bifurcation of Invariant Torus
fi(0,P,).,8) = O(pi),
237
i
= 1,2.
Changing the variables by
p = r exp (fo" A(t,)., 8)dt - ).a()., 8)0) ,
o
-t
T T(h o ) 0,
(4.5.17) is converted into
T(~o) + O(lrl + 1).1 3 ), T ).a()., 8)r + O(lrl2 + 1).1 3 ).
iJ =
r=
Then the conclusion follows from Theorem 4.2.4.
o
The theorem says that the bifurcation diagram of (4.5.1) near Lho is the same as that of (4.5.11). Remark 1. Theorem 4.5.5 is still valid for system (4.5.2). Furthermore, if (4.5.3) is a cylinder system and Lh is a family of nonzero-homotopic periodic orbits on the cylinder, then Theorem 4.5.5 still holds for the systems (4.5.1) and (4.5.2). 4.5.2.
Case B
In this part, we consider the case with (4.5.9). From Lemma 4.5.3, if (4.5.14) holds the following system,
x=
f(x)
+ ).fl(X,8)
(4.5.18)
has a unique limit cycle
f\,8 : x = u( t, )., 8)
°: ; t ::; T()', 8)
with u(t,O,8) = q(t,h o ). Let v(t) = ,~H~:~:l" Then from Lemmas 4.3.2 and 4.3.3, the system (4.5.1) can be transformed into
iJ
= 1 + pQl(O)
+ B2 ,
(4.5.19)
Chapter 4.
238
Periodic Perturbed Systems and Integral Manifolds
where
A(O, 0, 8) 1
-= T
t
r io
= -
d
dO In If(q(O, ho))I,
A(0,A,8)dO
A
= T-=,( divf1(x,8)dt == Aa(A,8), Jt~,6
1~~~~~~~~)jf2 [Jv~ - f~(q(O, ho))Jv(O)],
Q1(0)
=
R 1(0)
= _~vT(O)J{){) [f~(q(O, ho) 2 p
Sl(t,0,8) Bi
Jv(O)p)]
(4.5.20)
Jv(O), p=o
= vT(O)Jh(t,q(O, ho),0,8),
= O(lp, Ali),
i
= 2,3.
Let
p = u exp
[foB A(O, A, 8)dO -
AaO],
°
=
T~:o)'
We have from (4.5.19).
+ O(IA, uI 2 ), a = Aau + u2 R 2(¢) + A2S2(t, ¢, 8) + O(IA, uI 3 ), ¢=
T(ho)/T + uQz(¢)
(4.5.21)
in which (4.5.22) The system (4.5.21) is 27f-perodic in t and T(ho)-periodic in ¢. Theorem 4.5.6. [72] Suppose that (4.5.9) and (4.5.14) hold. If D.(ho) = 27f/T(ho) is irrational, then for IAI > 0, 18-80 1small enough, Lho generates a unique invariant torus of (4.5.1): M>..
= {(t, x) : x = S(t, 0, A, 8), t
E [0, 27fJ,
°
E [0,
T(h o)]}.
Moreover, M>.. is asymptotically stable (unstable) if Au(h o,80 ) < (> 0).
°
4.5.
Poincare Bifurcation of Invariant Torus
239
Proof. Let
T(~o) T
=
1 + 0 0 (8)'\
+ 0(,\2).
(4.5.23)
Then the variable change u = '\r yields
¢ = 1 + ,\[0 (8) + rQ2(¢)] + 0(,\2), r = '\[a(O, 8)r + r2 R2(¢) + S2(t, ¢, 8)] + 0(,\2). 0
(4.5.24)
From the averaging theorem (4.5.24) can be transformed into
¢ = 1 + '\[0 (8) + Q2r] + 0(,\2), r = '\[a(O, 8)r + R2r2 + 52(8)] + 0(,\2), 0
(4.5.25)
where
Take ,\ =
(4.5.26)
°in (4.5.21). We have du 2 d¢ = u R 2 (¢)
+ O(u 3 ),
which has a family of periodic orbits. Hence R2 conclusion follows from Theorem 4.2.4.
= O. Then the D
Now we give an expression for 0 0 (8) in (4.5.23). Lemma 4.5.7. Suppose (4.5.14) holds. Then
o (8) = o
471"2 [T'(h o )B(8) _ (27r (Q(() h 8) T(h o ) M{h(h o , 8) Jo ,0,
T'(h o ) R(() 8)~ d()]
+ T(h o )
'')
,
(4.5.27)
Chapter 4.
240
Periodic Perturbed Systems and Integral Manifolds
where R(0,8) =
fo Bp (s,h o,8)ds,
P(O, h, 8) = f(G(O, h)) /\ h(G(O, h), 8), Q(O, h, 8) B(8)
= n(h)G~ /\ h(G(O, h), 8),
= fo27r
[PQ
+ R(P~ + ~f~? p)] h=ho dO.
Proof. From Lemma 4.5.1, system (4.5.18) can be changed into
h = )..P(O, h, 8),
iJ
= n(h) -
)..Q(0,h,8).
Set r = h - h o , then
r = )..P(O, r + h o , 8),
iJ = n(r + ho )
-
)..Q(O, r
+ ho , 8),
(4.5.28)
and
It is easy to see that the solution of the above equation with r(O, r o,).., 8) = r o is given by
where rl(0,O,8)
M(ro
+ ho,8).
R(0,8)
B(8)
= n(ho)' r2(27l",O,8) = n 2(h o)'
and rl(27l",ro,8)
It follows that the equation r(27l", r o,).., 8)
equivalent to
which has a unique solution
=
=
r o is
4.5.
Poincare Bifurcation of Invariant Torus
241
Hence, we have
and ) n( r * + ha) = n(ha ) + AnD/(h 2 (h a) a
[
B(8)] R(O, 8)n(ha) - M{h(h a, 8)
+ O(A 2 ),
+ ha, 8) = AQ(O, ha, 8) + O(A2).
AQ(O, r* Then, from (4.5.28),
f _
dO
(27r
- 10 n(r* + ha) - AQ(O, r* + ha, 8)' and therefore
Now we can prove Theorem 4.5;8. [72] Suppose (4.5.9) and (4.5.14) hold. Let 21f
n(ha ) = T(h a ) =
n m
(4.5.29)
be rational, where (n, m) = 1. Then there exists a 21f-periodic function of the form
Nm/n(o, 8) = (2r::1f)2
[NaM~/n(o, 8) -
2m1fna(8)
tho div h(x, 8)dt] (4.5.30)
Chapter 4.
242
Periodic Perturbed Systems and Integral Manifolds
with
Aho Ij~3[JD C~I)
C~I)
No = f - DfJ ]dt, 2m7r m/n r M2 (0,8) = 10 f(q(t, h o» A /2(t - 0, q(t, ho), 0, 8)dt, such that: (i) if for 8 near 80 Nm/n(o,8) keeps the sign and has only finitely many roots when N m/ n ¢ 0, then for 1,\1 =f and 18 - 80 1small, Lho generates a unique invariant torus of (4.5.1) of the form
°
M)..
= ((t,x): x = S(t,O,,\,8),t E
with S(t, 0, 0, 8) = q(t (ii) if
[0, 27rm],
°
E [O,T(h o)]}
+ 0, h o);
for some 00 , then for 1,\1 > 0 and 18 - 80 1small, (4.5.1) has a subharmonic solution of order m of the form X m(t,,\,8)
= q(t, h o) + 0('\).
Proof. From (4.5.29), we may change variables by (4.5.24), and obtain
iJ
= '\[00 (8)
°
+ rQ2(O + t)] + 0(,\2),
= if> - t to
(4.5.31)
Let 1 So(O,8) = -2-
la2m7r
S2(t,O
+ t,8)dt.
m7r 0 Then applying the method of averaging to (4.5.31) yields
iJ
+ Q2r] + 0(,\2), '\[a(O, 8)r + So(O, 8)] + 0(,\2),
= '\[00 (8)
r=
(4.5.32)
(4.5.33)
4.5.
Poincare Bifurcation of Invariant Torus
243
where Q2 is given in (4.5.26). The system (4.5.33) is 2m1f'-periodic in t, and T(ho)-periodic in B. From (4.5.20), (4.5.22), (4.5.26), and (4.5.32), it is easy to see that
Q- _ If(q(O, ho))1 N 2 -
T(ho)
0,
Thus, from (4.5.30), (4.5.20), and (4.5.15)
Nm/n(B,8) = Q2 So(B, 8) - no(8)0'(ho, 8).
(4.5.34)
If So(B, 8) is independent of B, the conclusion follows from Theorem 4.2.4. We then suppose that So(B, 8) is not a constant. If Q2 = 0, then letting u( B, 8) be the unique T(ho)-periodic solution of the equation
du n o(8) dB = O'(ho, 8)u - So(B, 8), and making the transformation p
= r+u(B, 8) we obtain from (4.5.33)
+ 0(.\2), P= '\O'(ho, 8)p + 0(.\2). iJ
=
.\no(8)
In this case, the conlusion follows from Theorem 4.2.4. If Q2 f:. 0, then noting (4.5.34) we introduce p = (hr (4.5.33) and obtain
iJ =
.\p + 0(.\2),
P= '\[O'(ho, 8)p + Nm/n(B, 8)] + 0(.\2).
+ no(8)
to
(4.5.35)
The associated averaged equation of (4.5.35) is
B = p,
p = O'(ho, 8)p + Nm/n(B, 8).
(4.5.36)
Since 0'(ho,8)pp > 0 for Ipllarge, and noting that the divergence of (4.5.36) is O'(ho, 8), it is easy to see that if Nm/n(B,8) keeps its sign for 8 near 80 , then (4.5.36) has a unique periodic orbit (B(t, 8), p(t, 8)).
Chapter 4.
244
Periodic Perturbed Systems and Integral Manifolds
t=
From (4.5.36) we have Ii = p o. Hence, the periodic orbit can be represented by p = p*(e,8) with p*(0,8) = p*(T(ho), 8). Applying Theorem 4.3.4 and the Remark 1 after it we know that for IAI > 0 small system (4.5.35) has a unique invariant torus of the form
(e, p)
= S(t, cP, A, 8) = (e(cP, 8), p(cP, 8)) + O(A),
which can be represented by
p = 51(t, e, A, 8) = p*(e, 8)
+ O(A),
with 51 2m7r-periodic in t and T( ho)-periodic in e. Conclusion (i) follows. Next, we suppose that Nm/n(e,80 ) has a simple root eo. For 8 = 80 , (e, p) = (eo, 0) is a hyperbolic critical point of (4.5.36). Therefore, for IAI > 0 and 18 - 80 small enough, (4.5.35) has a 2m7r-periodic solution near (eo,O), which gives a sub harmonic solution of (4.5.1) of order m. (ii) is proved. 0 1
We remark that (4.5.1) may not have any invariant torus near Lh o x 51 even if (4.5.9) and (4.5.14) hold. 4.5.3.
Case C (General case)
In this part, we consider the general case, for which we mean
g(t, x, A, 8) has the form of (4.5.10). First, we discuss the bifurcation of subharmonic solutions. For this purpose, suppose (4.5.29) holds. Then making the transformation
we obtain from (4.5.7) i = IAI1/2sgnA[P(t, cP + n(ho)t, ho, 8) + P~(t, cP + n(ho)t, ho, 8)IAI1/2r + O(A)],
¢=
IAI1/2[n'(ho)r + (n"(h o)r2/2 - Q(t, cP + n(ho)t, ho, 8)sgnA)IAI1/2 + O(A)],
(4.5.37)
4.5.
Poincare Bifurcation of Invariant Torus
245
where
P(t, 0, h, 8) Q(t, 0, h, 8)
= f(G(O, h)) A h(t, G(O, h), 8), = O(h)DhG(O, h) A h(t, G(O, h), 0).
(4.5.38)
The system (4.5.37) is 2m7r-periodic in t and 27r-periodic in . Denote the right-hand side functions of (4.5.37) by 1/ F( t, , r, 1/, 8) with 1/ = 1.AI I / 2 . We may assume that .A > 0 in (4.5.37). Then for 1/ = 0
of
Ph(t, + O(ho)t, ho, 8)r
(
)
O"(ho)r2/2 - Q(t, + O(ho)t, ho, 0) ,
01/ =
of
= (
o(r, 0, 18 - 80 small enou9h, (4.5.1) has a subharmonic solution x m(t,).,8) of order m with xm(t, 0, 80) = q(t + 00, ho). 1
4.5.
Poincare Bifurca.tion of Invariant Torus
247
We remark that the stability property of subharmonic solutions can also be determined by using (4.5.43). It is obvious from (4.5.39) that
Since 1 (T(h o ) T(ho) io f(q(t 1
(T(h o )
+ B, ho)) /\ f1(t, q(t + B, ho), 8)dB
= T(h o)
io
=
i
f(x) /\ ft(t, x)dB,
(¢
)
T
(1h) 0
f( q( B, h o)) /\ ft (t, q( B, ho), 8)dB
Lho
we have from (4.5.46) 1 {27r 27r io M1
D(h o )' ho, 8 d¢
1 {2m7r 1 =-(~)Jn dt ll
T ho
0
(4.5.47)
f(x)/\ft(t,x,8)dB.
Lho
Notice that M 1 (B,h o ,8) is 27r-periodic in B. It follows from (4.5.46), (4.5.47), and Theorem 4.5.2 that a necessary condition that Lh o can generate a "large" T2>.,6 is
M- 1(h o , 8) == -1
27r
127r 0
M1(B, h, 8)dB = 0,
(4.5.48)
for some 8 E JR, where "large" means lim
(>.,6)->(0,6 0
Ti6
=
) '
Lh
X Sl. '"
If (4.5.44) and (4.5.45) are satisfied, then for ,\ = 0, (4.5.43) has a family of zero-homotopic periodic orbits given by
rc: H(r,¢,8) = C,
C E Jo
c JR,
Chapter 4.
248
Periodic Perturbed Systems and Integral Manifolds
which surround a center point on the r-axis. If (4.5.48) and (4.5.44) are satisfied, then for A = 0, (4.5.43) has two families of non-zerohomotopic periodic orbits given by
r6' r = (n'~ho) [2~" fo" M, (n(~o)' ho, /j) d4> - c =- r(..) to be determined. Let T(>") = 211"(1 + 0") and
f(x, >..)
= A(>")x + h(x, >..) + h(x, >..) + o(lxI 3 ),
for (x, >..) near (0,0), where fj(x, >..) is a homogeneous polynomial of degree j in x, j = 2,3. Then (5.1.29) can be rewritten as
x=
(1 + 0" )A(>")x + (1 + O")h(x>,,) + (1 + O")h(x,
>..) + o(lxI 3), (5.1.30)
1(>..,0")1 is small. As in Sec. 4.1.3, let Eo be the null space of A 2 e 1l" o - In, and El a complementary space with lRn = Eo (JJ E l . Also, let Pj : lRn - t E j , j = 0,1 be the project. Then from Lemma 4.1.2, there exists a nonsingular matrix H such that where
(5.1.31) Let
3
x(t, h, >.., 0") =
L
Xj(t, h, >.., 0")
+ o(lhI 3)
(5.1.32)
j=l
be a solution of (5.1.30) with initial value h = x(O, h, >..,0"). Here Xj is homogeneous in h of degree j, j = 1,2,3. Obviously,
Xl(O, h, >..,0")
= h,
Xj(O, h, >.., 0")
= 0,
j
= 2,3.
(5.1.33)
Substituting (5.1.32) into (5.1.30) and using (5.1.33) we obtain
Xl = e(1+17) At h == Xl(t), X2
= (1 + O")e(1+17)At lot e-(1+17)As h(Xl(S), >..)ds == X2(t),
X3 = (1
(5.1.34)
+ O")e(1+17)At lot e-(1+17)AS[h(Xl(S), >..) + i;(Xl(S), >")x2(s)]ds.
Let
Vj(h, >.., 0") = xj(211", h, >.., 0") == vj(h).
(5.1.35)
Then x( t, h, >..,0") is 211"-periodic if and only if
(e 271" Ao _ In)h
+ (e 21l"(1+17)A -
e21l" Ao)h + v2(h)
+ v3(h) + o(lhI 3) =
0.
5.1.
Methods of Bifurcation Functions of Periodic Orbits
261
From (5.1.31), the above equation is equivalent to
Set a = Poh, b = P 1h = h - a. Then from (5.1.31), it is easy to see that (5.1.36) is equivalent to the equations
PoH (e 27r (1+u)A - e21r Ao) h+ PoH[V2( h) +V3 (h)] + Pol o( Ih1 3)] = 0, (5.1.37) b + P1H(e 21r (1+u)A - e27rAo )h + P1H[V2(h)
+ v3(h)] + PI [o(lhI 3)]
=
o.
(5.1.38) Use of the implicit function theorem yields a unique solution b = b*('x,u,a) of (5.1.38). Let (5.1.39) where bj is homogeneous in a with degree j, j = 1,2,3. Then we have
(5.1.40) Inserting (5.1.39) into (5.1.37) and (5.1.38), we obtain bl = -[In
+ P1H(e 27r (1+u)A - e27rAo)t1P1H(e27r(1+u)A - e27rAo )a,
+ P1H( e27r (1+u)A -
e27rAo )]-1 P1H OV2 lo2v2 2 ·[V2(a) + oa (a)b1 + "2 oa2 (a)b1), b3 = -[In + P 1H(e 27r (1+u)A - e27rAo)]-lP1H OV2 3 ·(V3(a) + oa (a)b2 + o(lal )), b2 = - [In
(5.1.41) and
PoH (e27r (1+u)A - e27rAo )(a + b*) + PoH[V2(a + b*) + V3(a + b*)]
+ Po[o(laI 3)] = o.
(5.1.42)
262
Chapter 5.
Bifurcation of Higher Dimensional Systems
We denote the left-hand side of (5.1.42) by G(a,'x, CT), which is said to be a bifurcating function of (5.1.1). Then from (5.1.39) and (5.1.40) G(a,'x,O') = G1(a,'x, 0') + G 2(a,'x, 0') + G3(a,'x, CT) + Po[o(laI 3)], (5.1.43) where G 1 = PoH[e 27!'(l+a)A - e27!' Ao](a + bI),
G2 = PoH [(e 27!'(l+a)A - e27!' Ao)b2 + v2(a) G 3 = PoH [(e 27!'(l+a)A - e27!' Ao)b3 +
+ ~~(a)bl + ~ ~2:; (a)b~]
~~(a)b2 + v3(a)]
,
.
(5.1.44) Clearly, G j is homogeneous in a with degree j, j = 1,2,3. By using the formula of variation of constant to the linear equation x = (1 + O')A('x)x, it is easy to prove that e27!'(l+a)A _ e27!' Ao = Qo(O') + Ql(O')'x + O(,X2), (5.1.45) where
Qo( CT) = e27!'(l+a)Ao - e27!' Ao, Ql(CT) = (1
+ CT)e 27!'(l+a)Ao 102
71'
e-(l+a)AotAle(1+a)Aotdt.
Hence, from (5.1.41), (5.1.45), (5.1.34), and (5.1.35), we obtain
b1(a, 'x,CT) = Rlo(O') + Rll(O'),Xa + O(,X2a), b2(a,'x,CT) = R20(a) + O(laI 2 (1,X1 + 10'1)), where Rlo(O')
= -(In + P1HQo(0')t 1PIHQo(O') ,
Rll(CT) = -(In + P1HQo(0')t 1P1HQl (0') (In
+ P1HQo(0')t 1 ,
R 20 (CT) = -P1 He 27!' Ao10{27!' e-Aosj2 (eAosa , O)ds . Therefore, from (5.1.44), (5.1.4S)l"5.1.31), we deduce that
G1(a,'x,0') = G1o(CT)a + Gll(O'),Xa + O(,X2a),
+ G21 (a),X + O(laI 2 1'x1(1'x1 + ICTI)), G30(a) + O(laI 3 1'x1(1'x1 + 10'1)),
G 2(a,'x,0') = G 20 (a, 0') G3(a,'x,0') =
(5.1.46)
5.1.
263
Methods of Bifurcation Functions of Periodic Orbits
where
+ P1HQo(0-))-I, Gn(o-) = PoH[Ql(o-)(In + Rlo(o-)) + Qo(o-)Rn(o-)], GlO(o-) = PoHQo(o-) (In
G 20 (0-) = PoH fo27r e-Aoth(eAota,O)dt, aG2 G21 (0-) = a>. (a,O,O), G (0-) = P. H[ [27r e-Aotah(eAota O)eAotdtR 30
0
Jo
'(h(eAota, 0)
ax'
20
+ Jo[27r e-Aot
+ i:(eAota,O)eAot lot e-AoSh(eAosa,O)ds)dt]. (5.1.47)
Then from (5.1.43) and (5.1.46)
G(a, >., 0-) = G1o(0-)a + Gn(o-)>.a + G 20 (a, 0-) + G 21 (a)>. + G30(a) + Po[O(I>.2al + laI 2 (1)'0-1 + lal 2 + Io-al))]. (5.1.48) Summarizing the above, we obtain the following bifurcation theorem [75] Theorem 5.1.4. The system (5.1.1) has a periodic solution x(t, >.) with period T(>.) = 27r(1 + 0-(>')) satisfying
x(t, >.)
-t
0,
0-(>')
-t
° as>. ° -t
if and only if there exists a function a = a(>.) with a(O) = Osuch that G(a(>.), >., 0-(>')) == 0.
5.1.3.
Bifurcation at non-semisimple eigenvalues
In this part we suppose that in (5.1.28)
Chapter 5.
264
Bifurcation of Higher Dimensional Systems
Then from the proof of Lemma 4.1.2, it is easy to see that
Po = diag (J, 0),
PI
= diag (0, I n- 2 ),
H= diag ((H~I~) ,(e"C - In_"r I), HI = (2~I ... : ~)
I':t .
Hence
PoH PIH
= diag ( ( ~ ~)
= diag ((H~-I~)'
,0) ,
(5.1.49)
(e 27rC - I n _2k fl ).
Then we have
a
= Poh = (at, ol,
eAota
= (eJtal' ol,
al
Em?
Let where hj E IR?, j yields that G 2o (a, 0)
=
1"" ,k. From (5.1.47), a direct computation
= Jo(27r PoH e- Aot h( eAota, O)dt T
= ( fa27r e -Jt hk(e Jt al,O)dt,O ) = (0,0) From (5.1.45) we have
QI(O)
= e27rAo fa27r e- Aot AIeAotdt. ~
Suppose that
AI
_(~I.I ::: ~I.k :: :J
-
Akl ... A kk ···
............
(5.1.50)
5.1.
265
Methods of Bifurcation Functions of Periodic Orbits
with each Aj being a 2 x 2 matrix, i, j (5.1.47), (5.1.49), and (5.1.31) G ll (O)a
= PoHQl(O)a = PoH =
=
1,··· ,k. We have from
Jr e-AotAleAotadt
(Jr e- Jt ~k1eJtaldt) == ( Ak~al ) .
Noting that Qo((J) (5.1.49)
=
(5.1.51)
e21rAo(e21ruAo - In), we have from (5.1.31) and (5.1.52)
and
P1HQo((J)
= diag [ (H~-l~) e21rB(e21ruB -
hk), (5.1.53)
(e 21rC _ In_2k)-le21rC (e 21rC - I n- 2k)]. It is easy to see that
and
(5.1.54)
It follows that
PIHQo((J)a = (Sa,Ol =
(0, 2~(e21rUJ - I)al,O,··· ,O)T,
(5.1.55)
Chapter 5.
266
where
a = (aI, O)T
Bifurcation of Higher Dimensional Systems
E IR2k. Then from (5.1.53)-(5.1.55) we have
. = (Sj71) [PlHQo(a)j1a 0 ,j ~ O.
(5.1.56)
From (5.1.55) and by induction it is easy to prove that
sj71 = S(sj-l71) = (0,712, ... ,71j+l, 0,··· ,O)T, where
71j+l =
(2~(e211"UJ
-I)f
aI,
j
(5.1.57)
= 1,··· ,k-1.
It implies from (5.1,52), (5.1.56) and (5.1.57) that
PoHQo(a) [PlHQo(a)]la = (( =
~ e211"U~ -
I) sj71,
0) T
o {(
for j ~ k - 2 1 (211"uJ (27r)k-;-1 e - I)k al,O )T £or J. = k - 1.
(5.1.58)
Therefore, we have from (5.1.58) and (5.1.47)
Glo(a)a = PoHQo(a) "L,(-l)j(PlHQo(a))ja j?o = (-1)k-l(27ra kJk al , of + Po[O(ak+la)], since e211"uJ - I
(5.1.59)
= 27raJ + 0(a 2). If we write
G 2o (a,a) = (F2(al,a),Of, F2 E IR2, G 3o (a) = (F3(al),0)T, F3 E IR2,
(5.1.60)
then from (5.1.48), (5.1.51) and (5.1.59), we have G(a, A, a) and only if
= 0 if
+ -AklAal + F2 (al;f ) a + F3 (al) + O(lall(lalk+l + A2 + IAal + lalAI + laII3 + lalI 2 Ial)) = O.
(-1) k-l 27ra kJ kal
(5.1.61) From (5.1.32), (5.1.34), and (5.1.39), and using the implicit function theorem it is easy to see that if x(t, A) = (XI(t, A),··· ,xn(t, A)f is a
5.1.
Methods of Bifurcation Functions of Periodic Orbits
periodic solution of (5.1.1) with period T(>.) exists a unique ti E [0, T( >.)) such that
X1(ti, >.) = 0,
267
= 27r + 0(1), then there
(-1)i+1 X2 (t i , >.) > 0,
i
= 1,2,
and
(5.1.62) Then, without loss of generality, we may suppose in (5.1.61) that a1 = (O,rl, for r > (or < 0). Then (5.1.61) becomes
°
r[( -1)k- 127r J ke2a k + A k1 >'e2 + r F 2( e2, a) + r2 F 3(e2)
+O(lalk+l + >.2 + I>.al + Ir>'1 + Ir31 + Ir 2al)] where e2
= (0, ll. If F 2(e2, a) to,
F2(e2,a)
= 0,
(5.1.63)
we can suppose
= F2(e2)a m + O(am+l),
m ~ 1,
F2(e2)
i= 0.
Then letting
(( -1 )k- 1 27r Jk)-l Ak1
=
(( -1 )k- 1 27r Jk)-l Fj ( e2)
(~~~ ~~~) , = (9j1) ,
9j2
(5.1.64) j = 2,3,
we can see that the equation (5.1.63) is equivalent to d 12 >' + 921ram + 931r2 + h.o.t = 0, a k + d 22 >' + 922ram + 932r2 + h.o.t
= 0,
(5.1.65) (5.1.66)
where
Now we can prove [75] Theorem 5.1.5. Let d12921 Then:
i=
°and
~
= (d22931 -d12932)/d12 i= 0.
268
Chapter 5.
Bifurcation of Higher Dimensional Systems
(i) if k is odd, (5.1.1) has a unique local periodic orbit if and only if d 12 g31 ).. < 0; (ii) if k is even, (5.1.1) has precisely two local periodic orbits (resp. no periodic orbits) for d12 g31 ).. < 0 (resp. d 12 g31 ).. ~ 0) provided ~ > 0, and has no periodic orbits for small 1)..1 provided ~ < O.
:t
Proof. We first suppose that F2 (e2, 0') o. Then we have (5.1.65) and (5.1.66) in which Ig211 + Igd =f O. We claim that m > k/2. Since d 12 =f 0, we can solve from (5.1.65)
g21 rO' m - -d g31 r 2 + h .0 .t. /\, = /\, * ( r,O' ) = --d 12
(5.1.67)
12
Substituting it into (5.1.66) we have
O'k
+ N rO'm -
~r2
+ h.o.t = 0,
(5.1.68)
where N = g22 - d22g21/dI2. If m = k/2, then from (5.1.68) we get
r
= r±(O') =
1 2~ (N ± JN2
+ 4~)O'm + o(O'm+l)
== R±O'm + o(O'm+l).
(5.1.69)
From (5.1.64), (5.1.60), (5.1.46), (5.1.44), and (5.1.41), we know that g21 and g22 are independent of Al and h(x, 0). Note that d 12 and d22 depend only on Al and that Ig211 + Ig221 =f O. We may choose suitable Al and h(x,O) such that
N 2 + 4~ > 0,
R == max{R±} > 0,
Then (5.1.67) and (5.1.68) have a solution lim r~(O'()..))
>.---.0 r~(O'()..))
r
Rg31
= O'()..) satisfying
= R+ =f 1 R_
'
in contradiction to (5.1.62). If m < k/2, then from (5.1.68) we can get
r
+ g21 =f O.
= r~(O') = ~ O'm + o(O'm+1),
5.1.
269
Metbods of Bifurcation Functions of Periodic Orbits
and
1
r = r;(O") = - N20"k-m
+ o(O"k-m+l).
Choose Al and h(u,O) such that N6. > 0 and 6.g 21 + Ng 31 =1= O. Then (5.1.67) and (5.1.68) have a solution 0" = 0"(>.) with lim'x-->or 2(0")/rr(0") = 0, a contradiction also. Hence, we have m> k/2. Then (5.1.68) has a unique positive solution
r = r(O") = JO"k/6.(l for 6.O"k
+ 0(1))
> O. Substituting it into (5.1.67) we have >. = -g3W k/(d I2 6.) + o(O"k).
Therefore, the conclusion follows easily. If F 2(e2, 0") == 0, the proof is much easier.
o
We now suppose further d 12 = O. Then from (5.1.58) and (5.1.59),
G 1o (0")a = (_l)k-l PoHQo(u) (P1HQo(O" ))k-1a
+( _l)k PoHQo(0")(P1HQo(0") )ka + Po[O(0"k+ 2a)] = (( -7r /2)k-1(~211"(1J
- I)ka1)
+ (_1)k+10"k+l PoHQ~(O)
. (P1HQ~(0))ka + po[O(O"k+2a)]
= (( _1)k-127rJ kO"ka1 + (_1)k-1k7rJ k+lO"k+la1' O)T +( _1)k+lO"k+1 PoHQ~(O)(PtHQ~(O) )ka + po[O(O"k+2a)]
== (( _1)k-127rJkawk + L1awk+l + O(O"k+2 a1 ), O{, where Q~(O) to see that
= 27rAoe211" Ao.
(5.1. 70) Also, from (5.1.47) and (5.1.45), it is easy
G~l(O)a = PoH[Q~(O) - 27rQl(0)P1He 211" AoAo
-27rA oe211" AoPIHQ1(0)]a == (L 2 a1,0)T,
(5.1.71)
where Q1(0) = e211" Ao10211" e-AotA1eAotdt,
Q~(O)
=
(In
+ 27rAo)Q1(0) + e211" Aot11" e-Aot(A1Ao -
AoA1)eAottdt.
270
Chapter 5.
Bifurcation of Higher Dimensional Systems
Then, instead of (5.1.61) we have (-I)k- l 27ra k Jk al + L l aw k+1 + Akl..\al +F3(at) + O(lall(lalk+2 +..\2 + la 2..\1 +
+ L2al..\a + F2(aI,a) lal..\1 + lall 3 + lalI 2Ial)) = O. (5.1.72)
Let (( _1)k- l 27rJ k )-l Lje2
= (hjl' hj2f,
j
= 1,2.
(5.1. 73)
Using (5.1.64), (5.1.72) becomes
dl2 ..\ + 92lra m + 93lr2 + hlla k+1 + h2w..\ + h.o.t. = 0, d 22 ..\ + a k + 922ram + 932r2 + hl2a k+1 + h22 + h22a..\ + h.o.t.
= 0,
where h.o.t. = O(lalk+2 +..\2 + la 2..\1 + Ir..\1 + Ir 31 + Ir 2al + Ira m +1I). In a similar manner, we can prove [75]
Theorem 5.1.6. Let dl2 = 0, 93ld22 i= 0, and ~l (d22 hn - h 2l )/(93ld22) i= O. Then: (i) if k is odd, (5.1.1) has a unique local periodic orbit (resp. no local periodic orbit) for all 1..\ I i= 0 small provided ~ 1 > 0 (resp. ~l < 0); (ii) if k is even, (5.1.1) has a unique local periodic orbit if and only if ..\d22 < O. More generally, we have
Theorem 5.1. 7. Suppose that Id 12 1+ld22 1i= 0 and d2293l-dl2932 i= O. Then for 1..\1 i= 0 small enough, (5.1.1) has at most two local periodic orbits.
(
As an example, we consider the following 4-dimensional system, x=Jx+y,
iJ =
Jy
+ Jlxl 2x + ..\(aJ + /3J)x.
(5.1.74)
By applying Theorems 5.1.5 and 5.1.6 we can prove that if a/3 > 0 then (5.1.74) has precisely two (or no) local periodic orbits for 1..\1
5.2.
271
Zero and Pure Imaginary Eigenvalues
small and (3A < 0 (or (3A > 0), and if 0.(3 < 0 then (5.1.74) has no local periodic orbits for alllAI small. If (3 = 0 and a =f 0, then (5.1.74) has precisely one (or no) local periodic orbit for IAI small and o.A < 0 (or o.A ~ 0). In fact, we have
From (5.1.47), (5.1.51) and (5.1.64), it is easy to see that
= d 22 = a,
d ll
d12
= -d22 = (3,
931
= 1,
932
= O.
When (3 = 0, we have
Q1 (0)
=
e7l"0o.1 271"~o.1),
Q~ (O)a =
e7l"o.(I ~ 271" J)a1 ) .
Then from (5.1.70) and (5.1.71), it is easy to get
L1 = 271"1 + 271"(271" + 1)J, and from (5.1.73) hll
= 1 + 271",
h21
=
-a,
h12
=
1,
h22
= O.
The conclusion follows from Theorems 5.1.5 and 5.1.6.
5.2.
Zero and Pure Imaginary Eigenvalues
Consider the 3-dimensional system (5.2.1) 0 10) ( -100 . By o 00 adding up the 2-parmeter linear part diag (A1, AI, A2)Z we obtain
where Z2(Z) = O(lzI2) is Coo in z
Ern?,
and D
(5.2.2)
Cbapter 5.
272
Bifurcation of Higber Dimensional Systems
where
It is easy to verify that (5.2.2) has the following normal form [78]:
x= iJ =
A(At)X + A1xy + A 2xlxl 2 + A3xy2 + X(x, y), A2Y + cllxl 2 + d1y2 + c21xl 2y + d2y3 + Y(x,y),
(5.2.3)
where x E JR2, Y E JR, and A ;• = (
-
aib bi ) , i ai
= O(lx, yI4). Changing to polar coordinates by setting x = (pcos e, -p sin e), (5.2.3) X
= O(lx, yI4),
i = 1,2,3,
Y
becomes
iJ = 1 + b1y + b2p2 + b3y2 + S(e,p, y),
p = AlP + a1PY + a2p3 + a3Py2 + p(e,p, y), iJ = A2Y + C1p2 + d 1y2 + c2p2y + d2y3 + Y(e,p, y),
(5.2.4)
where S, P and Yare 27r-periodic in e, and pS, P, Y = O(lp, yI4). By the scaling
p
-t
EP,
Y
-t
EY,
Al
-t
E81,
A2
-t
E82 ,
E > 0,
1811 = 1,
(5.2.4) becomes
iJ = 1 + E[b 1y + E(b 2p2 + b3y2)] + p-1 E30(lp, yI4), p = Ep[8 1 + a1Y + E(a2p2 + a3y2)] + O(E 3),
(5.2.5)
iJ = E[8 2y + C1p2 + d 1y2 + E(C2p2y + d2y3)] + O(E3~ We obtain from (5.2.5)
dp de
-
= Ep[!(p, y) + E!o(p, y) + !(e,p, y, E)],
dy de = E[g(p,y)
_ + Ego(p,y) + g(e,p,y,E)],
(5.2.6)
5.2.
Zero and Pure Imaginary Eigenvalues
for p
> 0, where
273
f(p, y) = 81 + alY, g(p, y) = 82y + d l y2 + Clp2,
= -8 l b1 y + a2p2 + (a3 - bl ady2, go(p,y) = -b 182y2 + (C2 - cl bl )p2y + (d 2 - bld1)y\ fo(p, y)
1,9= O(E2). Then, letting v = y order E2, we have
+ 81/al,
f}
-+
E-lf} and truncating the terms of
(5.2.7)
where
h(v) = (a3 - blal)V 2 + 8l (b l - 2a3/adv + 8ra3/ar, g2(V) = dlv 2 + (82al - 2d18l )v/al + 8l (d 18l - a 1 82)/ai,
= (C2 - clbd(v - 81/al), g3(V) = (d 2 - bldl)(v - 81/al)3 - b182(v - 81/al)2. gl(V)
Observing the property of critical points of (5.2.7), we necessarily have 8 E JR, (5.2.8) and alcl < 0, cld l > 0 if (5.2.7) admits a limit cycle. Without loss of generality, we may suppose (5.2.9) (5.2.7) then becomes
~: =
pv
~~ =
Cl(p2 + v 2 - 1) + E[gl(V)p2
+ Ep[a2p2 + h(v)],
(5.2.10)
+ 93(v) + O(E)],
274
Chapter 5.
Bifurcation of Higher Dimensional Systems
where
h(v) = (a3 - bdv 2 + (b 1 - 2a3)81v + a3, gl ( v)
= (C2 -
93(V)
= (d 2
Cl bd (v
- 8d,
c1b - 1)v 3 + 81(b1Cl - 3d2 )V 2 + (8 + 3d2 + b1c - l)v - (8 + d2 + b1Cl)81. -
The phase portrait of (5.2.10) for
t =
0 is shown in Fig. 5.2.1.
\I
Fig. 5.2.1 In what follows, we first investigate the limit cycle bifurcation for the truncated system (5.2.10), we then get back to the full system (5.2.6) or (5.2.3) to obtain the existence of a unique invariant 2-torus. 5.2.1.
For t
Bifurcation analysis for limit cycles
> 0 small, (5.2.10) has a focus A((p(t),v(t)) with p(O) = 1,
v(O) = 0,
~5.2.11)
Let x = p - p(t), y = v - v(t). We have from (5.2.10)
dx
dO = ax
+ by + f(x, y, t),
dy dO = cx + dy
+ g(x, y, t),
(5.2.12)
5.2.
Zero and Pure Imaginary Eigenvalues
275
where
a = 2a2E + 0(E2), b = 1 + E(p'(O) + (b l - 2a3)8d + 0(E2),
C = 2CI + 2E(CIP'(0) - (C2 - cIbd8d + 0(E2), d = E(8 + 3d2 + C2 - 2Cla3 - 2a2cd + 0(E2), I(x, y, E) = 3a2Ex2 + (1 + E(b l - 2a3)81)xy + E(a3 - bl )y2 +W2X3 + E(a3 - bl )xy2 + 0(E 2Ix,yI2), g(x, y, E) = (CI - E(C2 - cl bd8dx 2 + 2E(C2 - clbdxy + E(C2 - cIb l ) +(CI + E(bICI - 3d2)81)y2 + E(d 2 - b1CI)y3 + 0(E2Ix,yI2). The eigenvalues of (5.2.12) at the origin have the real part
There exists a unique function 8 = 8;(E) = 80 + O(E) such that a( E, 8;) = 0, where
(5.2.13) Suppose a + d = 0 and let x (5.2.12) then becomes
= (wu + av)/c, y = v, w = J-a 2 - bc.
it = -wv + F(u, v),
v=wu+G(u,v),
where F(u,v) = [cl(X,y,E) - ag(x,y,E)]/W, G(u,v) = g(X,y,E). In order to determine the stability properties of the origin, we apply a formula given in [60] to calculate the first focal value. The formula is
WI =
_l_[F~v(F~u + +F~v) - G~v(G~u + G~v) - F~uG~u + F~vG~v] 16w + 116 [F~uu +
F~vv + G~uv + G~vv]'
where all partial derivatives are evaluated at (0,0). It is not hard to
Chapter 5.
276
obtain WI
Bifurcation of Higher Dimensional Systems
= c~o + O(c 2 ), where ~o =
1
S[3d 2 - 2a2(1 -
cd -
C2 -
2cla3].
Therefore, for 8 = 8~(c), and c > 0 small, the origin unstable) for (5.2.12) if ~o < 0 (or> 0). Note that a + d = 0 if and only if 8 - 8~(c) = O. It (5.2.10) that there exists Co > 0 such that for 0 < c < co, the system (5.2.10) has a unique limit cycle L(c, 8) near if and only if
(5.2.14) is stable (or follows from
18 - 80 1 < co, the point Af (5.2.15)
We next study the limit cycle with "large" amplitude. Making the coordinate change u = p-2c1 (2: 0), we have from (5.2.10)
du dO
], = - 2C IUV - 2cCIU [ a2 Uk + h(v)
~~ = CI(U k + v 2 -
(5.2.16)
1) + c[gl(V)U k + Ih(V)
+ O(c)],
where k = -l/cI > 0 . Now for c = 0 the system (5.2.16) is Hamiltonian with the Hamiltonian function
H(u,v) = -CIU [1- v 2
+ - CI- u k] .
Denote Lh: H(u,v)
= h,
0
1- CI
-CI
< h < ho ==--
1 - CI and let (u( h), 0) be the intersection point of Lh and the positive u-axis satisfying 0 < u(h) < 1. Then we have the Poincare map of (5.2.16) as follows,
P(h,c,8) - h = cM(h, 8)
+ O(c 2 ),
~.2.17)
where
Obviously, M is linear in 8 and M6 # O. Thus, there exists a function 8 = 8(h, c) such that for c > 0 small, (5.2.16) has a limit cycle through
5.2.
Zero and Pure Imaginary Eigenvalues
277
°
point (u(h),O) if and only if 8 = 8(h,E). Notice that u = is an integral line of (5.2.16). We have that (5.2.16) has a heteroclinic loop if and only if 8 = 8(0, E) == 8r(E). Also, from (5.2.15), there exists a function h = h(E) = ho + O(E) such that 8~(E) = 8(h(E), E). Therefore, for E > small (5.2.16) has a limit cycle if and only if 8 lies between 8~(E) and 8r(E). We now use the equation M(0,8) = to compute 8r(0) == 8~. Since along Lo
°
°
v2 =
1 + _C_l-uk, 1 - Cl
(5.2.18)
we have
where g is a functon of u. Then applying Green's theorem and integrating by parts, we have from (5.2.18)
J gdu ~ J ul+kdv
fLo
= 0, =
J udv = - J vdu,
~
~ -(1 + k) J ukvdu, fLo
J uvdv = 0, ~ J v 3du = -3 J v 2udv,
fLo
fLo
J v 2udv = J udv + _C_l_ J ul+kdv fLo fLo 1- Cl fLo = _ J vdu _ Cl(1 + k) J ukvdu. 1-
fLo
Cl
fLo
It follows that
M(0,8) = fLo J (C2 - c1b1)vukdu + (d 2 - b1ct)v 3du +(8 + 3d2 + b1ct)vdu + fLo J 2cla2ul+kdv +2cl(a3 - b1)v 2udv
= (8 + 6d 2 +(C2
J vukdu = (8
~
(5.2.19)
4cla3) J vdu fLo
+ 2(1 -
+ 6d 2 -
+ 2cla3udv
cl)a2
+ 2cla3 -
3d2) J vukdu fLo
4cla3) J vdu - 8~o J vukdu ~ ~
Chapter 5.
278
Bifurcation of Higher Dimensional Systems
where Do o is given by (5.2.14). Let j
U1
> 0 satisfy 1 + 1~ICI uk
=
o. Then
c1 _uk ) 1/2 U k- 1du r luI + __
vukdu = 2
U
(
Jo
fLo
1-
C1
Ioul (1 + - C1- uk) 3/2 du 0 1 - C1 4 [ Ioul C1 = -(1- cd - - u k(1 + - C1 - u k) 1/2 du 4 3
= -(1 - C1)
3
0
+
1-
1-
C1
Iooul (l +1--C1-C1u k) 1/2 du]
- 2C1 3
i
Lo
vu kd u+ 2(1 3
cd
i Lo
C1
v d u.
This yields j
fLo
vukdu = 2(1 - cd j vdu. 3 - 2C1 fLo
(5.2.20)
From (5.2.19) and (5.2.20), we have M(0,8)
=
[8 + 3 - 22C1 (c2(1 +3d2(2 -
Hence, M(0,8)
8~ =
+ 2a2(1 - C1)2 cd - 2c1a3(2 - cd)] Ao vdu. C1)
= 0 has a unique root
2 [(2-c1)(2c1a3-3d2)-(1-cd(c2+2a2(1-cd)]. (5.2.21) 3 - 2C1
It is seen directly from (5.2.13), ( 5.2.14) and (5.2.21) that
80
_
8' = a
8Do o 3 - 2C1
~
..
Summarizing the above, we obtain the following theorem. Theorem 5.2.1. Suppose C1 < 0 and Do o i= o. Then there exist Eo > 0 and functions 8~(E) = 80 + O(E), 8i(E) = 8~ + O(E), such that for 0 < E < Eo (5.2.10) has a limit cycle L( E, 8) if and only if Do 0 8i (E) < Do 08 < Do08~ ( E), which is stable (or unstable) if Do o < 0 (or
5.2.
Zero and Pure Imaginary Eigenvalues
279
> 0) .
Moreover, L( E, 8) becomes the critical point of index heterochinic loop) when 8 = 8~(E) (resp., 8 = 8i(E)).
+1 (or
a
Next, we discuss the uniqueness of limit cycles. Notice that along Lh we have (5.2.22) Then, in a way similar to the deducation of (5.2.19) we have M(h,8)
= (8 + 6d 2 -
4cla3) 1 vdu - 8.6. 0 1 vukdu.
ho
ho
Denote
1 = ho vdu,
Po () h
Pl
(h)
1 kd = ho vu u,
P(h) __ Pl(h) Po(h)·
Then M(h,8) = Po(h)(8
Hence, M(h,8)
= 0 if
+ 6d 2 -
4cla3 - 8.6. oP(h)).
and only if
8 = -6d2 + 4cla3
+ 8.6. oP(h),
and if M(h, 8) = 0, then (5.2.23) For the property of P, we have Lemma 5.2.2. P(O) = 2(1-cd/(3-2cd, P(h o) = 1, and P'(h)
>
o for 0 < h < h o . Proof. Obviously from (5.2.20), P(O) = 2(1 - cl)/(3 - 2Cl). Also, 8v 1 from (5.2.22) we have 8h = - - . Hence 2ClUV
P'(h) o
-
~ 1 du
2Cl JL h uv'
1 k-l P{(h) = 1 ~du. 2Cl hh v
(5.2.24)
Chapter 5.
280
and thus
Bifurcation of Higher Dimensional Systems
P(h) P(h) = 1 - -3-'
.
-
wIth P(h)
P2 (h)
= Po(h)·
To prove P' (h) < 0 for 0 < h < ho, we follow the idea used in [25]. Evidently, P(h) > 0 for 0 < h < ho. Notice that P(h) = 3 - 3P(h) and P(ho) = 1. It follows that P(ho) = o. Suppose P'(h I ) = 0 for some hI E (O,h o), then
P(h) - P(h I )
=
P2 (h)/ Po(h) - P(hd
=
[P2 (h) - P(hdPo(h)]/ Po(h)
= Q(O)(h -
(5.2.25)
hd/ Po(h),
where Q(h) = P~(h) - P(hl)P~(h), and 0 lies between h and hI. Since P'(h I ) = 0, we have Q(hd = 0 from (5.2.25). Then noting
P~(h) = ~ 1 vdu 2CI hh
U
and using (5.2.24), we obtain
Q(h)
=
_~ 1 P(h l ) - 3v 2 du, 2CI hh
and hence
Q(h)
=
_~ 2CI
[1
hh
P(hd - 3v 2 UV
du _
UV
1
hhl
P(h l ) - 3v 2 UV
dU].
(5.2.26)
5.2.
281
Zero and Pure Imaginary Eigenvalues
For h > hl close to hl we have Lh C int.L hl . Since Q(h 1) = 0, LhJ intersects the lines v = ±/P(h1 )/3. Then the annulus surrounded by Lh and Lhl is divided into four regions D 1 , D2, D 3 , and D4 with D2 on the right and D3 upper, see Fig. 5.2.2. Let ODi denote the boundary of Di with counterclockwise orientation. Then by the Green's theorem and (5.2.26) we have
Q(h) - __ 1 1 -
P(h 1 )
2Cl J8D 1 U{}D3
Ii; _~ I r Cl
3v 2 du _ ~ 1 Cl J8D 2 U8D 4
UV
P(h 1 ) 1-
uk -
3v 2 dv v2
'
2
= __1_ 2Cl
-
P(hd - 3v dudv uv 2 k 1 2 u - (P(hd - 3v ) dudv > O. (1 - uk - v 2 )2
DJ uD 3
JD 2 UD 4
Similarly, we can prove that Q(h) < 0 for h < hl close to h1. Hence, Q(h)(h - h1) > 0 for Ih - hll > 0 small. Thus from (5.2.25) we have (5.2.27) since (0 - h1)(h - h1) > O. Notice from the above discussion that (5.2.27) holds for any hl E (0, ho ) satisfying P' (hd = O. It follows that P'(h) > 0 for hl < h < ho . Therefore, P(h o ) > P(hd > 0, in contradiction to P(ho) = O. The proof is completed. 0
or-+-;----r--+--r----
-Fig. 5.2.2
Chapter 5.
282
Bifurcation of Higher Dimensional Systems
From (5.2.15), (5.2.17), and Theorem 5.2.1, we have immediately Theorem 5.2.3. Suppose C1 < 0 and boo =1= O. There exist constants Eo> 0, E1 > 0, such that for 0 < E < Eo, boobi(E) - E1 < boob < boob~(E), the system (5.2.10) has a unique limit cycle.
Fig. 5.2.3 Recently, the last author proved that if Ao =1= 0 the heteroclinic loop can bifurcate at most one limit cycle. Hence, Theorem 5.2.3 remains true for E1 = o. For the phase portraits of (5.2.10) see Fig. 5:~. 5.2.2.
Qualitative results for full system
Suppose boo =1= o. Then (5.2.16) has a focus near the point (1,0) for lEI =1= 0 small enough and it is stable (resp. unstable) if E(b-b~(E)) < 0 (resp. > 0). Then noting (5.2.8), (5.2.15) and Corollary 4.4.3, we have immediately Theorem 5.2.4. Suppose that (5.2.9) holds and boo
=1=
o.
Then
5.3.
Two Pairs of Pure Imaginary Eigenvalues
283
there exist an Eo> 0 and a C 1 function 'I) = 2Cl).1 +80)'~+O()'V, where 80 is given by (5.2.13), such that for 0 < ).~ +).~ < Eo the system (5.2.3) has a unique periodic orbit near the origin, which is stable (resp. unstable) if ).2 < 0 and a C 1 function 'I) = 2Cl).1 + 80 + O().V, 0,
j
= 1,2.
The matrix D is invertible with codimension-2 ([18]). Then we consider the 2-parameter perturbation of (5.3.1)
z=
D(>q, A2)Z + Z2(Z),
(5.3.2)
where D(AI' A2) = diag (AI (AI)' B I(A2)), AI(Ad
( AI WI), and -WI Al
B I(A2) = ( A2 W2). -W2 A2 Applying the normal form theory (see [9], [78], [181]), one can prove that if
nWI
+ mW2 =1= 0,
for 1 ~
Inl + Iml
Inl, Iml
~ 6,
E
N,
then (5.3.2) has the following normal form,
x=
AI(AI)X + A2xlxI 2 + A 3 xlyl2 + A4xlxI 4 +A5Xlx121Y12 + A 6 xlyl4 + X(x, y), iJ = B I(A2)y + B2ylyl2 + B 3 ylxl 2 + B4ylyl4 +B5ylxl 2lyl2 + B 6 ylxl 4 Y(x, y),
+
Bi
=
(
i d d ) '
Ci
-
i Ci
.
~
= 2, ... ,6.
(5.3.3)
5.3.
Two Pairs of Pure Imaginary Eigenvalues
285
Changing the variables by x
= (PI cos 01, -PI sin 01 ),
Y
= (P2 cos O2, -P2 sin O2),
we obtain from (5.3.3)
PI = pdAl + a2pr + a3P~ + a4Pi + a5prp~ + a6P~ + Pl(Ol, 02,Pl,P2)], P2 = P2[A2 + C2P~ + C3pr + C4P~ + C5prp~ + c6Pi + P 2(01, 02,Pl,P2)], ih = WI + b2pr + b3P~ + b4Pi + b5prp~ + b6P~ + 8 1 (01, 02,Pl,P2), 82 = W2 + d2P~ + d3PI + d4P~ + d5PIP~ + d6Pi + 8 2 (01 , 02,PI,P2), where P i ,8i
= p;lO(lpl,P21 6),
i
=
1,2. By the scaling
we obtain for Pi > 0 (i = 1,2),
+ E2 Pl(Pl,P2) + E3 P2(01, O2, PI, P2, E)], P2 = E2p2[Qo(Pl,P2) + E2Ql(Pl,P2) + E3Q2(01, 02,Pl,P2, E)], . 01 = WI + E2[ So(Pl, P2) + E2Sl (PI, P2) + E3 S2( 01, O2, PI, P2, E)], 82 = W2 + E2[To(pl, P2) + E2Tl (PI, P2) + E3T 2(01, O2, PI, P2, E)].
PI
= E2pdPo(Pl,P2)
(5.3.4)
where
= 81 + a2pr + a3P~, PI (PI, P2) = a4Pi + a5prp~ + a6P~, Qo(PI,P2) = 82 + C2P~ + C3pr, Ql(Pl,P2) = C4P~ + C5prp~ + c6Pi, So(Pt,P2) = b2pr + b3P~' Sl(Pl,P2) = b4Pi + b5prp~ + b6P~, To (PI, P2) = d2P~ + d3PI, Tl (PI, P2) = d4P~ + d5PIP~ + d6P~. Po (PI, P2)
Further, letting ri = p~, i = 1,2, rescaling the time t - t (2E 2)-lt, and truncating the higher order terms E3 P2 and E3 Q2, we obtain from the first two equations of (5.3.4)
7\ = rd8l + a2rl + a3r2 + E2(a4rr + a5rlr2 + a6r~)], r2
= r2[82 + C2r2 + C3rl + E2(c4r~ + c5rlr2 + c6rD]·
(5.3.5)
Chapter 5.
286
Suppose a2c2
i= r1
Bifurcation of Higher Dimensional Systems
°
in (5.3.5). By the rescaling
---+
rdla21,
r2
---+
rdlc21,
t
---+
(sg na 2)t
we have from (5.3.5),
+ r1 + br2 + E2(err + f r 1r 2 + gr~)L r2[1L2 + cr1 + dr2 + E2(hrr + j r1r2 + krDL
7-1 = rdIL1 7-2 =
(5.3.6)
where
= 81sgn a2 = ±1, 1L2 = 82sgn a2, b = a3/la2Isgna2, c = c3/la21, d = cdlc21sgna2 = ±1, e = a4/a~sgna2' f = a5/l a2c2Isg na 2, 9 = a6/ c§sgn a2, h = c6/a§sgn a2,j = c5/la2c2Isgna2, k = C4/c§sgna2' ILl
When
E
= 0, (5.3.6) becomes
7-1
= r1(1L1 + r1 + br2),
7-2
= r2(1L2 + cr1 + dr2)'
(5.3.7)
It is easy to see that a necessary condition for (5.3.7) having a center in the region r1 > 0, r2 > is that
°
d=-l,
A=:-l-bc>O,
(b+1)1L2=(c-1)1L1'
(5.3.8)
In this case, (5.3.7) has a family of periodic orbits in the region r1 > 0, r2 > if one of the following conditions holds: (1) b i= -1, c = 1, ILl = 1, 1L2 = 0; (2) c > 1, b + 1 < 0, ILl = 1; (3) c> 1, b + 1 > 0, ILl = -1; (4) c < 1, b + 1 < 0, ILl = 1; (5) c < 1, b + 1 > 0, ILl = -1. Suppose (b + l)(c - 1) i= 0. Then (5.3.7) has a first integral
°
F(r1' r2) = r?r~(1L1 where a suppose
= (1 - c)/A,
(3
+ r1 + ,r2),
= (b + l)/A, ,= (3/a.
From (5.3.8), we may
(5.3.9)
5.3.
Two Pairs of Pure Imaginary Eigenvalues
287
in (5.3.5), so that (5.3.6) becomes
1'1 = rl[lLl
.
+ rl + br2 + E2(err + f rlr2 + grD]'
[e -
rl = rl b + 11 ILl + erl - r2
+ E2( uc + hr 2 l +·J r lr2 + kr22)]
(5.3.10)
.
For definiteness, we consider the case e
< 1,
b> -1,
A
=
-1 - be > 0,
ILl = 1.
(5.3.11)
It is easy to see that (5.3.10) has a unique critical point P(E) (rl(E 2 ),r2(E 2)) with index +1, where
ri(E2) = - 1b +1
+ r~(0)E2 + (E4),
i
= 1,2,
r~(O) = ~ [b8+ (b:1)2(e+ f + g +b(h+ j +k))], r~(O)= ~ [-8+ (b:1)2(c(e+ f + g )-(h+ j +k))]. The divergence of (5.3.10) at P(E) is
~E2[(2 + c)r~(O) + (b - 2)r~(0) + 8 + (b: 1)2(3e + h + 3k + 9 +2(1 + j))] Since ~~
=
b~l
8 = 8~(E) = 80 80 =- (b
+ 0(E4) == ~E2G(8, (2).
> 0, the equation G( 8, (2) = 0 has a unique solution
+ 0(E2)
with
1
+ 1)3 [(1 - c + 2A)e + (1 - c + A)f + (1 -
+ (b
e)g + (b + l)h
+ 1 + A)j + (b + 1 + 2A)k].
(5.3.12)
Thus, we obtain the Hopf bifurcation curve 8 = 8~ (E). To discuss the heteroclinic bifurcation, we make the change of variables u = r l , v = rg so that (5.3.10) becomes
it = O::U[ILI
+ un + bv m + E2(eu 2n + funv m + gv2m)],
288
Chapter 5.
Bifurcation of Higher Dimensional Systems
where n = 1/Ci, m = 1/13. Notice that for E = 0, (5.3.13) is Hamiltonian with the Hamiltonian function uv H( u, v) = -[lt1 + un + ,vm]
,
and with a family of periodic orbits L>. : H(u, v) = A,
Ao == (1
+ bc)(b + c)-(2+a+.B) < A < 0.
Then the Melnikov function for (5.3.13) is given by
M(A,8) =
A>. f3v(8 + hu 2n + junv m + kv2m)du _1
1£->.
Ciu(eu 2n
+ funv m + gv2m)dv == M 1 (A,8)
Notice that in the region u > 0, v the heteroclinic loop Lo and thus
. n m h U 2n + JU V
k
2m _
~
"
> 0, we have un + ,vm = 1 along
j , - 2k n
+ V - 2+ eu 2n + funv m + gv 2m = e + (f -
- M 2(A,8).
k - j , + h,2) 2n
+ 2q)v m + (q2 2
U
,
2
f,
U,
+ g)v 2m .
Then integrating by parts yields
= -Ci Ao (ev +
1!
+_f3_(q2 _ f,
2+13
f3(f - 2q)v1+m
+ g)v1+2m)du
1 ((e + f3(f - 2q) + f3(q2 - f, + g))v 1£0 (1 + f3h (2 + f3h 2 _(2f3(q2 - f, + g) + f3(f - 2q) )vu n (2 + f3h 2 (1 + f3h + f3(e,2 - f, + g) 2n)d (2 + f3h2 vu U.
= -Ci
.. ~
5.3.
Two Pairs of Pure Imaginary Eigenvalues
289
Therefore, we have
M(0,8) = [,88 +
~ + (1+J)(~+t3)
(2!~)'Y2 J£'o vdu /a.(j2 + ~J 1 nd (1+t3)(2+t3h (2+t3h fLo VU u,
+ (1+t3)(f+ t3h +
-
['!:Ei -y2
+
[~ - J:f + h,8 + ~ - (2~a.~'Y + (2!~)'Y2 J £'0 vu 2n du.
-
i1l.
-y -
2eat3
(1+13)(2+13) -
2
(5.3.14) Similarly to (5.2.20) we can show that
1 vundu
fLo
i
vu 2n du
Lo
= = =
a,8 1 vdu,
1 +a
+
1+a
2 + a +,8
fLo
10 1 (,-1(1 - un))"undu a 0
a(l+a) 1 vdu. (1 + a + ,8)(2 + a + ,8) fLo
Substituting the above into (5.3.14) we obtain
M(O 8)
,
= ,8{8 +
1 [k,8(l + ,8)(2 + ,8) ,8(1 + a + ,8)(2 + a +,8) '"'(2
+ja,8(l + ,8) + ea(l + a)(2 + a) + fa,8(l + a) '"'(
'"'(
+ ga,8~2 + ,8) + ha,8( 1 + a)]}
£'0 vdu,
== ,8{8 - 81 } fLo 1 vdu. Hence, M(0,8)
= 0 if and only if 8 = 81 , where
[kb(l - c)2(b - 1 - 2bc) jb(l - C)2 1-(1+b)(b-c-2bc) (1+b)2 + l+b gb(l - c)3 J +c(l + c + 2bc)e - fc(l - c) + (1 + b)2 - hc(l + b) .
8 _
-1
(5.3.15) Summarizing the above we obtain the following theorem. Theorem 5.3.1. Under the conditions of (5.3.11), there exist Eo > 0 and two functions 8 = 8;(E) = 8i + O(E), i = 1,2" where
290
Chapter 5.
Bifurcation of Higher Dimensional Systems
80 and 81 are given by (5.3.12) and (5.3.15) respectively, such that if ~ == 81 - 80 =I 0, then (i) for ~8~(E) < ~8 < ~8i(E) and 0 < E < Eo, (5.3.9) has at least one limit cycle L(E,8), and for 8 = 8i(E), 0 < E < Eo, (5.3.9) admits a heteroclinic loop L f ; (ii) for 0 < E < Eo, L(E,8) ---t P(E) (resp. L f ) as 8 ---t 8~(E) (resp. 8i(E)). Remark 1. It was proved in [210] that (5.3.10) has at most one limit cycle generically. This suggests that (5.3.10) has a unique limit cycle if and only if ~8~(E) < ~8 < ~8i(E) when ~ =I 0 and (5.3.11) holds. The uniqueness of limit cycles near the heteroclinic loop La is difficult to prove in general case. The problem was not solved in [210] and needs further discussion. 5.3.2.
Existence of an invariant two-torus
We now discuss the existence of an invariant 2-torus for the original system (5.3.3). If
nWl
+ mW2 =I 0
for 1 ::;
Inl + Iml ::; 8, Inl, Iml
E N,
(5.3.16)
then we can further simplify (5.3.3) and obtain the following periodic 4-dimensional system from the reduction of (5.3.6):
e= W + ASo(r) + A2S1(r) + A3S2(r) + A7j 2S 3((), r, A), r = APo(r) + A2P1(r) + A3P2(r) + A7 j 2P3((),r,A), where A = E2, W = (Wi, W2), () 27r-periodic in (), and
=
(()ll ()2),
(5.3.17) -~
r = (rl,r2), S3 and P3 are
Suppose d = -1, b + 1 =I 0, and A = -1 - bc > O. Then (5.3.10) has a unique critical point P( E) in the first quadrant for lEI small. Let JL2
5.3.
Two Pairs of Pure Imaginary Eigenvalues
291
satisfy (5.3.9) and x = r - P(E). We obtain from (5.3.17)
+ AS~(X) + A2Si(x) + A3 S2(X) + A7/2Sj(O, x, A), x = AB(A,8)x + O(Alx12 + A7/2),
iJ =
w
(5.3.18)
where B(A,8) is a 2 x 2 matrix having eigenvalues with real part Aj2G(8, A), which has the same sign as A(b+1)j(2A)(8-8;(E)). Notice from (5.3.9) that
8=
E- 2
[1L2 -
~ ~ ~1L1]
= A12sgna 2 [A2 -
~ ~ ~ AI] .
Applying Theorem 4.2.4 to (5.3.18) we get immediately
Theorem 5.3.2. If (5.3.16), d = -1, b + 1 =1= 0, and A = -1 be > 0 are satisfied, then for El > 0 small enough and M > 0 (large) there exists Eo> 0 such that for 0 < Ai + A§ < Eo, the system (5.3.3) has an invariant 2-torus in a neighborhood of the origin provided e - 1 AI] - 81 < M. Ai b+ 1 0 Moreover, the torus is asymptotically stable (resp. unstable) if El
(b 5.3.3.
< 1sgn a2 [A2 _
+ 1)A-1sgna2 (A2 - ~ ~ ~ Al -
8o Ai) < 0 (resp. > 0).
Bifurcations of multiple periodic orbits
In this part, we are concerned with the bifurcations of multiple periodic orbits of the 4-dimensional system with one parameter:
x= iJ
=
+ X(x, y, A), mJy + Y(x, y, A), Jx
where A E JR, (x, y) E JR2 X JR2, mEN, X
(5.3.19)
= (Xl, X 2 ),
and
+ J3i(A)X2 + h(x, y, A) + O(lx, ylk+l), X 2 = -J3i(A)Xl + C¥1(A)X2 + gk(X, y, A) + O(lx, ylk+l), Y(x, y, A) = C¥2(A) + J32(A)Jy + Yk(X, y, A) + O(lx, ylk+l) Xl = C¥l(A)Xl
J3i(A) = J3l(A) - 1,
J32(A) = J3(A) - m,
k
=
2 or 3
(5.3.20)
292
Chapter 5.
Bifurcation of Higher Dimensional Systems
in which !k. gk and Yk are homogeneous polynomials in x and y of degree k. We introduce the transformation of variables
x
= p( cos 0, -
sin 0)
== ph( 0),
y =pv
to obtain from (5.3.19)
e= (31 -
+ O(pk), P = pal + pk Pk(O, v, A) + O(pk+1), v = mJv + a2V + (3;,Jv + pk- 1Vk(0, v, A) - [a1 + pk-1 Pk(0, v, A)]V + O(pk), pk- 1Sk(0, v, A)
where Sk( 0, v, A) = sin 0ik(h(O), v, A)
+ cos Ogk(h( 0), v, A),
Pk(O, v, A) = cos O!k(h(O), v, A) - sin Ogk(h(O), v, A),
(5.3.21)
Vk(O, v, A) = Yk(h(O), v, A).
Then we have the following 271"-periodic system dp dO = R(O,p,V,A),
dv dO
= mJv + V(O,p, v, A),
(5.3.22)
in which
(5.3.23) We first consider the case of m
= 2 and
k
= 2. Set
1 1211" e -2J8 V (0, 0, 0, )dO, 2 271" 0 1 r211" 2J8 B(b) = 271" 10 P2 (0, e b,O)dO.
Ao
= -
(5.3.24)
5.3.
Two Pairs of Pure Imaginary Eigenvalues
293
Then it follows directly from (5.3.21) that Lemma 5.3.3. Suppose
and
Then
(5.3.25) and B(b)
=
1 4(c l1
+ C22 + d 12 -
d 2i )b i
1
+ 4(C12 -
C2i -
d ll
-
d 22 )b2 , (5.3.26)
where b = (b i , b2 ).
Now applying Theorem 5.1.2 we can prove Theorem 5.3.4. ([73]) Suppose thatm = k = 2, a~(O) =1= 0, Ao =1= O. Then (i) if a~(O) = ,6HO) = 0, (5.3.19) has no periodic orbits satisfying Ix(t, A)I
=1=
0,
.
ly(t,A)1
hmsup Ix (t, /\')1 A-->O
< 00
(5.3.27)
and with period close to 27l' in a neighborhood of the origin for IAI small enough; (ii) if la~(O)1 + I,6HO)1 =1= 0 and the equation
=1= 0
(5.3.28) has exactly a pair of simple roots in JR2 (resp. no roots), then for IAI =1= 0 small (5.3.19) has precisely a periodic orbit (resp. no periodic orbit) near 0 satisfying (5.3.27) and with period close to 27l'.
294
Chapter 5.
Proof. Suppose G (5.1.26) that
Bifurcation of Higher Dimensional Systems
= (G 1 , G2 ) is given in (5.1.27). Notice from
p* = a + O(lal 2 + laAI), q* = b + O(lal
+ IAI)·
Then from (5.1.24),(5.1.26) and (5.3.23), we have * * ) 1 (27r G1(a,b,A)=-lc R(O,p,q,AdO
271'
= a[ad.81 G2 (a,b,A) = - 1 271'
=
(
0
5.3.29
)
+ B(b)a + B1a 2 + O(lal 3 + laAI)],
1027r -V( O,p* ,q *,A )dO 0
b(a2h.8~ .82J) + A(b)a - (;~ + B(b)a)b + O(a 2 + laAI), (5.3.30)
where B1 is a constant, B(b) is given by (5.3.26), and
A(b) = ~
(27r
271' 10
e- 2JO V2(O, e 2JOb, O)dO.
Similar to (5.3.25), we can verify A(b) = Ao. By applying the implicit function theorem to (5.3.29), we see that there exists a unique function (5.3.31) such that G 1 (a, b, A*) = 0, which gives ..
a1
2
.81 + B(b)a = O(lal + laAI).
(5.3.32)
Inserting (5.3.31) and (5.3.32) into (5.3.30), we get
G2 (a, b, A*) = (a~(O)h + .8~(O)J)A*b + Aoa + O(a 2) = -
I~ ) [(a;(O)h + .8~(O)J)B(b)b - Aoa~(O) + O(a)]
a 1d O
== -,-()Go(a,b). a1 0
~
5.3.
Two Pairs of Pure Imaginary Eigenvalues
295
Now the conclusion (i) is obvious. From (5.3.26), it is not hard to prove that the roots of (5.3.28) appear in pairs and are at most 2. Suppose that ±b(i) satisfy
aGo
(i)
det ab (0, ±b ) =f= 0. Then there exist functions b~)(a)
= ±b(i) + O(a) such that
(i)
Go(a, b± (a)) = 0. Substituting b~) into (5.3.31), we have
A=-
aB(±b(i») 2 _ '() +O(a )=A±(a). al
°
Since Ao =f= 0, we have B(±b(i») =f= 0. Therefore, the functions A±(a) have the inverse
a
=
*(
a± A)
= -
Aa~(O) ( 2) B(±b 0), the system (5.3.1 g) has a unique (resp. no) periodic solution satisfying X(t,A) = y(t, A)
-~~~io~(cost,-sintl +0(IAll/2),
= _ ai (O)A emJtbo + 0(IAll/2). Bm(bo)
Proof. Since a~ (0) =1= 0, it follows from the implicit function theorem that there exists a unique function A = - a2Bm(b) ai(O)
such that G 1 (a, b, A*) =
o.
+ O(a 3 ) =
A*(a b) -,
(5.3.37)
Therefore,
~: + Bm(b)a 2 = O(lal 3 + la 2AI). Substituting the above equality and (5.3.37) into G 2 in (5.3.34), we have
5.3.
Two Pairs of Pure Imaginary Eigenvalues
297
Suppose that bo satisfies (5.3.36). Then the equation Go(a, b) = 0 admits a unique solution b = b(a) = bo + O(a). Inserting it into (5.3.37) we obtain
whose inverse is given by
o
Then the conclusion really follows.
Obviously, if there exist bi Em?, i = 1"" ,k, such that F(bd = 0, det DF(bi ) =1= 0, Bm(bi)a~ (0) < 0, i = 1" .. ,n, and Bm(bj )a~ (0) > 0, j = n + 1" .. ,k, for some n :::; k, then for A > 0 (resp. < 0) small, (5.3.19) has exactly n (resp. k - n) periodic orbits with period close to 27r and satisfying (5.3.27). From Lemma 5.3.5, there must be k :::; 16 (for m = 1), k :::; 9 (for m = 3) and k :::; 1 (for m ~ 4). Remark 2. If we want to obtain a periodic solution satisfying Iy(t, A)I =1= 0, lim A-+ o Ix(t, A)l/ly(t, A)I = 0, then instead of (5.3.27), we may exchange x and y in (5.3.19) and then apply Theorem 5.1.2 (the Hopf bifurcation theorem) for m ~ 2 or Theorem 5.3.6 for m = 1. As an example, consider the system
x=
Jx
iJ
Jy + AY - Kylyl2 + xlxl 2 ,
=
+ AX -
Kxlxl 2 + ylyl2,
(5.3.38)
where K is a constant. Let N(A) denote the number of local periodic orbits of (5.3.38) of period close to 27r for IAI =1= 0 sufficiently small. Then we can show that for 0 < 1£1 « 1, (i) N(A) = 1 if IKI < 1; (ii) N(A) ~ 1 for KA > 0, if IKI = 1 or 2; (iii) N(A) = 2 (resp. 0) for KA > 0 (resp. < 0), if 1 < IKI < 2; (iv) N(A) = 4 (resp. 0) for KA > 0 (resp. < 0), if IKI > 2.
298
Chapter 5.
Bifurcation of Higher Dimensional Systems
In fact, from (5.3.38) we have k
=
2, m
=
1, and
13i = 132 = 1, Ql = Q2 = >., h = -Klxl2Xl + lyl 2 Yl, 93 = -Klx12X2 + ly1 2 Y2, Y3 = -Kylyl2 + xlxl 2. Hence, by (5.3.21)
P3 (O, v) = -K + IvI 2vT h(O),
lt3(O, v) = -Kvlvl 2 + h(O),
then noting that h( 0) = (cos 0, - sin of and (eJO)T h( 0) (1, of, it follows from (5.3.35) that
BI(b) = -K + Ibl 2bl ,
= e- JO h( 0)
=
AI(b) = -Kblbl 2 + (1, of,
where b = (b l , b2 f. Therefore, we have
It is easy to see that F(b) = 0 if and only if
b2 = 0,
P(bd == b1 + Kbf - K - 1 = O.
Notice that P(bd = (bi - 1)(bi + Kb l + 1). The conclusion follows from Theorem 5.3.6 and the symmetry of (5.3.38).
5.4.
Global Bifurcations of Large Periodic Orbits
In this section we are concerned with global bifurcations of periodic orbits and invariant tori near a large periodic orbit ~hree dimensional systems.
5.4.1.
Bifurcations of periodic orbits
Consider the following analytic autonomous systems:
x = f(x)
(5.4.1)
x = f(x) + >'F(x, >'),
(5.4.2)
and
5.4.
Global Bifurcations of Large Periodic Orbits
299
where x E IR3 , A E IR, f and F are analytic functions. Suppose (5.4.1) has a periodic orbit with period T:
o ::; t ::; T.
r : x = u(t),
Then from [62, Ch.6J, in a neighborhood of r we can introduce the local orthonormal coordinate transformation
x
= u(O) + Z(O)p,
O::;O::;T,
(5.4.3)
where p E IR2, and Z(O) = [6(0),6(0)] is aT-periodic 3 x 2 matrix, such that the 3 x 3 matrix [v(O) == f(u(0))/lf(u(0))1,6(0),6(0)] is orthogonal. Under (5.4.3), the system (5.4.2) becomes
iJ = 1 + O(lpl + IAI), p = A(O)p + AZT(O)Fo(u(O)) + O(lp, AI2), where Fo(x) = F(x, 0) and
A(O)
= ZT(O) [- ~! + Df(u(O))Z(O)] .
(5.4.4)
Then we obtain the T-periodic system (5.4.5) Let X(O) denote a fundamental matrix of the linear system
dp dO
=
A(O)p.
(5.4.6)
Then the solution p(O,Po, A) of (5.4.5) with p(O,Po, A) = Po satisfies
p(O,Po, A)
= X(O)X-l(O)po .+ fo8 X(O)X-l(S)[AZT(s)Fo(u(s)) + O(lp, AI2)]ds. (5.4.7)
It is clear that
p(O, 0, 0)
= 0,
(5.4.8)
Chapter 5.
300
Bifurcation of Higher Dimensional Systems
p(O,Po, A) = X(O)X-l(O)po
+ A fo8 X(O)X-l(s)ZT(s)Fo(u(s))ds + O(lpo, AI2). Therefore the Poincare map of (5.4.5) can be represented as (5.4.9) where (5.4.10) Hence, for IAI small (5.4.2) has a periodic orbit in a neighborhood of r with period close to T if and only if there exists Po sufficiently small, such that (5.4.11) where (5.4.12) From (5.4.11) it is clear that if B is invertible, then r generates a unique periodic orbit for IAI sufficiently small. If B is not invertible, then we may assume
B= (aed' b) Setting Po
=
(Pl,P2)T, K
=
ad
= be.
(Kl' K 2)y, (5.4.11) is equivalent to
+ bp2 + AKI + O(lpo, A12) = 0, epl + dP2 + AK2 + O(lpo, A12) = 0, apl
(5.4.13)
(5.4.14) -~
(5.4.15)
Theorem 5.4.1. If det B i- 0, then for IAI small enough (5.4.2) has a unique periodic orbit in a neighborhood of r. Suppose det B = 0, and let (i) aK2 - eK1 i- 0, or (ii) bK2 - dK 1 i- o. Then for IAI io small enough, (5.4.2) has no periodic orbits near r if r is nonisolated, and has a unique periodic orbit near r for all A i- 0 small or has precisely two (resp. no) periodic orbits for A lying on one side (resp. the other side) of A = 0, if r is isolated.
5.4.
Global Bifurcations of Large Periodic Orbits
301
Proof. We need only to consider the case of det B = o. And, because of the similarity, we may assume the condition (i) holds. Then, noting (5.4.13), we can solve from (5.4.14) and (5.4.15) that PI = R(P2), ). = E(P2), with R(O) = 0, E(O) = E'(O) = O. Notice that E(P2) is analytic in P2. We have either
(5.4.16) or (5.4.17) for some N m =1= 0 and m 2: 2. If (5.4.16) holds, then for small 1).1 =1= 0 the equations (5.4.14) and (5.4.15) have no solutions in (P1,P2), and for)' = 0 they have a family of solutions of the form (R(P2),P2) for Ip21 small, which shows that the periodic orbit r is non-isolated. If (5.4.17) holds, the function E has the inverse ). ) 11m
P2 = ( N m
+ 0(1).1 1I m),
for m odd,
= ± (NA) 11m + 0(1).1 1Im), for m even. m
Then the conclusion readily follows. 5.4.2.
D
Autonomous perturbations of linear system
Consider the following perturbed system,
x=
Ax
+ ).f(x, 8, ).),
where ().,8) E IR x IR, x E IR3, function, and A =
f : IR3 x IR x IR
(5.4.18) ---t
1R3 is a Coo
(~1o 0~ 0~).
By changing variables x = (pcos e, -psin e, z), we obtain from (5.4.18)
p = ).[cos ef1 - sin e12]' iJ = 1 + ).[cos e12 + sin efIlip, i = )'/3,
(5.4.19)
302
Chapter 5.
hf·
where f = (II, 12, 27r-periodic system,
Let
Bifurcation of Higher Dimensional Systems
°< 1).1 «
Ipl· We then have the following
dp dO = )'P(O,p, z, 0, ).), dz dO = ).Z(O,p, z, 0, ).),
(5.4.20)
where
P(O, p, z, 0, 0) Z(O,p,z,o,O)
°
= cos h(x, 0, 0) -
= h(x,o,O),
x
=
°
sin h(x, 0, 0),
(pcosO,-psinO,z).
(5.4.21)
By the method of averaging, (5.4.20) can be transformed into
(5.4.22)
where
H(p,z,o) = - 1 27r
!o27f 0
P(O,p,z,o,O)dO, (5.4.23)
1 (27f Zl(P, z, 0) = 27r 10 Z(O,p, z, 0, O)dO.
Noting (5.4.19) and applying Theorems 4.1.3, 4.3.1 and 4.3.12, we get immediately the following conclusions. Theorem 5.4.2. (1) If there exists a point (Po, zo) E Jll2 with Po > such that PI = 0, Zl = 0, and det D(Pl , Zl) = at (~) =
°
°
(Po, zo), then for 1).1 sufficiently small, system (5.4.18) has a periodic solution of the form
with period close to 27r. (2) If the following system
p = Pl(p, z, 0),
z=
Zl(P, z, 0),
p> 0
(5.4.24)
5.4.
Global Bifurcations of Large Periodic Orbits
303
undergoes a saddle-node bifurcation of singular points or a generic Hopf bifurcation at b = bo, then there exists a function b = b(>.) = bo + 0(>.) such that for 1>'1 small enough, system (5.4.18) undergoes a saddle-node bifurcation of periodic orbits or a generic Hopf bifurcation of invariant tori at b = b(>').
Similarly, from the results in Sec. 4.3.2, for the bifurcation of invariant tori of (5.4.18) we have Theorem 5.4.3. If (5.4.24) has a hyperbolic limit cycle, then for 1>'1 sufficiently small, (5.4.18) has a hyperbolic invariant torus. If (5.4.24) has a semi-stable limit cycle bifurcation at b = bo, then there exists b = b(>.) = bo+O(>.) such that for 1>'1 sufficiently small, (5.4.18) has a similar bifurcation of invariant tori at 8 = b(>'). As an example, consider a cubic system of the form
x= y iJ =
>.xz,
-x - >.yz,
(5.4.25)
Z = >'(2x2 - 1 - z3 - bz). Obviously, we have f(x, y, z, 8) = (-xz, -yz, 2x2 - 1 - z3 - bzf.
From (5.4.21), it is easy to see that P(O,p,z,b)
=
-pz,
Z(O,p,z,b)
= 2p2 cos Y 0-1-
z3 - bz.
Then (5.4.24) becomes
p = -pz,
z = p2 -
1 - z3 - bz, p
> O.
(5.4.26)
-2z.
(5.4.27)
Letting u = 2lnp, we obtain from (5.4.26) .
Z
= eu -
1 - 3 z -kuZ,
u=
From [78, Theorem 8.3 and Theorem 8.18], it is easy to see that (5.4.27) has a unique hyperbolic limit cycle for _3/4 1/ 3 < 8 < O. Furthermore, noting that around the only critical point (z, u) = (0,0),
304
Chapter 5.
+ {;Z
0, such that: (i) the system (5.4.25) has a unique hyperbolic invariant torus for -3/41 / 3 + Ao < {; < {;(A), < IAI < Ao, and no invariant torus for 1 3 {; ::; -3/4 / - Ao, < IAI < Ao or {; ~ {;(A), 0 < IAI < Ao; (ii) (5.4.25) has a unique periodic orbit for 0 < IAI < Ao and {; bounded, which is hyperbolic for {; =1= {;(A) and stable (resp. unstable) for A({; - {;(A)) < (resp. ~ 0). Z3
(_Z)3 -
{;z for {; ::;
°
°
° °
5.5.
Uniqueness and Stability of Bifurcated Periodic Orbits
From now on, we consider the homoclinic and heteroclinic bifurcation problems and the accompanying chaotic phenomena. In the remaining part of this chapter, we confine ourselves to study the homoclinic bifurcation to a hyperbolic equilibrium. And, in this section, under generic assumptions, we study the uniqueness and stability problem (i.e. the dimension problem of the stable and unsta~man ifolds) of the periodic orbits produced from homo clinic bifurcations. Consider the system i = F(z,o:),
z
E
lRm +n ,
0:
E
lRk,
(5.5.1)
where F: U x V is CT (r ~ 2 is adequate) for some open set U c lRm +n and some neighborhood V C lRk of the origin. Assume that (5.5.1) satisfies the following conditions. (H1) The origin z = is a hyperbolic equilibrium of system (5.5.1), DzF(O, o:) has m eigenvalues )'1," . ,Am with negative real parts, and --t lRm +n
°
5.5.
Uniqueness and Stability of Bifurcated Periodic Orbits
305
n eigenvalues J.tl, ... ,J.tn with positive real parts, in which Al and J.tl are real numbers. (H2) (5.5.1) has a homoclinic orbit Lo = {z*(t) : t E IR} connecting 0 to itself when a = O. Moreover, e-
= t-++oo lim i*(t)/Ji*(t)J,
e+
= t-+-oo lim i*(t)/Ji*(t)J
are unit eigenvectors corresponding to AI, J.tl, respectively. (H3) Al and J.tl are the principal eigenvalues of DzF(O,O), i.e., Re Ai < -A < Al < 0 < J.tl < J.t < Re J.tj for i = 2, ... ,m, j = 2, ... ,n and some constants A and J.t. Choose points p, q E Lo sufficiently close to the equilibrium z = 0, say p = z*(O), q = z*(T). Let W S , W U be the stable and unstable manifolds of 0 respectively, and
W SS = {z(t): W UU = {z (t):
lim eAtz(t) = O},
t-++oo
lim eJLt z (t) = O}
t-+-oo
be the strong stable manifold and the strong unstable manifold respectively. (H4) The nondegeneracy of Lo holds, i.e., codim(TpWu+TpWS) = 1. (H5) e- E ToWs \ ToWsS, e+ E ToW u \ ToW uu . Clearly, (H2) and (H3) imply (H5). If we denote Tz*(t) = Tz*(t) WS + Tz*(t)Wu, then, by the strong A-Lemma (see [22, 32]), (H5) is generically equivalent to
= To W SS EB To WU, lim Tz*(t) = ToWS EB ToWuu, t-++oo lim Tz*(t)
t-+-oo
(5.5.2)
when r > 5. (5.5.2) is called the strong inclination property ([22, 32]). More conveniently, we can rewrite (5.5.2) in the following form: lim Tz*W s = ToWsS EB Span {e+}, lim T z*W U= Span {e-} EB To WUU,
t-+-oo
t-++oo
(5.5.3)
306
Chapter 5.
Bifurcation of Higher Dimensional Systems
where e-, e+, defined in (H2), are the principal stable eigenvector and the principal unstable eigenvector, respectively. (H4), (H5) and (5.5.2) are generic assumptions. Our goal is to show the existence and uniqueness of the homoclinic orbit La and the periodic orbit La' near Lo when a i= 0, and to study the stability of La' if it exists. Theorem 5.5.1 ([201]). Suppose that n = 1 and the hypotheses (HI), (H2) are valid. Then La' is unique, if exists, and cannot coexist with La. Moreover, La' is stable when A* + J.Ll < 0, and La' has an m-dimensional stable manifold and a 2-dimensional unstable manifold when A* + J.Ll > 0, (H3) and the strong inclination property (5.5.2) hold, where A* = max{ReAi, i = 1, ... ,m},0 < lal «: 1. Corollary 5.5.2 ([201]). Assume m = 2, n = 1, Al + J.Ll i= 0 and (HI), (H2) hold. Then La' is stable when A2 + J.Ll < 0 and either Al + J.Ll < 0 or (5.5.2) is not valid; La' has a 2-dimensional stable manifold and a 2-dimensional unstable manifold when A2 < A1, Al + J.Ll > 0 and (5.5.2) is valid. Remark 1. Corollary 5.5.2 partly revises the stability criterion of [180, Th.3.2.12] and [181, ThA.8.1]. Theorem 5.5.3 ([201]). Suppose that the hypotheses (H1)-(H4) and the strong inclination property (5.5.2) hold. Then La' is unique if it exists and cannot coexist with La when 0 < lal «: 1. A~La' has an m-dimensional (resp. (m + 1) -dimensional) stable manifold and an (n + I)-dimensional (resp. n-dimensional) unstable manifold when Al + J.Ll > 0 (resp. Al + J.Ll < 0). In the following, we will first establish the Poincare map P near Lo. Then the stability criteria contained in the above theorems are proved separately. Next, using the Sil'nikov variables, we show the existence and uniqueness of La and La'. After that, we give some further remarks and references.
5.5.
Uniqueness and Stability of Bifurcated Periodic Orbits
307
Now, we want to construct the Poincare map in a neighborhood of the homo clinic orbit L o, which will be the composition of two maps: one given by an essentially linear flow near the equilibrium point z = and the other given by an essentially rigid motion along the homo clinic orbit outside a neighborhood of the equilibrium point. The whole construction will be accomplished in the following steps. Firstly, we may as well assume that the multiplicities of the eigenvalues of DzF(O,O) are one (for other cases the following proof only needs some modifications). And then, by utilizing a linear transformation, we can transform system (5.5.1) into the following system,
°
x= iJ
=
A(a)x + Fl(x, y, a), B(a)y + F2 (x, y, a),
where (x, y) E IRm x IRn , Fl and F2 are C r -
l
(5.5.4)
with
A and B are Jordan blocks such that all the diagonal entries have either negative or positive real parts, and e- and e+ are the directions of the negative xl-axis and the Yl-axis respectively. We further simplify (5.5.1) locally in some neighborhood Uo of 0 by using local stable and unstable manifolds as local coordinates and get
x = A(a)x + fl(x, y, a),
iJ
= B(a)y + h(x, y, a),
(5.5.5)
where (x, y) E Uo c IRm x IRn , a E V, ft, h E cr-I, ft(O, y, a) h(x, 0, a) = 0, fl' h = O(lxl 2 + IYI2). We choose the following two rectangles as the cross-sections: Xl = E, IX*I < E, Iyl < E}, Sl(E*) = {(x, y): Ixl < E*, Yl = E, ly*1 < E*},
So
= {(x,y):
where x* = (X2,' .. , x m), y* so that So, Sl C Uo.
= (Y2,"" Yn), E and E* are small enough
Chapter 5.
308
Bifurcation of Higher Dimensional Systems
Denote the flow of (5.5.5) by ¢(t, Xo, YO) = (x(t, Xo, yo), y(t, Xo, Yo)) and let T = T(xo, Yo) be defined by Yl(T, Xo, Yo) = c. Then define the map
Po:
5~
51
--t
(xo, Yo)
r-t
(x(T, xo, yo), y(T, xo, Yo)),
(5.5.6)
where 50 c 56 = {(x,y) E 50: Yl > o} is the domain of Po· By continuity, for sufficiently small c* with 0< c* < c and (xo, Yo) E 5~ = 5 1 (c*), there is a unique T(Xo,YO) such that (X(T,Xo,yO), y(T,xo,YO)) E 50. Thus we can define a map along Lo away from the origin z = as follows:
°
PI: 5~
--t
(xo,Yo)
50 r-t
(X(T,xo,YO),y(T,XO, yo)).
(5.5.7)
Taking 50 small enough such that Po(50) C 5~ and compounding the maps defined by (5.5.6) and (5.5.7), we get the Poincare map P
= PI 0 Po:
5~
--t
50.
Usually, P is so complicated that it is almost impossible to obtain the fixed point and consider the stability of the fixed point. But we can use the approximate Poincare map p L to replace the real map P, where p L = pF 0 PrJ':
Pi':
5~
--t
(xo, Yo)
pf: 5~
--t
(xo, Yo)
51 (eAT xo, e BT yo),
r-t
50 r-t
q(a)
+ D(xo, y~l,
(5.5.9)
where D = DPI = (d ij )(m+n-l)x(m+n-l), q(a) = (X(T(O,O),O,O), y( T(O, 0), 0, 0)). By [180, Proposition 3.2.8], Ip-pLI = O(c 2 ), IDP-DpLI = O(c 2 ). To consider the dimensions of the stable and unstable manifolds of
5.5.
Uniqueness and Stability of Bifurcated Periodic Orbits
309
the fixed point (x, fj) E So of P, it suffices to consider the numbers of the eigenvalues of DpL(x, fj) with quantities larger than 1 and those smaller than 1 respectively for E > 0 and lal > 0 sufficiently small. For convenience in the following, we give an explicit expression of Pt. We assume that A2, ... , Am, IL2, ... ,ILn are all real numbers to simplify the notations. Otherwise the following proof only needs a slight modification. \ -1 -1 W h D enot e uA = E-1-Yl = e -P,IT , (3i = - "iILl , Ii = - ILiILl. eave A{31 - Afh A{3m - A'Y2 - A'Yn)T p,oL(-X, Y-) -_ ( ELl. U ,X2 U , . . . , XmLl. ,Y2Ll. , ... , Yn .
(5.5.10)
Notice that PO,P1,Pt, pf, p,pL are all CT dependent on a and CT-l on E, and (x*(a),fj(a)) -+ 0 as a -+ o. If the multiplicities of the eigenvalues of DzF(O, 0) are not all unity, then
A
=
(~11J,
B
=
(~l ~J.
In this case, formula (5.5.10) is essentially the same, only with a slightly more complicated expression. Now we can prove the stability criterion of Theorem 5.5.1. Assume
La> exists, i.e., there exists (x, fj) E So, a fixed point of map P. Now consider the eigenvalues of the derivative DpL(x, fj) of pL. By (5.5.9), (5.5.10), and n = 1, we have
DpL(x,fjI) = D
( ~{32 ~ o
~ (32~\~:~~_1) :
0··· ~{3m {3mclxm~{3m-l
d12~{32 (
: ::
: ...:
d22~{32
:
PI)
dlm~{3m ... d2m~{3m P 2 ...
...
:
:'
(5.5.11)
dm2~fh ... dmm~{3m Pm
where Pj
=
djl{31~{31-1 +l:~2dji{3iE-lxi~{3i-l for j
=
1, ... ,m.
Chapter 5.
310
Firstly, assume ,\*
+ f..£l < O.
Bifurcation of Higher Dimensional Systems
It follows immediately that
for i = 1, ... , m.
(5.5.12)
Let (3 = -'\*f..£ll, ~1(0:) = ~f3-I. It is easy to check that the characteristic equation det( vI - DpL(x, iiI)) = 0 of matrix DpL(x, iiI) has the following expression: Vm
+ alUlv A m-l A m-l Am + ... + am-luI v + amUI =
0,
(5.5.13)
where ai = ai(O:) is bounded, and am = (_1)m-I{3I~f31+··+f3m-mf3+m-l. det D i= o.
Lemma 5.5.4. Denote by VI(O:),··· ,vm(o:) the roots of (5.5.13). Then Vi(O:) is C r - 2 with respect to 0:, Vi(O:) i= 0 for 10:1 i= 0 small enough, and Vi(O:) - t 0 as 0: - t 0 for i = 1, ... ,m. Proof. Let v
=
~1(O:)W.
Then (5.5.13) becomes (5.5.14)
Assume that WI(O:), ... , wm(o:) are the m roots of (5.5.14). From a m ( 0:) i= 0 for 10:1 i= 0 small enough, it follows that each Wi( 0:) i= 0 for 0: '# 0, and each Wi(O:) is bounded for 10:1 small enough. Noticing that fiI(O:) - t 0 as 0: - t 0, we have ~1(0:) - t 0 as 0: - t 0 for fixed E > o. Consequently, each Vi(O:) - t 0 as 0: - t o. The fact that Vi(O:) is C r - 2 with respect to 0: comes from that D pL is cr-2 with respect to (x, YI) and that DpL and (x, iiI) are C r with respect to 0:. 0 The stability criterion of Theorem 5.5.1 as ,\* + f..£l < O~ immediate consequence of Lemma 5.5.4. Next assume ,\* + f..£l > O. Now we have 0 < {3 < 1, and by (H3) {3
= {31 < f3z :S {33 :S . . . :S {3m-
Let M = ~f3-I, ~2 = ~f3. It is easy to see that the eigenvalues of DpL(x, yd are determined by the equation det (vI - DpL(x,
yd) = 0,
5.5.
Uniqueness and Stability of Bifurcated Periodic Orbits
311
which now has the following form,
vm + (alMl + bl ti 2)V m- l + ... + (am-lMm- l A km- 1) A m-2 + am M UA i32+···+i3m -- 0 , + bm-l U2 u2 V where al = dml i3l, am = (_I)m- l i3l det D
Ml
rv M as a Mi = O( M) as a
and k i
> 0 for
--+ --+
(5.5.15)
i= 0,
0, 0 for i = 2, ... , m,
(5.5.16)
i = 2, ... , m - 1.
Lemma 5.5.5. d ml are small enough.
i= 0 when (H3)
and (5.5.2) are valid and
lal, E
Proof. Let us assume dml(a, E) = 0 when a = 0, Eis small enough. We show this will lead to a contradiction. Using (5.5.9), we see that the Yl component of pf(xo, Yo) is Yl = qm(O) + dmlXl = 0 when Xo = (Xl, 0, ... ,0) and a = o. It means that
PlL(xO, Yo) C W S
when a
= 0,
or, equivalently,
{(x,yd: IXII < E,X2 = ... = Xm = O,Yl = E} C (pf)-lW S , a = Since
IH - pfl =
o.
O( E2) by [180, Prop.3.2. 7], we see that
{(X,E): IXII <E,X2=O(E2), ... ,Xm=O(E2)}nPl-lWsi=0. (5.5.17) On the other hand, by the strong inclination property (5.5.2)1 or (5.5.3h ,we must have x~+· ·+x~ »xI for any (x, E) E Pl-lws when Eis small enough. It contradicts (5.5.17). It follows that dml(O, E) i= 0 for E small enough which in turn implies that the lemma is true. 0 Lemma 5.5.6. Let vi(a), i Then
vl(a), ... , vm-l(a)
--+
= 1,···
,m, be the roots of (5.5.15).
0, vm(a)
if the subscripts are arranged properly.
--+ 00
as a
--+
0
312
Chapter 5.
= ~2W
Proof. Setting v
Bifurcation of Higher Dimensional Systems
and substituting it into (5.5.15), we get
+ (alMl + bl~2)M-lwm-l + ... + (am-1Mm- 1 + bm_l~~m-l )M-1w + am~() = 0,
~2M-lwm
(5.5.18)
where e = L:~2(,6i - ,6). Denote iii = lima--->o aiMiM-1 for i = 1, ... , m - 1, and iim = lima--->o am~(). When 0; -+ 0, (5.5.18) has the following limit expresSIOn:
m-l alw
+ ... + am-lw + am - =
0.
(5.5.19)
Since iiI i= 0 by (5.5.16) and Lemma 5.5.5, (5.5.19) has (m - 1) complex roots, say, WI, ... , Wm-l. Then (5.5.18) has m - 1 roots WI ( 0;), ... , Wm-l (0;) satisfying Wi ( 0;)
-+
Wi
as 0;
Let Vi(o;) = ~2Wi(0;) for i following form,
-+
0
for i = 1, ... , m - 1.
= 1, .... m
- 1. Rewrite (5.5.15) in the
(5.5.20) Comparing the coefficients of (5.5.15) and (5.5.20), we get Vm + ~2( WI + ... + wm-t) = -a1M1 - bl~2' It means that Vm is equivalent to -aIM = -dml,6l~/3-1. Then the lemma follows from the fact that ~2 -+ 0, M -+ 00 as 0; -+ O. 0 Now we have completed the proof of the stability criteriongiven in Theorem 5.5.1. In order to prove Corollary 5.5.2, it only needs to consider the case Al + ILl > 0, A2 + ILl < 0, and the strong inclination property is not valid which means ,61 < 1, ,62 > 1 and d21 = O. Due to (5.5.11), we can easily show that the eigenvalues of DpL vanish as 0; -+ O. To prove the stability criterion under the conditions of Theorem 5.5.3, it suffices to prove that the stable manifold of La' is at least m-dimensional if Al + ILl > 0, and at least (m + I)-dimensional if
5.5.
Uniqueness and Stability of Bifurcated Periodic Orbits
313
A1 + /11 < O. In fact, if A1 + /11 > 0 then the transformation t --t -t will reverse its sign, and the periodic orbit La' of the new system will have a stable manifold of at least (n + 1) dimensions, then by the dimension theorem it follows that La' of the original system has an (n + 1)-dimensional unstable manifold when A1 + /11 > O. In the case A1 + /11 < 0, the theorem can be proved in a similar way.
. D = (D1 D2) ' where Db D2, D3, D4 are m x m, m x Rewnte D3 D4
(n - 1), (n -1) x m and (n - 1) x (n -1) matrices respectively, and Dk = (a~j) for k = 1,2,3,4. A simple computation shows that
o ...
0
/1f32 . . .
0
o o
DpL(X*, y) = D O · .. /1f3m (3mC1xm/1f3m-1 0 o 0 12c1Y2/11'2- 1 /11'2...
o o 0 0
for k = 1,3, j = 1, ... , m - 1, and b~j = a~j/1 1'j+1 for k = 2,4. Denote Mk
=
(~11 ~~),
where
D~
=
(a~l"'" a~,n-1)'
D31 =
... , a n3)T - 1 ,! . The following lemma plays a crucial role in the proof of Theorem 5.5.3. 3 ( an'
Lemma 5.5.7. If n
>1
and (5.5.2) holds, then det Mm
i= o.
314
Chapter 5.
Bifurcation of Higher Dimensional Systems
Proof. Suppose that det Mm = O. Then we can deduce a contradiction. It suffices to do so for a = 0 by the continuity. det Mm = 0
= (6, ... , ~n)
implies that there exists a non-zero vector ~ Mm~ = O. Let p = La n So, q = LOnSl(E)"
(Yl,'" ,Yn), for i
=
2, ...
x = (Xl, ... , xm), Y = (Yb .. . , Yn),
,m,
Yl = E, Yj =
~j
for j
=
2, ...
+ L a;-l,k-l~k
x = (Xb""Xm), Y = where Xl
= 6, Xi = 0
,n, and Xl = E,
n
Xj = a}-16
such that
for j
= 2, ... , m,
k=2
Yi = 0 If we take
Si, and D
I~I
[~*]
for i = 1, ...
,n.
i= 0 small enough,
[~*].
then we have (x, y) E So, (x, y) E
Since detD
i=
0, we get
Ix*1 i=
i= 0,
we get
(x,y) E TpWs and (x,y) E Span{e-}. From
I~I
O. Then
(x, y) E (TqW S \ TqW SS ) EB TqW U and
(x, y) E TqW SS EB Span(e+). Now we can easily see that it contradicts the strong inclination property (5.5.3)1, This is because that, according to (5.5.3)1 and the local coordinates used in system (5.5.5), D-l should map e- into e+ and TpW s into ToW sS EB Span{e+} for a = O. 0 Lemma 5.5.B. Suppose that )'1 + J.Ll > 0, n > 1, and (5.5.2ris valid. Then det(vI - DPL(x,y)) = 0 can be expressed as
(5.5.21)
where ai is bounded for i = 1, ... ,m+n-1 as a
--t
0, 0 < -ai < -an,
5.5.
Uniqueness and Stability of Bifurcated Periodic Orbits
for i = 1, ... , n - 2, a n-1 = L:;']=2/j, an = a n-1 + /31 - 1, infinitesimal when a -+ 0, (}i > 0 for i = 1, ... ,m-1, (}m-1 and an i= 0, am +n-1 i= 0 when lal is small enough.
315 f1
is some
=
~J!=2
/3j,
Proof. We only show that an i= 0 and a m +n-1 i= 0, the rest being trivial. Using the property that if any two columns are proportional then the determinant vanishes, we can easily see that a m +n -1 = ( -1 )m-1 /31 det D i= o. Noting that Xjt::.f3j-f31 -+ 0 as a -+ 0, we obtain similarly an = (_1)n
(~::
:tll ... :t:~:)
A~l a~~l 1 ... a~_l n-1 -+
(-1)n/31 det Mm
as a
-+
0,
where
From Lemma 5.5.7, an
i= 0 holds for
o
lal small enough.
Lemma 5.5.9. Let n > 1 and (5.5.2) hold. Then det D4
i= O.
Proof. Assume det D4 = O. Then there is a vector (xo, Yo) E Si n Tq W U with Xo = 0, Yo i= 0 such that D 4 yo = O. Consequently,
pf(xo, Yo) n TpW uU = D(xo, y~f n TpW uU = {O} when a
= O.
On the other hand, by Lemma 5.5.7, the matrix
( DD24 ) has rank (n - 1), which means the Y1 component of ph xo, Yo) is not zero. It contradicts the strong inclination property (5.5.3h. 0 Lemma 5.5.10. Yl 1 li1i I -+ 0 as a the same as in (5.5.10).
-+
0 for 2 ~ j ~ n, where
Yi is
316
Chapter 5.
Proof. When It follows that
lal
IYjle JLjT
From T(a)
-t
IYjlYl 1
« 1, we have IYj(T,x*,y)1 «E for 2 ::; j ::; n. «
=
E
+00 as a
«
Bifurcation of Higher Dimensional Systems
-t
e(JLI-JLj)T
Yle JL1T
a,
we get
-t
a as a
for
-t
2::; j
::; n.
a for 2::; j
::; n.
o Lemma 5.5.11. If Al + J..ll < a, n > 1, and (5.5.2) is true, then det(vI - DpL(x,y)) = a can be expressed as v m +n - 1 + blb.aIVm+n-2
+ . . . + bm+n-I
A U
+ ... + bn_lb.O'n-lvm + bnb.O'n-IEflvm-l
a n _1 El0m
--
a,
(5.5.22)
where bi is bounded as a - t a, ai has the same property as in Lemma 5.5.8 for i = 1, ... ,n -1, El is some infinitesimal when a - t a, (}i > a for i = 1, ... ,m, (}m = 'L,j=l {3j - 1, and bn - 1 =1= a, bm +n - 1 =1= a when lal is small enough.
Proof. Still we only show bn - 1 =1= a and bm +n - 1 =1= a. Now b.tJj-l is an infinitesimal as a - t a for j = 1, ... , m. By Lemma 5.5.1a, it is easy to see that yjb.'Yj-l/b.'Yj = EYjy:;l - t a as a - t a. It follows that the main part of bn - 1 is (_l)n-l det D 4 , which produces bn - 1 =1= a when lal is small enough. The proof of bm +n - 1 =1= a is similar to that of am+n-l =1= a in Lemma 5.5.8. 0 We are now in a position to complete the proof of the s~ity criterion in Theorem 5.5.3. By Theorem 5.5.1, it suffices to consider just the case n > 1. First assume Al + J..ll > a. We show that the dimension of the stable manifold of La' is at least m. Let () = minl::;k::;m-l {k-1(}d, E2 = Ef, v = E2W' Then equation (5.5.21) becomes A U
-a nE nO W m+n-l 1
+ alU -a A
n
+0'1
El(n-l)O W m+n-2 +
•.•
+an-1b. -O'n+O'n-IEfwm + anw m - 1 + an+lEfl-OWm-2 Om_I-(m-l)O + am+n-2ElOm_2-(m-2)O w am+n-lEl = .
+
a
+ ...
(5.5.23)
5.5.
Uniqueness and Stability of Bifurcated Periodic Orbits
317
Since t:. -an ---t 0, t:. -an+ak ---t 0, E1 ---t 0 as 0: ---t 0 for k = 2, ... ,n 1, and fh - kO = k(k-10k - 0) 2: 0, (5.5.23) has the following limit expression, (5.5.24) -- l'1ma -+o a n +kE2-k+(h/() £or k -- 1, ... , m - 1. h an+k w h en 0: ---t 0 ,were Denote by WI, ... , Wm -1 the (m-1) roots of (5.5.24), then (5.5.23) has (m - 1) roots WI, ... , W m -1 satisfying Wi ---t Wi when 0: ---t O. Consequently, (5.5.21) has (m - 1) roots Vi = E2Wi with IVil < 1, i = 1, ... , m - 1, when 10:1 is small enough. This means that the dimension of the stable manifold of La' is at least m when >'1 + /11 > O. Now assume >'1 + /11 < O. Under the transformation v = E2W, (5.5.22) takes the following form,
(5.5.25)
And (5.5.25) has the limit expression -
b n - 1w
m
m-1 + bnw + ... + -bm +n - 2w + -b m +n - 1 =
0,
(5.5.26)
-b n+j -- l'1ma-+O bn+jE1()Hl-(j+1)() £or J. -- - 1 , ... , m - 1 an d uo {) -- 0 . h were In the same way as done for the case >'1 + /11 > 0 we can show that the dimension of the stable manifold of La' is at least (m + 1). Finally we need to prove the existence and uniqueness properties. For this, we use (5.5.10) to transform the variables (x, fj) E S; of P into the slightly modified Sil'nikov variable 0 = (x*, S, Y*), where s = 1 -. "Ij ,J. -- 2 , ... , n, Y1 - -- ES. uA -_ exp ( -/11 T) ,Y* -- (Y2"'" Yn1) , Yj1 -- YJS Consider the function (0,0:) = P(x*,fj) - (x*,y). Obviously, the zero point 0(0:) of (',0:) with s(o:) > 0 (resp. = 0) corresponds to a periodic orbit La> (resp. homo clinic orbit La) near Lo.
318
Chapter 5.
Bifurcation of Higher Dimensional Systems
We may assume Al + J..ll < 0, i.e., /31 > 1, otherwise let t --t -to By [22,32J or the following proof, we see (0, a) can be C 1 extended to the neighborhood of (0, a) = (0,0), and (0,0) = 0 corresponds to Lo. From (5.5.9),(5.5.10) and a simple computation, we get ~:(o, 0) = diag( -1, -E) for n = 1, and ~:(o, 0) = diag (-I, -E, D 4 ) for n > 1, where I is an (m - 1) x (m - 1) unit matrix. Then the existence and uniqueness of homoclinic and periodic orbits follow from Lemma 5.5.9 and the implicit function theorem. We end this section by making some further remarks and introducing some more references. Note that we can generically classify the homo clinic orbits satisfying (H5) and the strong inclination property into two classes: nontwisted and twisted. For the nontwisted one, the unstable manifold (or the stable manifold) undergoes an even number of half-twists before it joints itself along the strong unstable manifold (or the strong stable manifold). An analogous description but with an odd number of half-twists applies to the twisted homoclinic orbit. Equivalently, we say Lo is nontwisted (resp. twisted) if e- and e+ point to the same side (resp. opposite sides) of Tz*(s) and Tz*(t) for -s, t > 0 large enough (c.f. [22, 32]). Clearly, twisted homo clinic orbits can occur only in a space with dimension?: 3. Assume Al + J..ll = o. If Lo is nontwisted, then, as in planar flows, either a couple of periodic orbits or the coexistence of a periodic orbit and a homo clinic orbit may occur. If Lo is twisted, then a 2-homoclinic orbit or a 2-periodic orbit may be produced frQ[!l Lo. Here, an N-periodic orbit is a periodic orbit which is contained in a small tubular neighborhood U of Lo and has a winding number N in U. We can similarly define an N-homoclinic orbit. In particular, Lo itself is a 1-homoclinic orbit. For details, see [22J. An analogous definition applies to the twisted or nontwisted heteroclinic loop consisting of two heteroclinic orbits. In this case, the bifurcation pattern is much more varied (cf. [23,34]).
5.6.
Chaotic Dynamics Bifurcated by Symmetric Homociinic Loops
5.6.
319
Chaotic Dynamics Bifurcated by Symmetric Homoclinic Loops
From the analyses in the last section, we see that when there is only one orbit La homo clinic to the saddle 0, the unique possibility is the periodic orbit bifurcation arising from the break of the homo clinic orbit La, and no chaotic dynamics can occur in the neighborhood of La U {O}. Whereas, if there are a pair of orbits homo clinic to the saddle 0, the situation is completely different. In this case, small perturbation may lead to a homoclinic explosion (see O-explosion in Sec. 1.2): accompanying the break of the two homoclinic orbits, the corresponding Poincare map produces a horseshoe construction. In order to describe the chaotic dynamics and to show how to prove a given system has chaotic behavior, we need some preparations. 5.6.1.
Symbolic dynamics and Smale horseshoe
Symbolic dynamics and the idea of the horseshoe map will play a key role in the study of chaotic dynamical phenomena. Here we only sketch the main ideas and conclusions for the sake of simplicity. For further details, please see, for example, [115, 180, 181, 206J. Let SN = {1, ... , N}, N ~ 2, be a collection of N symbols, L,N = n~-oo S~ be the collection of all bi-infinite symbol sequences for S~ = SN, i.e., a point 8 E L,N iff 8 = {- .. Ln'" 8-18081' .. Sn ... } with Si E SN, i = 0, ±1,'" . The So (with· above) denotes the central symbol of 8. We define
d(
-) = 8, 8
=
la - bl,
Va, bE SN,
~ d(8 n, Sn) i=~oo 21nl'
"18,
d(a, b)
S E L,N.
(5.6.1) (5.6.2)
Then, equipped with the metric (5.6.2), L,N is a compact, totally disconnected and perfect space. Clearly, L,N is homeomorphic to a Cantor set. Now, we can define a shift map of L,N onto itself, denoted
Chapter 5.
320
Bifurcation of Higher Dimensional Systems
by cr, as follows:
cr(s) == {- .. s-n'" 8-1S081 ... sn'" } for s Si+l'
= {"'8-n'''8-180S1'''Sn'''}, or, abbreviated, by [cr(S)]i The map cr is often called a shift on N symbols.
=
Proposition 5.6.1. The shift map cr has the following properties: i) cr has a countable infinity of periodic orbits with all natural numbers as their periods, and the set of all the periodic orbits is dense in L,N; ii) cr has an uncountable infinity of nonperiodic orbits; iii) cr has a dense orbit; iv) for any nonempty open sets U, V C L,N, there is a k = k(U, V) such that crn(U) n V i= 0 for n > k. Remark 1. Property iv) is referred to as the topological mixing property which implies the property iii), and the latter is often referred to as the topological transitivity. In 1965, S. Smale constructed a horseshoe map f with very complicated behavior on its invariant set (also a nonwandering set) A. Precisely speaking, the limitation of f on A is topologically conjugate to cr : L,2 - t L,2 which means that there is a homeomorphism ¢ : A - t L,2, such that ¢ 0 f(p) = cr 0 ¢(p) for any pEA. Now let us introduce the map f. Let I = [0, 1] x [0, 1], 0 < A < 1/2, f.L > 2, ...~
Ho = {(x,y) Em?: HI = {(x,y) Em?:
Va
=
{(x,y) Em?:
Vi = {(x, y) E IR2:
O:S x:S 1,0:S y:S 1/f.L}, O:S x:S 1, 1-1/f.L:S y:S I}, O:S x:S A,O:S y:S I}, 1 - A :S x :S 1,0 :S y :S I}.
Consider a map f : I - t IR2 which contracts in the x-direction, expands in the y-direction, and satisfies f(Hi) = Vi, i = 0,1, f(1) nI = VouVi and f(x, y) = (AX, f.Ly) for (x, y) E H o, f(x, y) = (I-AX, f.L-f.LY) for (x, y) E HI.
5.6.
Chaotic Dynamics Bifurcated by Symmetric Homoc1inic Loops
321
We call Hi (resp. Vi) the horizontal (resp. vertical) rectangle. Obviously, f folds I around such that f(I) takes the horseshoe-like shape. Moreover, f maps the horizontal (resp. vertical) edges of Hi onto the horizontal (resp. vertical) edges of Vi, f- 1 (Vi) = Hi, i = 0,1, f-I(I) = Ho U HI, and for any horizontal (resp. vertical) rectangle H (resp. V), f-I(H) n I (resp. f(V) n I) consists of precisely two horizontal (resp. vertical) rectangles. Denote 00
A=
n
r(I)·
n=-oo
Then A is the maximal invariant set (nonwandering set) of any S E E 2 , let
n f-n(VsJ, 00
H
=
n=1
f. For
n r(Vs-J. 00
V
=
n=o
It is easy to show that H is a horizontal segment, V is a vertical segment, and H n V consists of a single point x = x( s) E A. We define '1/ >'2 in St. This geometry is shown in Fig. 5.6.2. Since P(+ is an affine mapping, and P(+(O, 1, E) = (b + x + ef..l, E, d + f..l), we see that when f..l = 0, corresponding to the cases d > and d < 0, the
S:
S:
°
5.6.
Chaotic Dynamics Bifurcated by Symmetric Homoc1inic Loops
327
image peS: US;;) exhibits different pictures as illustrated in Fig. 5.6.3. In the case d > 0, r + and r _ are nontwisted, while in the case d < 0, they are twisted. Clearly, no horseshoe behavior can occur when f..£ = 0. However, Fig. 5.6.3 tells us intuitively that horseshoelike dynamics appears when the parameter f..£ is varied in a suitable direction. s+
•
z
r"
•
Fig. 5.6.2
(a) d>
°
(b) d
0. For d < 0, the discussion is similar (the only difference is that f..£ is varied in a reverse direction). By (5.6.7), we see that when -1 « f..£ < 0, the homoclinic orbits break, and the image of peS: US;;) moves in the manner as shown in Fig. 5.6.4. Then, for fixed -1 « f..£ < 0, we can choose two f..£v-vertical strips Vi and V2 with Vi c peS:) and V2 c peS;;) for some f..£v such that the two horizontal edges of each Vi are parallel and sufficiently close to the x-axis, and the two vertical sides have preimages which are
328
Chapter 5.
Bifurcation of Higher Dimensional Systems
vertical segments in S: US;;. Let Hi = P-1(Vi), i = 1,2. It follows from (5.6.5)-(5.6.8) and (31 > 1, (32 < 1 that, for Izl « 1 and b =I- 0, we have
Consequently, we can easily verify that if we take /-Lv ~ Ibldl, 0 ::; /-Lh « 1 and 0 < /-L « 1 in (M1) and (M3) so that 0 ::; /-Lh/-Lv < 1 and o < /-L < 1- /-Lh/-Lv, then Hi is a /-Lh-horizontal strip, and the conditions (M1) and (M3) hold.
Fig. 5.6.4 When b = 0, the above conclusion still holds if we take 0 < /-Lv Then, the following theorem follows from Proposition 5.6.3.
«
l.
Theorem 5.6.4. Suppose that (H1)-(H4) hold and d =I- O. Then there exists /-La > 0 such that, when dz'(O) > 0 (resp. dz'(O) < 0) and /-L E (-/-Lo, O) (resp. /-L E (O,/-Lo)), the map P defined on S:US;; has an invariant Cantor set on which P is topologically conjugate to a shift on two symbols.
5.7.
Saddle-Focus Homociinic Bifurcation. Chaos
329
Remark 6. (HI) and d#-O insure that the map P has a strongly expanding direction and a strongly contracting direction. Remark 7. From the above section, we see d#-O is equivalent to that the strong inclination property is valid. Thus, d#-O is a generic assumption. And, under (HI), it is also generic that r + is tangent to the y-axis at the saddle O. Remark 8. When the dimension is greater than 3, this kind of homoclinic explosion phenomenon (the broken homo clinic orbits lead to the chaotic dynamics) can be discussed in a similar way. But, in this case, we need the version of higher dimensional horseshoe which can be found in [180].
5.7.
Saddle-Focus Homoclinic Bifurcation. Chaos
In this section, we consider the bifurcation and chaotic behavior near a homoclinic orbit .connecting a nonhyperbolic equilibrium point of weak saddle-focus type in 3-dimensional systems. In the case of hyperbolic equilibrium, this kind of dynamics was first studied by Sil'nikov in 1965 ([144]) and so it has become known as the BiZ'nikov phenomenon. And in the nonhyperbolic situation, we refer to it as the weak BiZ 'nikov phenomenon. Consider the 3-dimensional autonomous system
x=
px - wy + j(x, y, z), iJ = wx + py + g(x,y,z), i = AZ + hex, y, z),
(5.7.1)
where j, g, h E CS, and 0(2) at the origin, in which the notation O( n) denotes the terms with order n 2: 2 in its Taylor expansion at the origin. Under the hypotheses that s 2: 2, A > -p > 0 and (5.7.1) has an orbit r homoclinic to the equilibrium 0(0,0,0), it is shown that there exists chaotic dynamics near r. Precisely speaking, the Poincare map defined by the orbits near r possesses a countable infinity of horseshoes (see [35,144,180,181]). We now show that there also exists
Chapter 5.
330
Bifurcation of Higher Dimensional Systems
°
chaotic behavior in the neighborhood of r even when p = which means the origin is nonhyperbolic. In fact, accompanying the generalized Hopf bifurcation, a new variety of homo clinic and heteroclinic orbits and bifurcation phenomenon will appear, and the structure of the corresponding Poincare map will become more complicated. To describe the problem more precisely, we assume that (5.7.1) satisfies the following conditions. (H1) p = 0, A > 0, W > 0, and for the confined system on (x, y)plane, 0(0,0,0) is a stable fine focus with order k for some k ~ l. (H2) 8 > 2k + 2. (H3) There exists an orbit r homoclinic to O. For simplicity, we only treat the 3-dimensional system. The case of dimensions greater than 3 is discussed in [36].
5.7.1.
Normal form and Poincare map
We will first construct the Poincare map near r. To do this, we must change (5.7.1) into a local normal form to simplify the computation. By the theory of normal form and the results obtained in Chapter 2, we see there is a coordinate transformation such that (5.7.1) becomes
r = akr2k+1 + Rk(r, e, z),
+ b1r2 + ... + bkr2k + 8 k(r, e, z), Z = AZ + z(g1r2 + ... + gkr2k) + Hk(r, e, z),
iJ =
W
(5.7.2)
where ak < 0, Rk, 8 k, Hk E CS! for 81 = 8 - 2k - 1,~Hk = 0(2k+2), 8 k = 0(2k+1) for fixed e, and R k,Hk,8k are 27r-periodic with respect to e. Let we, W U be the local center manifold and unstable manifold of o respectively. Since they are CS!, we see there exists a neighborhood U1 of 0 and a CS! transformation such that the limitations of Rk and Hk in U1 satisfy (5.7.3) Rk(O, e, z) = Hk(r, e, 0) =
°
and the new system is CS! -1.
5.7.
Saddle-Focus Homoc1inic Bifurcation. Chaos
Proposition 5.7.1. Under a suitable C 81 -
331
transformation defined in some neighborhood U C U1 of 0, (5.7.2) satisfies the following conditions: e z) , R k = r2k+l R*(r (5.7.4) k " R'k = O(2k), 8 k = O(2k + 1), Hk = O(2k + 2), R'k E C 8 1- 2 and H k,8k E C8 1 -1. 1
Proof. Denote the right-hand sides of the e, z equations in (5.7.2) by 8(r,e,z) and H(r,e,z), respectively. Let
r=u+N(u,e,z),
(5.7.5)
where N satisfies {)
{)
8 (r, e, z) {)e N (u, e, z)
+ H (r, e, z) {) z N (u, e, z) = Rk (r, e, z), N(O, e, z) = O.
(5.7.6 ) (5.7.7)
Clearly, N E C8 1 -1 and
N = O(2k
+ 1).
(5.7.8)
By the implicit function theorem, (5.7.5) has the solution
u=r+M(r,e,z)
(5.7.9)
with M E C8 1 -1 and M(O, e, z) = O. From (5.7.5) and (5.7.9), we get
M
+ N(r + M,e,z) == 0, M = O(2k + 1).
(5.7.10) (5.7.11)
Differentiating (5.7.9) and (5.7.10), and using (5.7.6) and (5.7.7) we have 'Ii = ak(1 + trM(r, e, z))(u + N)2k+l == aku2k+l + u 2k+l Rk( u, e, z). Thus, (5.7.4) is valid. Owing to (5.7.8) and (5.7.11), the other conclusions can be easily verified. 0
Chapter 5.
332
Bifurcation of Higher Dimensional Systems
Now we begin to construct the Poincare map which will be used to prove the existence of the chaotic dynamics by checking the ConleyMoser conditions.
s+
•
Fig. 5.7.1 Consider two cross-sections
So S1
= =
{(x,y,z): E1::; x::; E, y {(x,y,z): lxi, Iyl ::; 8, z
= 0, Izl ::; E}, = E},
which are transversal to r, located in the neighborhood U, and pass through points A(x,O,O) and B(O,O,E), as shown in Fig. 5.7.1. Here, < 8 « 1 such that each orbit with we take < E1 < X < E « 1, initial point on S:; = {(x, y, z) E So : z > O} must hit S1 before returning to So, and that
°
°
(5.7.12) We construct a map Po : S:; ---t S1 by Po(x,O,z) = (Xo,yo,E), where (xo, Yo, E) is the first intersection point with S1 of the orbit ap~om (x, 0, z). Let T be the time from (x, 0, z) to (xo, Yo, E). By z(T) = E and (5.7.2), we have T
=
(>,-1
+ O(E2)) In(Ez- 1).
(5.7.13)
Change the coordinates (xo, Yo, E) to cylindrical coordinates (To, eo, E), we obtain To = x(l- 2k(,X-1 ak + O(E2))x2kln(Ez-1))-1/2k, (5.7.14) eo = (,X-1 w + O(E2)) In(Ez- 1).
5.7.
Saddle-Focus Homoc1inic Bifurcation. Chaos
333
Then we consider the map PI : Sl -+ So defined by PI (Xo, Yo, E) = (Xl, 0, zt), at which the orbit starting from (x, 0, z) first hit So. Denote
DPl(O, 0, E) = (:
~) .
The Poincare map can now be constructed by taking the composition of the above two maps P == PI 0 Po : S: -+ So with P(x, 0, z) (Xl, 0, zt), where ( Xl) = Zl
(x +To(ecosO TO(a cos 0 + ~sinOo») + O(T~), o + dsmO o) 0
(5.7.15)
and To, 00 are given by (5.7.14). In order to study the geometric structure and chaotic behavior of the map P, we divide S: into a countable infinity of rectangles R l , . .. , Rn, ... , where
Rn = {(x, 0, z)
E
S: : Zn+l ::; Z ::; zn}
and Zn = E exp( -2mI' AW- l ). Let M = max{l, IDI-l, IDI(e2 +d2)-1/2, (a 2 +b2)1/2(e2 +d2)-1/2}, D = ad - be, E = (1- 4akkml'w-lE2k)1/2k. For E > sufficiently small and fixed, we take n large enough such that
°
(5.7.16) M E-(2k+1)
«
1.
(5.7.17)
Due to (5.7.16), for any point (x, 0, z) ERn we have (5.7.18) Denote the upper, lower, left and right boundaries of Rn by h U , hi, ve,v T , respectively. And let h~ = Po(hU ), h~ = poChe), v~ = Po(v T ), v~ = Po(v e). From (5.7.14), we see that the image of Rn under Po has the horseshoe-like (or annulus-like) shape as shown in Fig. 5.7.2.
Chapter 5.
334
Bifurcation of Higher Dimensional Systems
Particularly, we have
= {(r,O,€): h~ = {(r,O,€): v~ = {(r,O,€): v~ = {( r, 0, 10) : h~
0= OlO,€lEo(€l) ::; r::; €Eo(€)}, 0= 011, €lEl(€l) ::; r ::; €El(€)}' OlO ::; 0 ::; 011 , r = €Eo(€)},
OlO::; 0::; 011,r = €lEo(€l)},
where Ei(X) = (1 - 4a kk(n + i)7fW- l X2k(1 + O(€2))tl/2k, Eo(x) = (1-2akkw-lx2k(1+0(€2))0)-1/2k, Oli = 2(n+i)7f(1+0(€2)), 011-0lO = 27f + 0(10 2 ). It is easy to verify that v~ and v~ are helixes with monotonously decreasing polar radius.
h"
vI
Rft
v ' - - -....
hi
Fig. 5.7.2 5.7.2.
Verification of Conley-Moser conditions
In this section, we show that the map P acting on Rn satisfies the Conley-Moser conditions (M1) and (M3) provided (5.7.16) and (5.7.17) are valid, which means P has horseshoe structureorr Rn. Since there are an infinite number of such n for which (5.7.16) and (5.7.17) hold, we see P has a countable infinity of horseshoes. Proposition 5.7.2. For 10 > 0 small enough and n satisfying (5.7.16) and (5.7.17), the inner boundary of P(Rn) intersects the upper boundary of Rn at at least two points, and the preimages of the vertical boundaries of P(Rn) n Rn are contained in the vertical boundaries of Rn.
5.7.
Saddle-Focus Homoclinic Bifurcation. Chaos
335
Proof. On the upper boundary of Rn, Z = Zn. The least polar radius of v~, the inner boundary of Po(Rn), is if = min{EIEe(EI)} > 3EE- I /4. Due to (5.7.12) and (5.7.18), (5.7.19) Moreover, PI is approximately an invertible affine map which is independent of n for E > 0 small enough. And the Poincare map expands the z-direction with a speed close to exp(27r AW- I ) as E -+ O. It follows that P( vi) n hU consists of at least two points when E > 0 small enough and n large enough. Since the vertical boundaries of P(Rn) n Rn belong to the union P(v i ) U P(VT), we see their preimages are contained in the vertical boundaries of Rn. 0 s+
R,. ///////
•
/
/
lL/////////
A.
Fig. 5.7.3 The geometry of P(R) is shown in Fig. 5.7.3. Denote the two vertical strips contained in p(Rn)nRn by Vi, 112, and Hi = P-I(Vi), i = 1,2. Let L j = AjBj be the horizontal edges of HI and H 2 , NI = A~A~, N2 = B~B~, N3 = B~B~, N4 = A~A~ be the vertical edges of VI and 112. We refer to Hi as the horizontal strip for i = 1,2. Before verifying the Conley-Moser conditions, we consider first the differential of P:
Chapter 5.
336
Bifurcation of Higher Dimensional Systems
where
D1 = rox(acosOo + bsinOo ), D2 = roz(a cos 00 + b sin ( 0 ) + e(a sin 00 D3 = rox(ecosOo + dsinO o), D4 = roz(ecosOo + dsinO o) + e(esinOo
-
b cos ( 0 ),
-
dcosO o),
+ 2kqln(Ez- 1))-1-1/2k, roz = qxz- 1r ox , q = -ak>.-l(l + O(E2))x 2k . Set p = (xp, 0, zp), Oo(xp, zp) = Oo(p), ro(xp, zp) = ro(p). e
= >.-l wroz -I,
rox = (1
Proposition 5.7.3. Suppose that and (5.7.17) are valid. Then
ecosOo(p)
E
>0
+ dsinOo(p)
is small enough, (5.7.16) ~
0,
Ie sin Oo(p) - d cos Oo(p) I ~ (e 2 + d2)1/2
(5.7.20) (5.7.21)
!orpEH1 UH2.
R;
Proof. Let L be the lower boundary of S:;, L1 = P 1- 1(L), = 1 P1- (Rn). Since PI is approximately an affine map, we see L1 may be regarded as a straight segment on Sl. Denote by 01 (resp. 0*) the angle bounded by L1 and the x-axis (resp. the polar radius of Po(p)), and by f the distance from Po(p) = P1- 1(P(P)) to L1. Then f ~ azp(p), where ZP(p) is the z-coordinate of P(p) and a is the expanding coefficient of P1- 1 in the z-direction. Obviously, a decreases with E decreasing. Moreover, owing to Zn+1 < zp < Zn and (5.7.14)~get ro(p) ~ x pE-1. It follows that 0* ~ sin 0*
Let
= f/ro(p)
~ ax;l zp(p)E« l.
13 be the expanding coefficient of PI along L1. Then by either = 01 ± 0* ~ 01 or Oo(p) = 7r + 01 ± 0* ~ 7r + 01, and P1(L 1) = L,
Oo(p)
we have (a
b) (c~s (1) ~ (13). sm01 0
ed
5.7.
Saddle-Focus Homoc1inic Bifurcation. Chaos
337
It turns out that
ccosOo(p)
+ dsinOo(p)
~
CCOSOI
+ dsinOl
~
o.
Consequently, (5.7.21) follows from the fact that (csin 00
-
d cos ( 0 )2
+ (ccos 0 + dsin ( 0
0
)2 = c2 + d2.
We are now in a position to give the first main result.
o
Theorem 5.7.4. ([209]) Suppose that (H1)-(H3) are valid. Then, for n sufficiently large, the limitation of P on Rn has an invariant Cantor set An, on which P is topologically conjugate to a shift on two symbols. Proof. By Proposition 5.6.3, it suffices to show that the conditions (M1) and (M3) are valid. We first prove P acting on Rn satisfies (M1) if (5.7.16) and (5.7.17) hold. By the definition of the horizontal strip Hi, P maps HI and H2 homeomorphically onto the vertical strips Vi and 1/2, and maps the horizontal (resp. vertical) edges of Hi onto the horizontal (resp. vertical) edges of Vi. Thus, it suffices to show that there exist J.Lh, J.Lv ~ 0 such that L j (resp. N j ) is a J.Lh-horizontal (resp. J.Lv-vertical) curve, and 0 :S J.LhJ.Lv < l. Let (u,v) (resp. (x,y)) be vectors tangent to L j at p (resp. N j at q). Then they are parallel to the following vectors,
(1)
-1 D4 ) DPp(p) 0 = ~ -1 ( -D3
DPp -l(q)
(~)
=
(Z:) +
+ h.o.t.,
(5.7.22)
h.o.t.,
(5.7.23)
respectively, where ~ = D 1 D 4 -D 2 D 3, the right hand sides of (5.7.22) and (5.7.23) take values at p E L j and P- 1 (q) E P- 1 (Nj ) respectively. Now we define (5.7.24)
Chapter 5.
338
Bifurcation of Higher Dimensional Systems
(5.7.25) where D 2, D3 and D4 take values in HI U H 2· Then max{
M}
0 sufficiently small, n sufficiently large, and (x, 0, z) E Hi, we have r o '" ...c E-l 'ox r '" E- 2k - 1'oz r '" -a k ). -I E2k+l E- 2k - 1z-1 • Due to (5.7.21), we have
D 3 ", (ccosO o + dsinO o )E- 2k - 1,
(5.7.26)
D 4 ", hE). -IW(C2 + d 2)1/2(EztI,
(5.7.27)
+ b2)1/2(Eztl,
(5.7.28)
iD2i '" where h
~
2E). -lw(a 2
1 is a constant. It follows from (5.7.18) that
o < /lh/lv < AE- 1E- 2k Z « 1, 16).w- 1(a 2 + b2)I/2(c2 + d 2)-1/2h- 2.
where A = Now we show that there exists /l satisfying
o < /l < 1 -
(5.7.29)
/lh/lv
such that the condition (M3) is valid. Let (5.7.30) where b. and Di take values in HI U H 2 • Using the expressions of D i , rox, r oz , and (5.7.24)-(5.7.28), it can be verified that .~ A L..l ' "
\
-EA
-1
W
DE-2k-2 Z -1 ,
(5.7.31) (5.7.32)
iD 41 - /lhiD2i '" I D 4i· Then, owing to (5.7.27) and (5.7.17), we obtain
/l
< 8M E- 2k - 1 exp(27r ).w- 1 )
which in turn means that (5.7.29) holds.
«
1,
5.7.
Saddle-Focus Homoc1inic Bifurcation. Chaos
339
Let S~ = {(xp, Yp) : Ixpl :S J.lvIYpl} be the vertical sector at p. For p E Hi, denote (X,y)T = DPp(Xp,Yp)T. Then by J.lvlDll «: max ID21, J.lv1D31 «: min ID41 for E > 0 sufficiently small and n sufficiently large, we have Ix/yl = I(D1xp + D2Yp)/(D3xp + D4Yp)1 + h.o.t. :S (J.lvIDll + ID21)/(ID41- J.lv1 D31) + h.o.t. < 2 max ID21 . max ID41- 1 = J.lv. So, we have shown DPpS~ c Sp(p). By (5.7.30)-(5.7.32) and (5.7.18), it follows that IYp/yl < 2(I D41 - J.lvI D31)-1 < 4 max ID41- 1 < 8 max(ID41 - J.lhID21)-1 < I~I max(ID41 - J.lhI D21)-1
< J.l. Now denote by S; = {(xp, Yp) : IYpl :S J.lhlxpl} the horizontal sector at p, and let (X,y)T = DPp-1(xp,Yp)T. Using J.lhlDll «: maxlD31 and (5.7.32), we get Iy/xl = I( -D3Xp + D 1Yp)/(D 4xp - D2yp) I + h.o.t. :S (I D 31 + J.lhI D d)/(ID41- J.lh1D21) + h.o.t. < 2 max ID31 . max ID41- 1 = J.lh, i.e., DPp-1S;
c SJ-l(p). Moreover, IXp/xl :S 21~1(ID41 - J.lhI D21)-1 < J.l.
Thus, (M3) holds and the theorem is proved.
0
Remark 1. If (Hl)-(H3) hold, then we can show that, for any given integer N 2:: 1, S: contains an invariant Cantor set, AN, on which the Poincare map P is topologically conjugate to a shift on 2N symbols. The strategy attacking this problem is to consider the intersection Ui= 1 P (Rn+i) n Ui=l Rn+i' For further details see [209]. Remark 2. By using the version of subshifts of infinite type and the method in [180] we can show that P moreover, has an invariant set, Aoo , which is homeomorphic to the space of symbol sequences
'E~
= {s = {Si}~_oo : Si is a nonzero integer, ISi+11 > a-1Isil}, where a > 1 is a finite number.
340
Chapter 5.
5.7.3.
Bifurcation of Higher Dimensional Systems
Complicated behavior with Hopf bifurcation
Now we consider the perturbation of system (5.7.1) with p = O. We will show that, accompanying the generalized Hopf bifurcation, the homoclinic and heteroclinic bifurcations and the chaotic behavior will become increasingly complicated. In cylindrical coordinates, the perturbed system has the following generic form:
r=
/-lor
+ ... + /-lk_1r2k-1 + akr2k+1 + Rk(r, e, z, /-lk),
e= w + b1r2 + ... + bkr2k + 8 k(r, e, z, /-lk), Z = AZ + Z(91r2
(5.7.33)
+ ... + 9kr2k) + Hk(r, e, z, /-lk),
where R k , 8 k and Hk satisfy (5.7.3) and the conclusion in Proposition 5.7.1 for fixed /-lk' For conciseness, we define /-lk as the distance from the intersection point B of WU with S1 to L1 = P1-1(L). In view of (5.7.3), we see that, confined to wcnu, system (5.3.33) is reduced to the form r = /-lor + ... + /-lk_1r2k-1 + ak r2k+l + O(r 4k+1), (5.7.34) = w + b1r2 + ... + bkr2k + O(r 2k +1).
e
It follows from the Poincare-Bendixson theorem that (5.7.34), and hence (5.7.33), has a nest of limit cycles Ck-i C ... C C 2 C C 1 for o ::; i ::; k - 1 and
O = /-lo = ... = /-li-1
-1 -1 < -/-li/-li+1 < ... < -/-lk-2/-lk-1 < -/-lk-1 a k-1
« 1.
(5.7.35) Here, the notation Cj+l C Cj denotes that the cycle Cj+l is situated in the interior of Cj . Cj and Cj+l have the opposite stability, aI!ql~\ is stable. In order to estimate the radius of Cj , we need the following two propositions.
Proposition 5.7.5. Suppose that (5.7.35) holds. Then the function F(x) = akxk + /-lk_1Xk-1 + ... + /-lo has exactly (k - i) positive zero points. Proof. By induction over k, the detail being omitted.
D
5.7.
Saddle-Focus Homoc1inic Bifurcation. Cbaos
341
Proposition 5.7.6. If (5.7.35) is valid, then the positive zero points of F( x) are as follows, Xl = -J.tk-lak"l(l + al(J.ti,·· . ,J.tk-l)), X2 = -J.tk-2J.tk"2 l (1 + a2(J.ti,··· ,J.tk-l)), Xk-i = - J.tiJ.t"ii1 (1
where lajl
«
+ ak-i (J.ti, . ..
(5.7.36)
,J.tk-l)),
1, j = 1, ... , k - i.
Proof. We only consider the case i = 0 and verify Xk-3. The others can be treated similarly. First assume J.to = J.tl = J.t2 = 0 and J.t3 i= o. Then F has (k - 3) positive zero points. Let
and v = Xk-3 be the least positive root of G = o. We have v ~ 0 as J.t3 ~ o. From J.t4 i= 0 and the implicit function theorem, there exists a smooth function v = V(J.t3, ... , J.tk-I) with v(O, J.t4,···, J.tk-l) = O. Since
8
-1
-8 v(O, J.t4,· .. ,J.tk-d = -J.t4 , J.t3
we get v = -J.t3J.til(1 + Pk-3(J.t3, ... , J.tk-d), where Pk-3 is smooth and Pk-3(0, J.t4,· .. ,J.tk-d = o. Therefore, we have the expression Xk-3
= -J.t3J.til(1 +
qk-3(J.tO, ... , J.tk-d) + J.t2q2(J.tO, J.tl, J.t2) +J.tiql(J.tO, J.tl) + J.toqo(J.to) == -J.t3J.til(1 + ak-3),
where ak-3 = qk-3 - J.t:;1J.t4(J.t2q2 + J.tlql + J.toqo), qk-3 = Pk-3 as J.to = J.tl = J.t2 = o. Then the estimate lak-31 « 1 follows from (5.7.35) and the smoothness of qi for i = 0,1,2 and k - 3. 0 Denote by rj the polar radius of the limit cycle Cj. Corollary 5.7.7. Suppose that (5.7.35) is valid. Then rl
~ (-J.tk_l ak"1)1/2, r2 ~ (-J.tk-2J.tk"2 l )1/2, ... , rk-i ~ (-J.tiJ.t"iil) 1/2.
342
Chapter 5.
Bifurcation of Higher Dimensional Systems
Proof. Consider the truncated system
r = JLor + ... + JLk_lr2k-l + akr2k+\ iJ = w + bl r2 + ... + bkr2k.
(5.7.37)
By Proposition 5.7.5, system (5.7.37) has exactly (k - i) limit cycles Gk- i C ... C Gi, which are all circles with radii y'xl, ... , v'Xk-i, where Xj is given in (5.7.36). Since Gj is sufficiently close to GJ, their approximate radii follow from (5.7.36). 0 Let WS (G j ) and WU( Gj ) be the stable and unstable manifold of Gj . Now we state our second main result which is concerned in the homoclinic and heteroclinic bifurcation. Theorem 5.7.8. Suppose that (H1)-(H3) and (5.7.35) hold for o ::; i ::; k - 1. Then, in the neighborhood of the origin, system (5.7.33) has exactly (k - i) limit cycles G l , .. . , Gk-i with radii given approximately by Corollary 5.7.7. Moreover, in the parameter space, there exist (k - i) bifurcation surfaces
JLk-l = hl(JLi, ... ,JLk-2,JLk) ~ -akJL~, JLk-2 = h2(JLi, . .. , JLk-3, JLk-l, JLk) ~ -JLk-lJL~,
(5.7.38)
such that, for i ::; j ::; k - 1, WS(G l ) n WU(Gk_j ) contains at least two heteroclinic orbits when IJLjl < Ihk-jl; WS(G l ) n WU(Gl ) consists of at least two homoclinic orbits when 0 < JLk-l < hI; W S( G l ) .r;;;;[ WU( Gk- j ) are tangent to either a heteroclinic orbit for j i= k - 1 or a homoclinic orbit for j = k - 1 when JLj = hk-j. Proof. From the proof of Corollary 5.7.7, we see that the system
(5.7.39)
5.7.
Saddle-Focus Homoc1inic Bifurcation. Chaos
343
has exactly (k - i) periodic orbits C~, ... ,Ck- i and the cylinder Tj
= {(r,O,z):
r
= r; = yIxj, Izl :S c}
C WU(Cj)
intersects the cross-section 51 in a circle 6j = Tj n 51 with radius ri ~ (-J.Lk-la,/ )1/2 for j = 1 and r; ~ (-J.Lk-jJ.L"k2j+1)1/2 for j l. On the other hand, by the proof of Proposition 5.7.3, we have that Ll = Pl-l(L) C WS(C l ) is approximately a straight line segment on 51. Then, due to the definition of J.Lk, 6j is tangent to Ll if and only if r; = J.Lk, which defines the bifurcation functions (5.7.38). Thus the proposition follows from the fact that (5.7.33) is sufficiently close to its truncated system (5.7.39) in U. 0
t=
Now suppose that we have constructed a parameter-dependent Poincare map for (5.7.33) by a process similar to that for (5.7.2), and we use the same notations for these two maps. We see that, compared to those defined by (5.7.2), the new quantities have the following characteristics. The polar radius ro will increase, and roz will decrease. h~, h; are line segments approximately, and v~, v~ are still helixes with monotonously decreasing polar radius but with a slow down in speed. Hence Fig. 5.7.2 is qualitatively unchanged. The above conclusions can be verified simply by the facts that the dynamics is only qualitatively changed in the neighborhood of the cylinder Tl and in its interior when (5.7.35) holds, and that
when rl :S r :S c. The most important thing is that, with the appearence of the limit loop C l and its unstable manifold near T l , we have ro(O) - t rl(O) ~ ri as n - t 00. Consequently, we obtain that ro > rl and rox - t 0 as n - t 00; IDll, ID31 « 1 for n sufficiently large; and there exists m > 0 such that the inner boundary of P(Rn) intersects the upper boundary of Rn at at least two points for all n 2: m.
344
Chapter 5.
Notice that
Bifurcation of Higher Dimensional Systems
ID21 < 2wro(Az)-1y'a2 + b2, ID41 wro(AZ)-1y'c2 + d2, f"V
Ll = wroroxD(Azt1, and that the proof of Proposition 5.7.3 is still valid for /-Lk « ri. Then, there is an N ~ m such that the proof of Theorem 5.7.4 is still true for n ~ N if we define /-Lh and /-Lv still by (5.7.24) and (5.7.25) respectively, but /-L by
The verification will be easier if we realize /-Lh/-Lv = O( r oxz), /-L = O( z) and /-L = O(rox) in this case. We construct two /-Lh-horizontal strips H 2i +1, H 2i +2 C RN+i as above, such that their images under P are two /-Lv-vertical strips N +j - Un=N V2i+1, TT V2i+2 C X j =
TT
1D
.!Ln·
Then the following main result can be shown in a similar way as for Theorem 5.7.4.
Theorem 5.7.9. Suppose that (Hl)-(H3) and (5.7.35) are true, E > 0 and -ak/-L~/-Lk~l are small enough. Then Xj contains an invariant Cantor set Aj with Aj C Aj+1 for j = 1,2, ... , acting on which the Poincare map P is topologically conjugate to a shift on 2j symbols. Remark 3. Let Aoo = UY;=l Aj . Then Aoo is the maximal invariant set of P on Xoo = U~NRn. By analogy with the proof of ProposittOii5.6.2 (cf. [180], Th.4.1.3 or [206],Th.4.4.1), we can show that, confined on Aoo , P is topologically conjugate to a shift on an infinite number of symbols if the assumptions of the above theorem are valid. Remark 4. By Sec. 5.6, Remark 4, the invariant sets An, Aj and Aoo , obtained in Theorems 5.7.4, 5.7.9 and Remarks 2 and 3, are all hyperbolic. Remark 5. If, instead of C 1 , the largest limit cycle produced from the generalized Hopf bifurcation is C j (in this case we must
5.S.
Period-Doubling Bifurcation in Weak Sil'nikov Phenomenon
345
have /Lk-j+l, ... , /Lk-l SO, /Lk-j > 0) for j > 1, then, almost the same results as those given in the above two theorems can be deduced in a similar way. And, it is easy to see that, if no limit cycles are produced from the generalized Hopf bifurcation, then Theorem 5.7.4 is valid. This case only can occur when /Li S 0 for i = 0, ... ,k - 1.
5.8.
Period-Doubling Bifurcation in Weak Sil'nikov Phenomenon
We have seen from Sec. 5.7 how complicated dynamics .occurs in the neighborhood of an orbit homoclinic to a saddle-focus, which is either hyperbolic or nonhyperbolic. The corresponding Poincare map has a countable infinity of horseshoes, and each horseshoe contains infinite periodic orbits including those with arbitrarily high periods. However, with the break of the homoclinic orbit but without the occurence of the generalized Hopf bifurcation, the dynamics becomes simpler and simpler: first there are finitely many horseshoes, then no horseshoe at all. Consequently, only a few periodic orbits may survive in the neighborhood of the original homo clinic orbit. Now a natural problem arises. What kind of phenomenon happens which leads to the complicated horseshoe structure when these periodic orbits approach the original homo clinic orbit in a reversed process? In this section, we will reveal some facts to help one to get a comparatively intuitive understanding of how this chaotic behavior is created. In fact, we will describe explicitly how a given system follows a countable infinity of saddle-node bifurcations and period-doubling bifurcations so that it behaves more and more chaotic. For the case of hyperbolic saddle-focus, Glendining and Sparrow have given an insightful exposition in this regard with arguments combining theoretical, numerical, and intuitive ideas in [52], which, emphasizing understanding over rigor, considers the perturbation of a single parameter. Almost at the same time, [50] studies the same Sil'nikov phenomenon deeply in a two-parameter space, but they used
Chapter 5.
346
Bifurcation of Higher Dimensional Systems
the version of a countable set of tangent (i.e., saddle-node) bifurcations, followed by period-doubling bifurcations and cusp bifurcations with bistability and hysteresis phenomena. In this book, we restrict ourselves to considering the bifurcation process and chaotic mechanism in the framework of the weak Sil'nikov phenomenon. The following results can be found in [208]. We consider the perturbed system of (5.7.2):
r = akr2k+1 + Rk(r, e, z, IL),
e=
+ bl r2 + ... + bkr2k + 8 k(r, e, z, IL), z = )'z + z(glr 2 + ... + gkr2k) + Hk(r, e, z, IL),
where ak
w
(5.8.1)
< 0, Hk = O(2k + 2), 8 k = O(2k + 1), Hk(r, e, 0, 1L)lu
= 0,
(5.8.2) (5.8.3)
R'k = 0(2k), R'k, H k, 8 k E C 2. We choose parameter IL so that the hypotheses (HI) and (H3) given in Sec. 5.7 are valid for IL = 0, and the Poincare map P : S6 --t So has the form
P(x, 0, z) = (Xl, 0, zd, ( Xl) = (x+elL+ro(acoseo+.bsine o)) +0(2), Zl IL + ro(ccoseo + dsme o)
(5.8.4)
where we assume the homo clinic orbit r intersects So and Sl at A(x, 0, 0) and B(O, 0, E) respectively. The term 0(2) denotes t~ order infinitesimal with respect to ro and IL. By (5.7.14) and (5.7.15) we have
ro = x(1 - 2k().-lak + 0(E2))x2k In(EZ- l ))-1/2k, eo = ().-lw + 0(E2)) In(Ez- l ).
(5.8.5)
It is the purpose of the present section to show that, accompanying the three countable sets of saddle-node bifurcations, period-doubling bifurcations and 2-homoclinic bifurcations (see the definition given in
5.B.
Period-Doubling Bifurcation in Weak Sil'nikov Phenomenon
347
Sec. 5.5), the number and periods of the bifurcated periodic orbits of the Poincare map P tend to infinity as IL tends to zero. Moreover, we calculate the asymptotic ratio of the IL values at which two saddlenode bifurcation points have the neighboring z components, and the asymptotic ratio of two neighboring saddle-node bifurcation values.
5.8.1.
Existence and stability of bifurcated periodic orbits
To study the periodic orbits of system (5.8.1), it suffices to consider the periodic points of the Poincare map P. However, we will concentrate mainly on the simplest fixed points of P, partly because a general consideration is very complicated, but mainly because this study will be able to give us a surprising amount of valuable information for a good understanding of the formation of the chaotic mechanism. For conciseness, now let
acosOo + bsinOo = acos(Oo + cPr), c cos 00 + d sin 00 = /3 cos(Oo + cP2), 0= _.\-lw, T = (-2k.\-lak)1/2k, 01 = cP1 - olnE, O2 = cP2 - olnE, E = (1 + T2kx2kln(Ez-1))1/2k. Since the quantities O( (2) and 0(2) are not essential to the folllowing study, we neglect them so that the map P will be easier to work with. Then, by (5.8.4) and (5.8.5), the fixed points can be found by solving x = axE- 1cos(oln z + Or) + x + elL, (5.8.6) z = /3xE- 1 cos( oln z + O2 ) + IL. Notice that we have E Tx(ln(Ez-1))1/2k as T tends towards zero, so (5.8.6) can be rewritten approximately in the following form, f"V
x
= aT- 1(In( EZ- 1) r 1/ 2k cos( oln z + Or) + x + elL, z - IL
= /3T- 1(ln( EZ- 1) )-1/2k cos( oln z + O2).
(5.8.7) (5.8.8)
348
Chapter 5.
Bifurcation of Higher Dimensional Systems
Now we want to seek solutions for (5.8.7) and (5.8.8). Obviously, (5.8.8) is independent of x. Hence, it follows that the existence of fixed points is equivalent to the existence of small solutions for (5.8.8). But a direct calculation of these solutions and their number is generally rather complicated. Thus we restrict our attention to the region o < z « 1, and turn to considering the intersections of two curves reduced by (5.8.8). Let
F(z) = z - J.L, G(z) = H cos(8lnz + (h), with H = ,BT- 1(ln(Ez- 1))-1/2k. Then the two curves can be denoted by
L1 : W
= F(z),
L2 : W
= G(z),
where L1 is a straight line and L2 a curve. They are all defined for o < z « 1, and their intersection points give the z components of the fixed points of the Poincare map P. We consider the amplitude function H(z) of the curve L 2 . Its derivative satisfies {)H
-
{)z
= ,B(Tz)-1(ln(Ez-1)t1-1/2k/2k --+ 00
as
z
--+
o.
So, we see L2 is a wiggly curve (see Fig. 5.8.1). Moreover, there is a finite number of intersections of the two curves for 0 < IJ.LI « 1, each representing a fixed point of P lying on some I-periodic orbit of the original system (5.8.1); and a countable set of intersections for J.L = 0, representing a countable set of I-periodic orbits of the originatsystem. The definition for N-periodic orbit is given in Sec. 5.5. From (5.7.13), the time of flight from (x, 0, z) E to (xo, Yo, E) E Sl is given by (5.8.9)
st
And we see, by the continuous dependence, the time spent from (xo, Yo, E) to So is approximately a constant. It means that the periodic orbits represented by fixed points of P with smaller z coordinates twist around the z-axis more often and hence have longer periods. As
5.B.
Period-Doubling Bifurcation in Weak Sil'nikov Phenomenon
349
f.t decreases to zero, there are fixed points with z coordinate decreasing
to zero, and the period of the orbit passing through the corresponding fixed point increases to infinity. We may imagine that the homoclinic orbit occurring at f.t = 0 is a periodic orbit of period infinity, and is accompanied by a countable infinity of I-periodic orbits with arbitrarily large period in its small tubular neighborhood.
Ldp. > 0)
w
Fig. 5.8.1 Now we study the stability properties of these periodic orbits. Using (5.8.4)-(5.8.7), we obtain (5.8.10) where
= r ox ( a cos 80 + b sin 80 ) = 0(E-1) cos(8lnz + 8d, D2 = roz(a cos 80 + b sin 80 ) - r o8oz (a sin 80 - b cos 80 ) = af( TZ)-l(Apk cos(8ln z + 81 ) - 8 sin(8ln z + 81 )), D1
Chapter 5.
350
Bifurcation of Higher Dimensional Systems
= rox (e cos 00 + d sin ( 0 ) = O(E- 1 ) cos( 8ln z + ( 2 ), D4 = roz(ecosOo + dsinOo) - roOoAesinOo - dcosOo) D3
f
= f3f( Tztl(Af2k cos(8ln z = (In(Ez- 1))-1/2k.
+ ( 2) -
8 sin(8ln z
+ ( 2)),
We assume that (xp, 0, zp) is a fixed point of P.
°
Theorem 5.8.1. D3 = if and only if G(zp) = 0. Moreover, (xp, 0, zp) is a saddle if G(zp) = and < Zp « 1.
° °
Proof. Because G(zp) and D3 contain the same trigonometric function cos( 8ln zp + ( 2 ) and the remaining factors are all nonzero, it should be clear that G(zp) = if and only if D3 = 0. Now assume G(zp) = and < zp « 1. From the above analysis, we see ID41 = 8f3f(Tzt 1 and the eigenvalues of DP are Dl and D 4. By the fact that E :» 1 for < z « 1, it follows that IDII « 1, ID41:» 1, which in turn implies that (xp, 0, zp) is a saddle. 0
° ° ° °
Theorem 5.8.2. Suppose that zp is an extreme point of G(z) and 0< zp« 1. Then D4 = and (xp,O,zp) is an unstable fixed point
°
( source).
°
Proof. We have G'(zp) = if zp is an extremum of G(z). Then D4 = follows simply from the fact D4 = G'(z). On the other hand, an elementary calculation shows that
°
det(DP) = DID4 - D2D3 = (ad - be)roroxOoz ~ (ad - be)8x(zE2tl. Then D4
=
~
(5.8.11)
°
means that
Noting that the diffeomorphism induced by the flow maintains the orientation and that ad - be 1= 0, we have
(5.8.12)
5.B.
Period-Doubling Bifurcation in Weak Sil'nikov Phenomenon
351
°
< z «1. Since /D1/ « 1 can be neglected compared to J-D2D3, we see that the eigenvalues of DP are
for
A± = (D1 ± VDr By (5.8.11), /A±/
»
+ 4D2D3)/2 ~ ±J-D2D3i.
1 and the theorem follows.
Theorem 5.8.3. Suppose J.L is a saddle.
=
° °< and
o
zp «1. Then (xp, 0, zp)
Proof. Owing to Theorem 5.7.4, the Poincare map P possesses a countable infinity of invariant sets, An C Rn, and acting on each An, P is topologically conjugate to a shift on two symbols. Now let n be sufficiently large, and zp sufficiently small, then (xp, 0, zp) E An. This is simply because that An is the maximal invariant set in Rn. Finally, (xp,O,zp) is a saddle by Remark 4 in Sec. 5.7. 0 5.8.2.
Bifurcation diagram and asymptotic ratio of bifurcation values
Now we want to draw the bifurcation picture and make some further analysis of the bifurcation parameter values. For this, let us first summarize the information contained in Theorems 5.8.1-5.8.3 and Fig. 5.8.1. If zp is the z-coordinate of points B, C, H, G, then (xp, 0, zp) is a saddle; if zp is the z component of the extreme points, E, J, of the wiggly curve L 2 , then (xp, 0, zp) is an unstable source; and if zp is the abscissa of points A, D at which L1 touches L 2, then (xp, 0, zp) is a saddle-node. Let J.L decrease continuously from J.L > to J.L < 0, such that one of the intersection points of L1 and L2 goes along L2 first from E to G, then from G to H, and at last from H to J. Since the fixed points corresponding to E and G (resp. Hand J) have different types of stability, we conjecture that there must exist an additional point F (resp. 1) between E and G (resp. Hand J) which corresponds to a period-doubling bifurcation point. When we pass by F along L2
°
352
Chapter 5.
Bifurcation of Higher Dimensional Systems
as J.L decreases, the fixed point representing a I-periodic orbit of the original system will lose its stability in a period-doubling bifurcation and become a saddle, while the reunstabilization of J must occur through a reverse period-doubling bifurcation at I. We now show that our conjecture is true.
Theorem 5.8.4. (xp, 0, zp) is a period-doubling bifurcation point when zp is the z-coordinate of either F or 1. Proof. By the above analysis, we see F and I must correspond to bifurcation points, which entails that DP has at least one eigenvalue with modulus one at (xp, 0, zp). Now we show that it is exactly one. Assume that there are two eigenvalues ..\± with modulus one. Then they must be a pair of conjugate eigenvalues. Due to ..\± = (tr (DP) ± J(tr (DP))2 - 4det(DP))/2, we obtain I..\±I = (det(Dp))1/2. It follows from (5.8.11) that I..\±I » 1 if < zp « 1, which contradicts the assumption I..\±I = 1. We may assume ..\+ = ±1 and 1..\_1 i- 1. If ..\+ = 1, then (xp, 0, zp) will be either a saddle-node, or a transcritical point, or a pitchfork point, or another kind of bifurcation point from which more than 3 fixed points can be produced. However, the number of intersection points near F and I keeps unchanged as the parameter J.L varies. So, we must have ..\+ = -1, 1..\_1 i- 1. Equivalently, we know that (xp, 0, zp) is a period-doubling bifurcation point. /U
°
We summarize the above information to give the diagram in Fig. 5.8.2, where the T coordinate represents the period of the corresponding orbit passing through the fixed point of the map P. The wiggly curve has vertical tangent lines at points such as A and D, each of which corresponds to a J.L value at which the saddle-node bifurcation takes place, i.e., at each such value J.L a pair of periodic orbits (one unstable, one saddle-like) appears or disappears.
5.B.
Period-Doubling Bifurcation in Weak Sil'nikov Phenomenon
353
T
•...
-
~-
-
-"'"
,--- --'" J'-...-- ----.! ] H G F 'D ___ --.;tI
'------B
C A'--___ ____
~
__ ---- ___ -'fIIIJ>' o Fig. 5.8.2 The points on the dashed line of the wiggly curve give the J..l values at which the periodic orbit with given period T has a 2-dimensional stable manifold and a 2-dimensional unstable manifold. And those on the real line part produce a pair of values (J..l, T) which corresponds to an unstable periodic orbit. Finally the junctions of the real lines and imaginary lines, say F and I, provide the period-doubling bifurcation J..l values. Let J..ll, • .. ,J..ln,··· be all the saddle-node bifurcation values starting from some J..ll, that is, at each of them Ll is tangential to L 2. Here we choose J..li such that J..liJ..li+1 < 0, the corresponding z-coordinate Zi of the tangent point tends monotonously to zero. Moreover, Zi and Zi+l are the closest neighboring pair of points. Theorem 5.8.5. The saddle-node bifurcation value J..li tends to zero and the ratio lJ..li+1/ J..li I tends to 1 as i tends to infinity. Proof. Let Zi be the Z coordinate of the point at which Ll is tangent to L2 with J..l = J..li. Then, Zi and J..li satisfy the following equations: (5.8.13)
Chapter 5.
354
Bifurcation of Higher Dimensional Systems
(5.8.14) where ai = oln Zi. In view of the fact that the slope of the straight line L1 is fixed and equal to one, while the slope of the amplitude function H(z) of L2 tends to infinity as z tends to zero, it follows that these tangencies must occur nearly at the maxima and minima of the wiggly curve L2 for small Zi, which in turn means that
+ (2 )1 ~ 1,
1cos(ai
Remembering that 0 = z· 1 ~ Zi
_).-l W ,
= exp(o-l(ai+1 -
ai+1 - ai ~ 7r.
(5.8.15)
we have ai)) ~ exp( -).w- l 7r).
(5.8.16)
Then, using (5.8.13)-(5.8.16), we get J.Li+1/ J.Li = (Zi+1 - H(Zi+d cos(ai+1 + ( 2))/(Zi - H(Zi) cos(ai ~ -((In(Ez;1)/ln(Ez;+\))1/2k ~-l.
Finally, owing to
zi
-t
0 as i
-t
00,
lim J.Li+ Ii J.Li
~-->oo
+ (2 ))
we obtain
=
-1,
which implies the conclusion of the theorem.
o
Corollary 5.8.6. The ratio of two neighboring saddle-node bifurcation values tends asymptotically to one. Proof. It is clear that J.Li and J.Li+2 are a pair of neighboring saddlenode bifurcation values. Due to the proof of Theorem 5.8.5, we have
o Remark 1. Comparing with the asymptotic ratio
lim lJ.Li+Ii J.Lil = exp(pw- l 7r)
~-->oo
5.S.
Period-Doubling Bifurcation in Weak Sil'nikov Phenomenon
355
given by [52] for p < 0, we see that the ratio given in Theorem 5.8.5 is larger for p = O. It turns out that, comparing with the case p < 0, the decreasing rate of the wiggles (Le., amplitudes) of the wiggly curve in the (IL, T)-plane will be smaller in the case p = O. Hence the number of I-periodic orbits will be much larger for IILI « l. Remark 2. By (5.8.9) and (5.8.16), it is easy to see that the growth of periods of orbits corresponding to saddle-node bifurcation points is
5.8.3.
Subsidiary homo clinic orbits
Now we consider the existence of the simplest kind of subsidiary homoclinic orbits, 2-homoclinic orbits, which sometimes are referred to as double-pulse homoclinic orbits. When IL > 0, the original homo clinic orbit is broken, and the unstable manifold intersects the section S6 first at the point (x, 0, z) = (x + elL, 0, IL). Then, by (5.8.4) and (5.8.5), there is a 2-homoclinic orbit if and only if (5.8.17) where
= (x + elL)/(1 + (r 2k + O(€2))(x + elL)2k In(€1L- 1))-1/2k, eo = (- that IX(t,s)Ps(s)l::; Ke-a(t-s), t 2: s in J, IX(t, s)Pc(s)1 ::; K eo- It - sl , t, s in J,
(J
> 0 such
t 2: s in J.
IX(s, t)Pu(t)1 ::; K e-a(t-s),
The constants a and (J are called the exponents of the trichotomy, and the projection spaces ~Ps(t), ~Pc(t), ~Pu(t) are called the stable space, centre space and unstable space respectively. We say that (6.1.1) has an exponential dichotomy in J if it has an exponential trichotomy with Pc(t) = 0 and Ps(t) + Pu(t) = I. Consider a C r autonomous system x = f(x) with r 2: 1. Suppose that it has an equilibrium x = Xo' Then it should be clear that the associated linear variational equation
has an exponential dichotomy in IR if Xo is hyperbolic, and an exponential trichotomy in IR if Xo is nonhyperbolic. Moreover, the corresponding projections are independent of t. Now we consider the adjoint system of the linear homogeneous equation associated with (6.1.1)
x=
-A*(t)x,
where the sign A* denotes the transposition of A.
(6.1.2) ~
Proposition 6.1.1. Suppose that (6.1.1) has an exponential trichotomy in J with constants K, a, (J, and projections Ps(t), Pc(t), and P u(t). Then the adjoint system (6.1.2) also has an exponential trichotomy in J with the same constants and the corresponding projections P:(t), Pc*(t), and Ps*(t). More precisely, for the solution map Y(t,s) of (6.1.2), we have Y(t, s)P;(s)
=
P;(t)Y(t, s),
t 2: s in J, v = s, c, u,
6.1.
Exponential Trichotomies
361
JY(t,s)P:(s)J ~ K e-a(t-s), JY(t, s)P;(s)J ~ K e17lt-sl, JY(s, t)Ps*(t)J ~ K e-a(t-s),
t ~ s in J,
t, s ~
s
in J, in J.
Proof. It is well known that Y(t,s) = X*-l(t,S) Then, by
X*(s, t).
X(t, s)Pv(s) = Pv(t)X(t, s) and taking transposes, we obtain
Y(t,s)P:(s) for v
=
= X*(s,t)P:(s) = (X(s,t)Pv(t))* = P:(t)Y(t,s)
s, c, u, and
JY(t, s )P:(s)J = J(X(s, t)Pu(t))*J ~ K e-a(t-s), t ~ s in J, JY(t, s)Pc*(s)J = J(X(s, t)Pc(t))*J ~ K e17lt-sl,
t, s
in J,
JY(s,t)Ps*(t)J = J(X(t,s)Ps(s))*J ~ Ke-a(t-s), t ~ s in J. o The following four propositions can be found in [40]. Proposition 6.1.2. Let X(t, s), t ~ s, have exponential trichotomies in both IR- and IR+, with projections Ps± (t), Pc±(t), P;;= (t), t E IR±. Suppose that the exponents in IR- and IR+ are the same, the unstable spaces in IR- and IR+, and the center spaces in IR- and IR+ have the same dimensions: ~Pu-(O)
n {~Ps+(O) EB ~Pc+(O)}
{~Pc-(O) EB ~P;(O)}
=
0,
n ~Ps+(O) = 0,
Then, X (t, s) has an exponential trichotomy in IR = IR- U IR+. Proposition 6.1.3. Let X (t, s) be defined in ( -00, to] and have an
exponential trichotomy in (-00, r], to> r. Suppose that X(to, r)(4)l + 4>2) f. 0 for 4>1 E ~Pu(r), 4>2 E ~Pc(r), and 4>1 +4>2 f. O. Then X(t, s) has an exponential trichotomy in (-00, to] with the same exponents,
362
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
and the projections ps(t), Pe(t) and pu(t) approch Ps(t), Pe(t) and Pu(t) exponentially as t ---t - 00, respectively. Proposition 6.1.4. Let X (t, s) be defined in [to, +00) and have an exponential trichotomy in [7, +00), 7 > to. Suppose that
for'l/Jl E RP;(7), 'l/J2 E RP;(7), and 'l/Jl + 'l/J2 i= O. Then X(t,s) has an exponential trichotomy in [to, +00) with the same exponents, and the projections ps(t), Pe(t) and pu(t) approach Ps(t), Pe(t) and Pu(t) exponentially as t ---t +00, respectively. Proposition 6.1.5. Suppose that (6.1.1) has an exponential trichotomy in J = (-00, OJ or [0, +00), or ( -00, +00) with projections Ps(t), Pe(t) and Pu(t), and exponents a, (J'. Then the differential equation (6.1.3) x = (A(t) + B(t))x
has an exponential trichotomy in J, with projections ps(t), pc(t) and Pu(t), and exponents ii > (j > 0, provided that B is continuous in J and 8 == SUPtEJ IIB(t)11 < 80 for some sufficiently small constant 80 > O. Furthermore, ps(t) ---t Ps(t), Pe(t) ---t Pe(t) and Pu(t) ---t Pu(t) uniformly in t and (ii, (j) ---t (a, (J') as 8 ---t O. Under the same hypotheses as for (6.1.1) with J = (-00, OJ [0,00)), and IIB( t) II ---t 0 as t ---t -00 (or t ---t 00), there is a 7 > 0 such that (6.1.3) has an exponential trichotomy on (-00,-7J (or [7,00)) and ps(t) ---t Ps(t), Pe(t) ---t Pe(t), and pu(t) ---t Pu(t) as t ---t -00 (or
0w-
t---too). The proof of Proposition 6.1.5 is given in [66J. Here, we use a modified statement of the original proposition so that it is more suitable to ordinary differential equations. We now turn our attention to considering the number of the linearly independent bounded solutions of the adjoint system (6.1.2) and
6.1.
Exponential Trichotomies
363
the space spanned by these solutions. For this, we need the following lemma. Lemma 6.1.6. Suppose that P is a projection operator in a Hilbert space H. Then RP = (R(I - P*))1.. Here the sign ..1 denotes orthog-
onal complement. Proof. For any x E RP, we have Px = x. It follows that
(x, (I - P*)y)
(x, y) - (x, -P*y) = (x, y) - (Px, y) =
= O. Thus we have shown RP C (R(I _ P*))1.. To prove the lemma, it suffices to show that (R(I - P*))1. C RP. It should be clear that, for any z E (R(I - P*))1. and any y E H, the following equalities are valid.
o = (z, (I -
P*)y) = (z, y) - (Pz, y) = (z - Pz,y).
By the arbitrariness of y we obtain pz = z, which means (R(I p*))1. c RP. Thus the lemma follows. 0 Now set
E(b,J)
= {x
E(b,r,J)
=
{x
E
Co: sup{lx(t)le b1tl } < <Xl}, tEJ
E Cr
: x, ... ,x(r)
E
E(b,J)}.
Then E(b, J) and E(b, r, J) are Banach spaces with norms Ilxll o
= sup{lx(t)lebltl } tEJ
respectively. Denote dim RP;(O) = si,
and
364
for i
Chapter 6.
= +,-, d= n
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
+c-
c+ - s+ - c- - u-,
d 1 = d + c- ,
d2
= d + s+ + c-,
d3 = d + c+
d4
= d + s+ + u-.
+ c-,
Proposition 6.1. 7. Suppose (6.1.1) has exponential trichotomies in both JR+ and JR- with constants K i , ai, eJi and projections P;(t), P~(t), P~(t), respectively, for i = +,-. Let a = min{a+,a_}, eJ = max { eJ +, eJ _}. Then the following five conclusions are valid. (i) If dim(~(Ps+(O) + Pc+(O)) n ~(Pc-(O) + Pu-(O))) = c, then dim(~P:*(O)
n ~Ps-*(O))
=
d,
(6.1.4)
that is, (6.1.2) has exactly d linearly independent bounded solutions 1fJl(t), ... ,1fJd(t) in E(a, 1, JR). (ii) If dim(~(Ps+(O) + Pc+(O)) n ~Pu-(O)) = c, then dim(~P:*(O)
n ~(Ps-*(O)
+ Pc-*(O)))
=
d1 ,
(6.1.5)
that is, (6.1.2) has exactly d 1 linearly independent bounded solutions 1fJl(t), ... , 1fJd1 (t) in E(a, 1, JR+) n E( -eJ, 1, JR-). (iii) If dim(~Pc+(O) n ~P;(O)) = c, then (6.1.6)
that is, (6.1.2) has exactly d2 linearly independent bounded solutiors 1fJl(t), ... ,1fJd2 (t) in E( -a, 1, JR+) n E( -eJ, 1, JR-). (iv) If dim(~Ps+(O) n ~Pu-(O)) = c, then (6.1.7)
that is, (6.1.2) has exactly d3 linearly independent bounded solutions 1fJl(t), .. . , 1fJd3 (t) in E( -eJ, 1, JR+) n E( -eJ, 1, JR-). (v) If dim(~Pc+(O) n ~Pc-(O)) = c, then (6.1.8)
6.1.
Exponential Trichotomies
365
that is, (6.1.2) has exactly d4 linearly independent bounded solutions 'l/Jl(t), ... ,'l/Jd4 (t) inE(-a,l,IR). Proof. We need only show the conclusion (i). The others can be proved in a similar way. By the fact that
we have
Then, it follows from Lemma 6.1.6 that
dim(RP:*(O) n RPs-*(O)) =dim((R(I - Pu+(O))).l. n (R(I - Ps-(O))).l.) = dim((R(Ps+(O) + Pe+(O))).l. n (R(Pe-(O) + Pu-(O))).l.) = codim (R(Ps+(O) + Pe+(O)) EEl R(Pe-(O) + P:(O))) =d. Therefore, by Proposition 6.1.1, (6.1.2) has exactly d linearly independent bounded solutions 'l/Jl(t), ... ,'l/Jd(t) in E(a, 1,1R). 0 If we notice that RP~*(t) is a linear subspace for i = v = s, c, u, and that
R( P
+, -
and
+ Q) = RP EEl RQ,
for any two projections satisfying PQ = 0, then, the following four propositions can be easily deduced from Lemma 6.1.6 and Proposition 6.1.7. Proposition 6.1.8. Suppose that the conditions contained in Proposition 6.1.7 are valid, and
dim(R(Ps+(O) + Pe+(O)) n R(Pe-(O) + P:(O))) dim(R(Ps+(O) + Pe+(O)) n RP:(O))
= c.
= c,
366
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
Then system (6.1.2) has exactly d linearly independent bounded solutions 'ljJl(t), . .. ,'ljJd(t) in E(a, 1, JR), and exactly c- linearly independent bounded solutions 'ljJd+ 1 (t), ... , 'ljJd 1(t) in E( a, 1, JR+) n (E( -CT, 1, JR-) - E(a, 1, JR-)).
Moreover, we can ch;ose 'ljJ1 (t), ... , 'ljJd 1(t) such that span{'ljJl(t), ... , 'ljJd1(t)} span{ 'ljJ1 (t), ... , 'ljJd(t)} span{ 'ljJd+1(t), .. . , 'ljJd 1(t)}
c OR(Ps+(t) + P/(t)))~ for t ~ 0, c OR(Pc-( t) + Pu- (t)))~ for t ::; 0, c (R(Ps-(t) + Pu-(t)))~ for t ::; O.
Proposition 6.1.9. Suppose that the conditions contained in Proposition 6.1.7 are valid and
dim(R(Ps+(O)
+ Pc+(O)) n RPu-(O)) = c,
dim(RPc+(O) n RP;(O)) = c. Then system (6.1.2) has exactly d 1 linearly independent bounded solutions 'ljJl(t), ... , 'ljJd 1(t) in E(a, 1, JR+) n E( -CT, 1, JR-), and exactly s+ linearly independent bounded solutions 'ljJd1+l(t), ... , 'ljJd2 (t) in (E( -a, 1, JR+) - E(a, 1, JR+)) n (E( -CT, 1, JR-).
Moreover, we can choose 'ljJ1 (t), ... , 'ljJd2 (t) such that span{ 'ljJ1 (t), ... ,'ljJd1(t)} span{ 'ljJd1+l(t), ... , 'ljJd2 (t)} span{ 'ljJl(t), ... , 'ljJd2 (t)}
c (R( Ps+(t) + Pc+( t)))~ c (R(Pc+(t) + P:(t)))~ c (RPu-(t))~ for t ::; O.
for t ~ 0, for t ~ 0, ~
Proposition 6.1.10. Suppose that the conditions contained in Proposition 6.1.7 are valid and
dim(R(Ps+(O)
+ Pc+(O)) n RPu-(O)) = c,
dim(RPs+(O) n RP;(O)) = c. Then, system (6.1.2) has e:.cactly d 1 linearly independent bounded solutions 'ljJl(t), ... , 'ljJd 1(t) in E(a, 1, JR+) n E( -CT, 1, JR-), and exactly c+ linearly independent bounded solutions 'ljJd1+l(t), ... , 'ljJd3 (t) in (E( -CT, 1, JR+) - E(a, 1, JR+)) n (E( -CT, 1, JR-).
6.1.
Exponential Trichotomies
367
Moreover, we can choose 'l/Jl(t), ... ,'l/Jd3 (t) such that span{'l/Jl(t), ... ,'l/Jd1 (t)} C OR(Ps+(t) + Pc+(t)))J. for t 2 0, span{ 'l/Jd 1+1(t), ... ,'l/Jd3 (t)} C (?R(Ps+(t) + Pu+(t)))J. for t 2 0, C (?RPu-(t))J. for t::; O. span{'l/Jl(t), ... ,'l/Jd3 (t)}
Proposition 6.1.11. Suppose that the conditions contained in Proposition 6.1.7 are valid and
dim(?R(Ps+(O)
+ Pc+(O)) n ?R(Pc-(O) + Pu-(O))) = c,
dim(?RP/(O) n ?RPc-(O)) = c. Then system (6.1.2) has exactly d linearly independent bounded solutions 'l/Jl (t), ... , 'l/Jd( t) in E( a, 1, JR), and exactly s+ + u - linearly independent bounded solutions 'l/Jd+ 1 (t), ... , 'l/Jd4 (t) in E( -0'.,1, JR) - E(O'., 1, JR). Moreover, we can choose 'l/Jl (t), ... , 'l/Jd4 (t) such that span{'l/Jl(t), ... , 'l/Jd(t)} span{'l/Jl(t), ... , 'l/Jd(t)} span{'l/Jd+l(t), ... ,'l/Jd4 (t)} span{'l/Jd+l(t), ... ,'l/Jd4 (t)}
C C C C
(?R(Ps+(t) (?R(Pc-(t) (?R(P/(t) (?R(Ps-(t)
+ Pc+(t)))J. + Pu-(t)))J. + P;;(t)))J. + Pc-(t)))J.
for for for for
t 2 t ::; t2 t::;
0, 0, 0, O.
At last, we consider the orthogonality condition associated with system (6.1.1). When A(t) and h(t) are T-periodic, this condition is known as the Fredholm Alternative Lemma (cf. [62]). [129J extends this alternative lemma to the case where (6.1.1) has an exponential dichotomy. In the following, we give a further generalization. Let L be the linear operator defined by
(Lx)(t) = x(t) - A(t)x(t) for x E C1(lR, lRn ), L 1 , L 2 , L3 be the restrictions of L in E( -(), 1, lR), E( -(), 1, lR+) n E(O'., 1, lR-), E(O'., 1, lR) respectively. Denote
Ef = E( -(), lR), E2 = E( -(), lR+) n E(O'., lR), Eg = E( a, lR),
= E(O'., 1, lR), E2 = E(O'., 1, lR+) n E( -(), 1, lR), E3 = E( -(), 1, lR),
El
368
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
h = dim~(Ps+(t) + P/(t)) + dim~(Pc-(t) + P;;(t)) - n, lz = dim ~(Ps+(t) + P/(t)) + dim ~Pu-(t) - n,
h = dim~Ps+(t) + dim~Pu-(t) - n. Definition 6.1.2. The linear operator L is referred to as a Fredholm operator if ~(L) is closed and has finite codimension. The index of L as a Fredholm operator is defined as dimN(L) - codim ~(L). Proposition 6.1.12. Suppose that the hypotheses of Proposition 6.1.7 hold. Then, h E ~(Li) if and only if h E Ef and
L: 1f;*(t) h(t) dt
= 0
(6.1.9)
for all bounded solutions 1f; (t) of (6.1. 2) in E i . Moreover, Li is a Fredholm operator with index Ii. Proof. We need only consider the case i = 1, the proof of the other cases being similar. The proof given below is an analogue and generalization of that for the case of having an exponential dichotomy in [129]. Assume that h E ~(Ld. Then there exists an x E E( -0',1, IR) satisfying
h(t) = x(t) - A(t)x(t). So, h E E( -0', IR). Now if 1f;(t) is a bounded solution of (6.1.2) in/ E(a, 1, IR), we have
L: 1f;*(t) h(t) dt
=
L:(1f;*(t)x(t) -1f;*(t)A(t)x(t)) dt
= L:(1f;*(t)x(t) + -0*(t)x(t)) dt = 1f;*(t)x(t)l~oo = o. The last equality holds since 1f;*(t)x(t) -+ 0 exponentially as It I -+ 00 owing to the fact that 1f;(t)eQ1tl and x(t)e- u1tl are bounded. Thus, we have shown that if h E ~(Ll)' then the orthogonality condition (6.1.9)
6.1.
Exponential Trichotomies
369
holds for all bounded solutions 1jJ(t) of the adjoint system (6.1.2) in El = E(a, 1,lR). Conversely, suppose that h E E( -0', lR) and that (6.1.9) is valid for all bounded solutions 1jJ(t) of (6.1.2) in E(a, 1, lR). It should be clear that, for each 1jJ(t) E E(a, 1, lR), there exists a vector 'r/ E lRn such that
t for t for
~
0,
~
O.
(6.1.10)
that is, Then, we obtain (6.1.11) Substituting (6.1.10) into (6.1.9) and using Y(t, s) get 'r/*v = 0,
= X*(s, t), we (6.1.12)
where
v = fo'XJ P:(O)X(O, t)h(t) dt
+ J~oo Ps-(O)X(O, t)h(t) dt.
(6.1.13)
By (6.1.11) and (6.1.12), it follows that there exists a vector ~ E lRn satisfying
(Ps+(O)
+ Pc+(O) -
P;(O) -
Pc-(O))~
Making use of (6.1.13) and Pj(O)X(O, t) +, -; j = s,u, we have
= v.
= X(O, t)Pj(t) for
+ Pc+(O))~ - 10 X(O, s)P:(s)h(s) ds = (P;(O) + Pc-(O))~ + J~oo X(O, s)Ps-(s)h(s) ds.
(Ps+(O)
i
=
00
(6.1.14)
Chapter 6.
370
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
Then it can be verified that the function x(t), defined for t
X(t, O)(Ps+(O)
-i
oo
~
0 as
+ Pc+(O))~ + fat X(t, s)(Ps+(s) + Pc+(s))h(s) ds
X(t, s)P;:-(s)h(s) ds
and for t ::; 0 as
X(t, O)(Pu-(O)
+ Pc-(O))~ + fat X(t, s)(Pu-(s) + Pc-(s))h(s) ds
+ J~oo X(t, s)Ps-(s)h(s) ds, is in E( -0',1, IR) and is a solution of the nonhomogeneous linear system (6.1.1). It means hE R(Ld, as expected. Now we show that the linear operator L1 is Fredholm. By (6.1.9), each bounded solution 'lj;(t) of (6.1.2) in E(a, 1, IR) defines a bounded linear functional on E( -a, IR) through
L:
h~
'lj;*(t) h(t) dt.
This correspondence gives an isomorphism between
RP;:-*(t) n RPs-*(t) = (R(Ps+(t)
+ Pc+(t)))l. n (R(Pc-(t) + Pu-(t)))l.
and a finite dimensional subspace of the dual space (E( -a, IR))*. This means that R(Ld is a subspace of E( -a, IR) with codim R(L1)
= dim((R(Ps+(t) + Pc+(t)))l. n (R(Pc-(t) + P;(t)))l.),
that is, R( Ld is closed, and then L1 is Fredholm. By definition, the index of L1 is
/
dimN(L 1) - codim R(L1)
= dim(R(Ps+(t) + P/(t)) n R(Pc-(t) + Pu-(t))) - dim((R(Ps+(t)
+ Pc+(t)))l. n (R(Pc-(t) + Pu-(t)))l.)
= dim(R(Ps+(t) + Pc+(t)) n R(Pc-(t) + Pu-(t))) -codim (R(Ps+(t)
+ Pc+(t)) EB R(Pc-(t) + Pu-(t)))
= dim(R(Ps+(t) + Pc+(t)) n R(Pc-(t) + Pu-(t))) -n + dim(R(Ps+(t)
+ Pc+(t)) EB R(Pc-(t) + P;(t)))
= dim R(Ps+(t) + Pc+(t)) + dim R(Pc-(t) + Pu-(t)) - n, as asserted.
0
6.2.
Melnikov Vector in Higher Dimensions
371
Remark 1. If (6.1.1) has an exponential dichotomy on both half-lines, then in the above propositions we have Pc+ = Pc- = 0, E(-0",1,J) = E(0,1,J), E(-O",J) = E(O,J) for J = lR+, lR-. For convenience of application in the following, we restate below results corresponding to the case of having an exponential dichotomy, which is a particular case of Proposition 6.1.12. Corollary 6.1.13. [129] Suppose that (6.1.1) has an exponential dichotomy on both half-lines. Then the linear operator
L: E(O, 1, JR)
--t
E(O, JR)
is Fredholm and has index dim~Ps+(t) + dim~P;(t) - n. Moreover, h E ~(L) if and only if h E E(O, JR) and the orthogonality condition (6.1.9) holds for all bounded solutions 7/J( t) of the adjoint system (6.1.2).
6.2.
Melnikov Vector in Higher Dimensions
We develop in this and the following sections the Melnikov method for higher dimensional systems. This method is based on measuring the separation of the stable and unstable manifolds associated with a Poincare map. As mentioned above, this method has been extensively studied. However, for higher dimensional systems, a relatively full development of this method is achieved only in recent years. In the case of higher dimension, along a singular orbit, i.e., a homo clinic or heteroclinic orbit, tangent bundles of stable and unstable manifolds may have an intersection which is of higher dimension (this dimension may be equal to or greater than that of the intersection of the stable and unstable manifolds which are referred to as a homoclinic or heteroclinic manifold). In this situation, many interesting bifurcation problems, such as the persistence of singular orbits under perturbation, the transversality and tangency of stable and unstable manifolds, and local and global bifurcations associated with orbits connecting nonhyperbolic critical points, may arise.
372
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
In this section, we only deal with the case of orbits homoclinic or heteroclinic to hyperbolic critical points, and present two geometrical methods developed in [182,199] to construct the Melnikov vector which measures the separation of stable and unstable manifolds. Further development of these methods will be introduced in the last section of this chapter, where nonhyperbolic critical points and local bifurcations are considered. 6.2.1.
A method using orthogonality condition and transversality theory
We consider the following system of ordinary differential equations,
x = f(x),
(6.2.1)
where f E C 2 • Assume that system (6.2.1) satisfies the following two hypotheses. (HI) System (6.2.1) has two hyperbolic saddles PbP2, and a heteroclinic orbit r : x = x(t) with x(t) --t PI (resp. P2) as t --t -00 (resp. +(0). (H2) Let WS(Pi) and WU(Pi) be the stable and unstable manifolds of Pi, TqWS(Pi) and TqWU(Pi) the tangent spaces of WS(Pi) and WU(pd at q for i = 1,2, respectively. Then,
dim TqWU(PI)
+ dim TqW S(P2))
=
n - d + e,
for q E r, d ~ 1. Hypothesis (HI) implies that the linear variational system of (6.2.1) along the heteroclinic orbit r,
i = A(t)z,
A(t) = Df(x(t))
(6.2.2)
has an exponential dichotomy in both IR+ and IR-. Denote the corresponding exponent and projections by Ct, PS+(t) , P;;(t), Ps-(t), P;;(t), respectively.
6.2.
Melnikov Vector in Higher Dimensions
373
Due to Proposition 6.1.1, the adjoint system of (6.2.2),
~ = -A*(t)1/;
(6.2.3)
has also an exponential dichotomy in both IR+ and IR - with exponent and projections a, Pu+*(t), Ps+*(t), Pu-*(t), Ps-*(t), respectively. It should be clear that z(t) is a bounded solution of (6.2.2) if and only if Zo = z(O) E Tzo WS(P2) n Tzo WU(pt} , which means that
Tzo W S(P2) n Tzo WU(Pl)
= 3tPs+(O) n 3tP;(O),
and we can choose
3tPs+(t) = T z(t)WS(P2), 3tPu-(t) = Tz(t)WU(Pl).
(6.2.4)
Thus, the following lemma follows from Proposition 6.1.7. Lemma 6.2.1. Suppose that hypotheses (HI) and (H2) hold. Then,
dim(3tPs+(t) n 3tPu-(t)) = e, dim(3tP:*(t) n 3tPs-*(t)) = d, that is, the linear variational system (6.2.2) has exactly e linearly independent bounded solutions Zl = i:(t), Z2(t), .. . ,ze(t), and its adjoint system (6.2.3) has exactly d linearly independent bounded solutions 1/;l(t), ... ,1/;d(t) in E(a, 1, JR). Let L: E(a, I,IR)
-+
E(a,IR) be defined by (Lx)(t)
=
x-
A(t)x.
Lemma 6.2.2. Suppose that hypotheses (HI) and (H2) hold. Then L is a Fredholm operator with index 1= e - d. Moreover, h E 3t(L) if and only if h E E(a, JR) and the orthogonality condition
L:
holds for i = 1, ... ,d.
1/;;(t)h(t) dt = 0
374
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
Proof. By Corollary 6.1.13, it suffices to show that dim~Ps+(t)
+ ~P;:(t) - n =
e - d,
which can be easily deduced from (H2) and (6.2.4).
o
Now we consider a perturbed system of (6.2.1),
x=
f(x)
+ g(t,X,J.L),
(6.2.5)
where J.L E U c IRm, g E C 2, g(t, X, 0) = 0, g is T-periodic in t, and U is a small open neighborhood of the origin in IRm. For i = 1, ... ,d, we denote (6.2.6) Theorem 6.2.3. Suppose that (HI) and (H2) hold, m > d, and there is a to such that (MI(t o), .. . , Md(to)) has rank d. Then, there exist a neighborhood V C U of the origin in JRm and an (m - d)dimensional hypersurface H C V such that, for J.L E H, system (6.2.5) has hyperbolic periodic solutions PI (t, J.L), P2( t, J.L) connected by a heteroclinic orbit r J1: x = x( t, J.L) satisfying
IPi(t, J.L) - Pil = O(IJ.LI), Ix(t, J.L) - x(t)1 = O(IJ.LI),
i
= 1,2, t E JR,
and the tangent space of Hat J.L = 0 has normal (MI(t o), ... , Md(t o)). Proof. By (HI), there exists a sufficiently small neighborhgod V C U such that, for J.L E V, (6.2.5) has two hyperbolic periodic solutions PI(t, J.L) and P2(t, J.L) very close to PI and P2 with PI(t, 0) = PI, P2(t,0) = P2· Let WU(PI(t, J.L)) be the unstable integral manifold of PI(t, J.L), and WS(P2(t, J.L)) the stable integral manifold of P2(t, J.L). We now consider the following extended system of (6.2.5),
x = f(x) + g((),X,J.L),
e= 1, jJ, = 0,
where x EIRn , ()ETI==={t: tEIR, t===t+T}, J.LEU.
(6.2.7)
6.2.
Melnikov Vector in Higher Dimensions
375
System (6.2.7) possesses two invariant sets:
Til
=
{(x,B,J.L):
for i = 1,2. Topologically, manifold W~
and
Ti
= {(x,B,J.L): x(t)
E
X=Pi, J.L=O, BETl}
Til
is a circle.
Tl
W (Pl(t,J.L)), B(t) U
E
has a center-unstable
TI, J.L
E
V,t
E lR},
has a center-stable manifold
W~ = {(x,B,J.L): x(t) E W S (P2(t,J.L)), B(t) E Tl, J.L E V,t E lR},
where B(t) = t + to. Owing to (HI), we see that system (6.2.7) has a heteroclinic orbit
t:
((x,B,J.L): J.L = 0, x = x(t), B = B(t) E Tl, t E lR} c W~ n W~
Ti.
connecting Tl and Let O. Set
= T- efaT ... faT g(x(y),y,z,/-L,0)dz1... dze.
The associated averaged system
iJ
= Eg(y,/-L)
(6.3.3)
has a hyperbolic saddle y = Y with j eigenvalues having negative real parts and the others having positive real parts for E > O. (H3) When m > 0, the equation mlWl (x(y), y)
+ ... + mewe(x(y), y) = 0
has no solutions for any integers ml, ... ,me which are not all zero. Let d = n + c - kl - k 2 . (6.3.4) Under the hypotheses that (Hl)-(H3) hold, kl = k2 = C = d, and that (6.3.2) is a completely integrable Hamiltonian system, (6.3.1) has been studied by S.Wiggins. Two sets of sufficient conditions for the persistence and transversality of homo clinic orbits are given in [180] and the references therein. These results are extended by [200] to the case of system (6.3.2) being non-Hamiltonian. In the following, we use the generalized Melnikov method developed in the previous section to study the same problem for system (6.3.1) satisfying only hypotheses (Hl)-(H3).
392
6.3.1.
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
Fenichel's invariant manifold theory
To develop the perturbation techniques allowing us to solve the problem metioned above, we need Fenichel's theory (cf. [45,46]) which treats invariant manifolds with boundary. Consider an autonomous system
x = f(x),
(6.3.5)
where f E C r , x E IRn. Denote the flow generated by (6.3.5) by Xt(p) with xo(p) = p. Let M == M u 8M be a compact, connected C r manifold with boundary. The following definitions will be useful. Definition 6.3.1. M is referred to as an overflowing invariant manifold of (6.3.5) if Xt(p) E M for all t ::; 0, P E M, and the vector defined by (6.3.5) points strictly outward and is nonzero on 8M. M is referred to as an inflowing invariant manifold of (6.3.5) if it is an overflowing invariant manifold of (6.3.5) under time reversal. Definition 6.3.2. M is called a locally invariant manifold of (6.3.5) if for each p E M, there exists an open time interval Ip containing 0 such that Xt(p) E M for all t E IpNow assume M is a C r overflowing invariant manifold. ;then, the subbund~ . TM = {(p,v):p E M, v E TpM} is invariant under DXt(p) for t ::; mentary to TpM, i.e.,
o.
Let Np be a subspace comple-
Then we have a decomposition of the tangent bundle of IRn restricted on M, TIRnl M as follows:
TIRnl M = TM EB N ==
U (TpM EB N p). pEM
(6.3.6)
6.3.
Orbits Heteroc1inic to Invariant Manifolds
393
Next, we take a subbundle NU c TIRnl M which contains T M and is negatively invariant under the linearized flow generated by (6.3.5). Let Ie NU be a subbundle complementary to TM, i.e.,
and J c TIRnl M be a subbundle complementary to NU. Then we obtain another splitting of TIRnI M : (6.3.7) Let 7rN, 7rI, 7rJ, 7rT be the projections onto N,I, J, TM, respectively. We define the following quantities.
l/N(p) = inf{a E IR+: atjwN O"N(p) = inf{ s E IR: vt/wN AI(p) = inf{b E IR+: btWI
--t --t
l/J(p) = inf{a E IR+: atjwJ O"J(p) = inf{s E IR: vt/w}
--t
0 as t 0 as t
--t
--t
0 as t
--t -00 --t -00
0 as t
0 as t
--t -00
v E TpM, W E N p}, W E Ip},
--t -00
--t -00
W E N p},
W E J p},
v E TpM,w E J p}.
Here, O"N(p) and 0" J(p) are well defined when l/N(p) < 1 and l/J(p) < 1, respectively. l/N < 1 is an asymptotic stability condition for M under the linearized flow of (6.3.5). The inequality l/J < 1 has an analogous meaning.
0"
Definition 6.3.3. The functions l/N(p), O"N(p) , AI(p), l/J(p) , and J(p) are called generalized Liapunov type numbers.
Owing to [45], we see that these functions are constant on the orbits and are bounded, achieve their suprema on M, and are independent of the choice of the metric and of N, N U , I and J, respectively. Moreover, they have the following more computable expressions: (6.3.8)
394
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
-.
O"N(p)
log IIDxt(p)1TTII
(6.3.9)
= hmt--+-oolog 111TN DX t(p) I '
Aj(p) = limt--+_ooI11TjDxt(p)lr1/t, -
I/J(p) = limt--+-ooI11TJDxt(p)11 O"J(p)
=
l/t
(6.3.10) (6.3.11)
,
-. log IIDxt(p)1TTII hmt--+-oolog 111TJ DX t(p) II '
(6.3.12)
where II . II is a matrix norm. An interesting problem is that when overflowing invariant manifolds are preserved under perturbations and have unstable manifolds which are structurally stable. The following two theorems due to [45,46J give an answer to this question. Theorem 6.3.1. Suppose that M is an overflowing invariant manifold of (6.3.5) with I/N(p) < 1 and O"N(p) < l/r for all p E M. Then for any CT vector field
x=
g(x),
x E JRn,
(6.3.13)
with g( x) C 1 close to f (x), there is a CT manifold with boundary M g, being C 1 close to M and having the same dimension as M such that ( M 9 is overflowing invariant under (6.3.13). \
Theorem 6.3.2. Suppose that M is an overflowing invariant manifold of (6.3.5) with Aj(p) < 1, I/J(p) < 1 and 0" J(p) < l/r for all p E M. Then the following conclusions are valid: i) There exists a CT manifold WU overflowing invariant under (6.3.5) such that WU contains M and is tangent to NU along M. ii) Assume g is C 1 close to f. Then there exists a CT manifold W; overflowing invariant under (6.3.13), which is C 1 close to WU and has the same dimension as W u . Theorem 6.3.1 is called the perturbation theorem for overflowing invariant manifolds, and Theorem 6.3.2 is called the unstable manifold theorem for overflowing invariant manifolds. They are proved by
6.3.
Orbits Heteroc1inic to Invariant Manifolds
395
means of contraction mappings. The manifold WU (resp. W;) is referred to as the unstable manifold (or more precisely a local unstable manifold) of M (resp. Mg ). The above two theorems are still valid for manifolds with corners, provided (6.3.5) points strictly outward on all the smooth surfaces of 8M. If M is a manifold with corners, then W U is a local unstable manifold with corners. Theorems 6.3.1 and 6.3.2 can also be applied to inflowing invariant manifolds. In that case, the generalized Lyapunov type numbers are computed for the time reversed flow with the limits taken as t --t +00. Then, all things stated in the above two theorems remain exactly the same except that the word overflowing should be replaced by inflowing, and the notations NU, WU and W; by N S , WS and W; respectively. Here, NS is chosen to be a subbundle which contains T M and is positively invariant under the linearized flow generated by (6.3.5) . Suppose now that M = M, i.e., M is a compact, boundaryless C r manifold. Then, Theorem 6.3.2 implies the following usual stable manifold theorem and perturbation theorem. Theorem 6.3.3. Suppose that M is a compact, boundaryless, C r invariant manifold of (6.3.5), and NU (resp. N S ) is a subbundle satisfying the hypotheses of Theorem 6.3.2 (resp. for the time reversed flow) and
Then there are c r manifolds W U , W S tangent to N U , N S along M with WU overflowing invariant, WS inflowing invariant under (6.3.5), with Wu n w s = M . Moreover, there are C r manifolds W; and W; C 1 close to WU and Ws, and having the same dimensions as WU and Ws, with W; n W; = M g , W; overflowing invariant and W; inflowing invariant under (6.3.13), provided that g is C 1 close to f. Definition 6.3.4. A manifold M satisfying the hypotheses of Theorem 6.3.3 is called a C r normally hyperbolic invariant manifold of (6.3.5).
396
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
Remark 1. Sometimes, for convenience, the manifold M with aM#- 0 is also referred to as a Cr normally hyper bolie invariant manifold if M is invariant and has an unstable subbundle NU and a stable subbundle NS satisfying the hypotheses of Theorem 6.3.3. In this case, the stable manifold theorem is still valid, but the perturbation theorem may be violated, which is simply because M 9 may not be invariant. In order to use the perturbation theorem, some modification technique is needed. For further details, one may refer to [45,180] or the proof of Proposition 6.3.9. We illustrate the above three theorems with the following example. Example 1. Consider a system
Xl = 2XI + xIx~, X2 = -X2 + x~ cos 27fX3/T, X3 = 1,
MI = {(XI,X2,X3) : M2 = {(Xl, X2, X3) : M3 = {(Xl, X2, X3) :
(6.3.14)
-1::; Xl::; 1, X2 = 0, X3 E TI}, Xl = 0, -1/2 < X2 < 1/2, X3 E TI}, Xl = X2 = 0, X3 E TI}. ~.
It should be clear that MI and M3 are overflowing invariant, while M2 and M3 are inflowing invariant under (6.3.14). And we have
TMI
N 7fN
=
= MI
x (lR,O,lR),
= MIX (0, lR, 0),
(~~~) 000
,
7fT
=
(
100) °001 °° ,
6.3.
Orbits Heteroc1inic to Invariant Manifolds
397
By (6.3.8) and (6.3.9), we get
It follows from Theorem 6.3.1 that Ml will be preserved under perturbations. Now we show that M3 is a normally hyperbolic invariant manifold. First, M3 is a compact, boundaryless manifold, and we have TM3 = M3 1= M3 J = M3
7fJ=
7fJ=
000) ( 010 , 000
X
(0,0, lR),
x (lR,O,O), X
(0, lR, 0),
100) ( 000, 000
7fT
000)
= ( 000
.
001
Due to (6.3.10)-(6.3.12), the generalized Lyapunov type numbers are given by
Next let the time be reversed. This causes I and J, 7fJ and 7fJ, and hence AI(p) and 1/J(p) to interchange their positions. Therefore, NU = TM3 EB I is an unstable subbundle, and NS = T M3 EB J is a stable subbundle. This means that M3 is normally hyperbolic. It follows from Theorems 6.3.2 and 6.3.3 that there exist a 2-dimensionallocal unstable manifold W U and a 2-dimensionallocal stable manifold WS tangent to NU and NS along M 3, respectively. Moreover, M 3, WU and WS are structurally stable.
398
6.3.2.
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
Existence and transversality of singular orbits
Let us now return to our original problem concerning the existence and transversality of singular orbits heteroclinic to invariant manifolds. We first continue to consider system (6.3.1) with hypotheses
(Hl)-(H3). Assume y E U c JRm . Notice that any orbit contained in WU(pd n WS(P2) can be parametrized by
Ca
:
x
=
x = x(t -
to,a,y),
where to E JR, a E W, and W is a C r (c - I)-dimensional compact manifold. Let A(t, a, y) = Dxf(x, y), it
v=
= A(t, a, y)u, -A*(t, a, y)v.
(6.3.15) ( 6.3.16)
Owing to Lemmas 6.2.1 and 6.2.2, we have the following proposition. Proposition 6.3.4. Suppose that (HI) holds. Then (6.3.15) has an exponential dichotomy in both R+ and R-, and exactly c linearly independent bounded solutions Ul(t - to, a, y), ... , uc(t - to, a, y). And the a~joint sys~m (6.3.16) has exa.::tly d linearly indep~ndent bounded
solutwns'ljJl(t to,a,y), ... ,'ljJd(t to,a,y). Moreover,"u~ and'ljJ] approach zero exponentially as t -+ ±oo for i = 1, ... , c; j = 1, ... ,d. Proposition 6.3.5. For i = 1, ... , c, j = 1, ... , d, the following
equalities are valid:
Proof. By Lemmas 6.2.1, 6.1.6 and 6.1.7, we have
'ljJj(t) E 1J?P;;*(t) n 1J?Ps-*(t), Ui(t) E 1J?Ps+(t) n 1J?P;(t), = (1J?(I - Ps+*(t)))J.. n (1J?(I - Pu-*(t)))J.. = (1J?P;;*(t))J.. n (1J?Ps-*(t))J... D
6.3.
Orbits Heteroc1inic to Invariant Manifolds
399
Proposition 6.3.6. There exist C r functions F l , ... ,Fd satisfying
(DxFi)*f
= 0,
Proof. It should be clear that there exist n - 1 linearly independent and locally defined C r functions F l , ... , F n - l with
(DxFi,J)
= O.
(6.3.17)
Differentiating (6.3.17) with respect to x we obtain
which means that DxFi(x, y) satisfies the adjoint system (6.3.16). By Proposition 6.3.5 and the fact that the vectors
DxFl(X, y), ... ,DxFn-l(x, y) span the orthocomplement of the vector Ul = f (x, y), we may as well assume DxFi(X, y) = 'l/Ji, i = 1, ... ,d. Then it is easy to get
i(DyFi(x, y)*) = f* . DxyFi = -(DxFi)* Dyf = -'ljJi Dyf(x, y).
o Now we denote
M(q) Mi(q)
=
= (Ml(q), . .. , Md(q)),
L:('l/J7 h (ql)
+ (DyFi(x(t),y))*g(qI))dt
- (DyFi(x(y), y))* L: g(ql) dt, where q = (y,zo,a,/-L), 'l/Ji
ql(t)
=
= 'l/Ji(t,a,y),
(x(t), y, Zo +
x(t)
= x(t,a,y),
l w(x(s), y)ds,
and Fi is obtained by Proposition 6.3.6.
/-L,
0),
(6.3.18)
400
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
Proposition 6.3.7. The improper integral in (6.3.18) converges absolutely and is equivalent to Mi(q)
=
L:
'I/J;(h(ql(t))
+ Dyf(x(t), y)
l g(ql(s))ds)dt.
(6.3.19)
Proof. Since x = x is a heteroclinic orbit, x(t), Dyf, g, and hare all bounded. Thus, by 'l/Ji ~ 0 exponentially as t ~ ±oo, we can conclude that integral (6.3.19) converges absolutely. Integrating by parts the second term in the integrand of (6.3.19) and using Proposition 6.3.6, we obtain the equivalence between (6.3.18) and (6.3.19).
o Now set ,:
x
= x(t,ii,y), Y = y.
Theorem 6.3.8. Suppose that (Hl)-(H3) hold and there exists q = (y, zo, ii, Jl) such that M(q) = 0, D(zo,a,/L)M(q) has rank d and D(zo,a)M(q) has rank b with p - d + b > O. Then, there is an EI :::; Eo such that for 0 < lEI < EI there exists a C r (p - d + b)-dimensional hypersurface Hf C V near f..L = fl, and system (6.3.1) has an orbit If with x and y components close to I and heteroclinic to two (homoclinic to one if m > 0) C r f.-dimensional normally hyperbrlic invariant tori when f..L E H f • Moreover, the stable and unstable m'anifolds of these invariant tori intersect transversally near (x(t, ii, y), y, zo) for all t in some bounded interval if there are d = n + c - kl - k2 column vectors of the matrix D(zo,a)M( q) which are all nonzero. Before proving Theorem 6.3.8, we make some necessary geometric preparations. Consider a manifold defined by Mi
= {(x, y, z) : x = Xi(Y),
Y E U, z
E T€},
i
= 1,
2.
Obviously, Mi is an invariant manifold of (6.3.1) with E = o. For = (x, y, z) E M I , the tangent spaces TpMI and Tpffin+m+e have the following decompositions:
p
TpMI = ffim
X
ffie,
6.3.
Orbits Heterociinic to Invariant Manifolds
T:p JRn+mH = E yS
X
E yU
401
X
JRm
X
JRe ,
where E~ and E: are the stable and unstable subspaces of the linearized flow of 'Ii; =
Dxf(XI(Y)'Y)w.
Denote by Au(Y) (resp. As(Y)) the smallest positive (resp. biggest negative) real part of the eigenvalues of Dxf(XI(Y), y) and
Then we can define the stable and unstable sub bundles of MI as follows: In fact, the generalized Lyapunov type numbers associated with NU now are given by AEU(p) = e-Au(Y) < 1, VE'(p) = eA,(y) < 1, (JE'(p) = 0; and those associated with NS under the time reversed flow are given by AE'(p) = eA,(y) < 1, l/E"(p) = e-Au(Y) < 1, (JEu(p) = 0, for any p E MI' Thus, MI is a C r (m + f)-dimensional normally hyperbolic invariant manifold. Similarly, M2 is also a C r (m + f)dimensional normally hyperbolic invariant manifold. Furthermore, MI and M2 have a C r (kl + m + f)-dimensional unstable manifold WU(Md and a C r (k2 + m + f)-dimensional stable manifold W S (M2) respectively. The intersection
r = WU(Md n W
S
(M2 )
= {(x,y,z): x = x(to,a,y), to is a C r (c + m
+ f)-dimensional
E
JR, a E W, Y E U, z = Zo E T e}
heteroclinic manifold.
Chapter 6.
402
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
We now consider the presistence of Mi.
Proposition 6.3.9. There are a compact neighborhood UI C U of the origin and an E~ > 0 such that for lEI < E~ :::; Eo system (6.3.1) has two c r (m + f) -dimensional normally hyperbolic locally invariant manifolds M:
= {(x, y, z) : x =
Xi(y, z, /-L, E)
=
Xi(Y)
+ 0(10),
y E UI ,
Z
E T e}
for i = 1, 2, and M fl has a C r (kl + m + f) -dimensional local unstable manifold WU(Mn and M f2 has a C r (k2 + m + f)-dimensional local stable manifold WS(Mf2). Moreover, WU(M}) and WS(Mn are C r close to WU(MI) and WS(M2) respectively.
Proof. When m = 0, the conclusion is a direct consequence of Theorem 6.3.3. When m > 0, MI = M2 == M is neither overflowing nor inflowing invariant. To apply Theorems 6.3.1-6.3.3, some technical treatment must be made. Let U I cUbe a compact neighborhood, and U2 , U3 be open neighborhoods with U I c U2 C U2 C U3 C U. Now we can choose a Coo function ¢> : lRm --t lR such that
¢(y) = 0 ¢(y) = 1 ¢(y) = -1
for for for
y E UI or y E BU2 , y E BU3 .
y E lRm
-
U,
Let WI, W 2, and W3 be the submanifolds of M with y restricted in U I , U2 , and U3 respectively. Then,
and W 2 (resp. W 3 ) is an overflowing (resp. inflowing) manifold of the following modified unperturbed vector field,
x = f(x, y), if = ¢(y)y, i = w(x,y).
(6.3.20)
6.3.
Orbits Heteroclinic to Invariant Manifolds
403
Notice that, for y E Ul , (6.3.20) is identical with the unperturbed system of (6.3.1). Then the proposition follows from Theorem 6.3.2 and Definition 6.3.2. 0 Remark 2. By Definition 6.3.2, M fl = M f2 == M f may not be really invariant when m > O. So, the manifold WS(Mf ) certainly need not be a stable manifold of M f in the usual meaning. Since some points on M f may leave Mf in finite time by crossing its boundary, we cannot expect that the points in WS(Mf ) will actually tend to any point on ME as t - t +00, although they approach M f in forward time. A similar illustration may be made for WU(Mf ). Thus we need some further study of the dynamics on M f • Let Xl = X2 == X. Consider the restriction to Mf of system (6.3.1):
if = Eg(X(y, z, j.L, E), y, z, j.L, E), .i = W(X, y) + EV(X, y, z, j.L, E).
(6.3.21 )
Assume (H2) and (H3) hold. By the averaging theorem (see [180] and the related references therein) and
g(X, y, z,
j.L,
E) = g(x(y), y, z, j.L, 0)
+ O(E),
system (6.3.21) possesses a C r .e-dimensional normally hyperbolic invariant torus with a CS (m- j +.e)-dimensional unstable manifold wu('t) and a CS U+.e)-dimensional stable manifold ws('t). Here the differentiability is with respect to y, z, j.L and E. Then, for E > 0 small enough, system (6.3.1) has a C r .e-dimensional normally hyperbolic invariant torus
tun
Tf(y) = {(x,y,z):
x = X, (y,z) E
t(Yn
having a CS (kl + m - j + .e)-dimensional unstable manifold WU(Tf ) C WU(Mf ) and a CS (k2 + j + .e)-dimensional stable manifold WS(Tf ) C WS(Mf ). Let
7rp
be a (n - 1 + m )-dimensional section passing through a point p = (x( -to, a, y), y, zo) E
r
404
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
and be spanned by the m-dimensional y space and a subspace Tx complementary to TpCa in the x space. By the hypothesis (HI), the dimension of Tx is n - 1. Let
S;
S;
Then, is (k l -1 +m )-dimensional and is (k2 -1 +m )-dimensional. The perturbation theorem implies that, for If I small enough, WU(Mn and WS(M;) intersect trp transversally in a (kl - 1 + m)-dimensional manifold: S;,f = WU(M}) n 7rp and a (k2 - 1 + m )-dimensional manifold: S;,f
= W S(Mf2 ) n 7rp,
respectively. The above intersections may have a countable number of disconnected sets. In this case, we always choose S;,f (resp. S;,f) to be the connected set of points which is closest to M; (resp. Mn in backward (resp. forward) time along WU(Mn (resp.W-S(Mn). When m > 0, we define
Clearly,
Proposition 6.3.10. There exists an fl ~ f~ small enough such that there are two points p~ and p: with y~ = Y: == Yf for < If I < fl'
°
7r;
Proof. Let be the restriction of trp on Y space, Wp~m(Tf) and W:,m(Tf ) be the projections of WpU(Tf) and W:(Tf ) onto These
7r;.
6.3.
Orbits Heteroc1inic to Invariant Manifolds
405
two projections have dimensions m - j and j, respectively. By the perturbation theorem, W;'m(TE ) and W;,m(TE ) are C 1 close to the unstable and stable manifolds of the hyperbolic saddle y = y of (6.3.3), and hence they must intersect transversally at some point YE near Y for f small enough. 0 Proof of Theorem 6.3.8. In the following, we assume f > 0. In the case f < 0, we only need to interchange the positions of j and m-J. From the above analysis, we see that
and
x s = {x: : (x:, YE) c S;,E} are (k1 - 1)- and (k2 - 1)-dimensional manifolds respectively, and the distance on 7rp between WU(Mn and WS(Mn is entirely determined by the distance between XU and XS. By Proposition 6.3.4 and the generalized Melnikov method developed in the above section (particularly Proposition 6.2.5), the separation between WU(Mn and WS(Mn is completely measured by their separations along the d different directions,
These separations are given by
di = di([j, zo, ct, f.L, f) = [?jJi( -to, ct, y)[-l (?jJi( -to, ct, y), x~ - x:), i = 1, ...
,d. Since dly, zo, ct, f.L, 0)
= 0, we have
(6.3.22) where
Chapter 6.
406
Let
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
qo(t) = (x(t, a, y), y, Zo + J; w(x(s, a, y), y)ds), q!(t) = (x~(t), y!(t), z!(t)), i = u, s, x~(t) = x(t - to, a, y) + Exi(t) + O(E2), y!(t) = y + Eyi(t) + O(E)
satisfy x~(O)
=
x~,
y!(O)
= Yf respectively. It is easy to see that
xi (t) = Dxf . xl + Dyf . yi + h(qo(t - to), I-l, 0), 1ii(t) = g(qo(t - to),I-l,O), where x = x,y = y. Denote .6.j(t) = ('l/Jj(t - to, a, y), xl(t)) , z = u, s. Then
Aj(t) = 'l/J;Dyf· yHt)
+ 'l/J;h(qo(t -
to), I-l, 0).
(6.3.23)
By Proposition 6.3.6, we have
ITi 'l/J;Dyf. yt(t)dt = -(DyFj(x, y))*yi(t)I;' a
~
T
+ fa ' DyFj(x, y)g(qo(t - toFP, O)dt,
(6.3.24)
for i = u, s; j = 1, ... , d. By integrating (6.3.23), and using (6.3.24) and the following facts:
yf(O) = yf(O),
yf (t)
is bounded for t
:s;
0,
yf(t)
is bounded for t
~
0,
DyFj (x(T1t - to, a, y), y)
~
DyF'.i(Xl(f}), y)
as T1t ~ -00,
DyFj(x(Ts - to, a, y), y)
~
DyFj(X2(Y), f})
as Ts ~ +00,
yf(Ts) - yf(T1t) ~
L:
g(qo(t - to), I-l, 0) dt,
we obtain (6.3.18) if we replace y and t by Y and t + to respectively. The existence of the orbit "if is a direct consequence of (6.3.22) and the implicit function theorem. To verify the transversality condition of "if' it suffices to show that W1t(Mn and W S (Mf2 ) (resp. W1t(Tf ) and
6.3.
Orbits Heteroc1inic to Invariant Manifolds
407
WS(Tf ) when m > 0) intersect transversally at Pf = (Xf' Yf' Zf) E "if when they are restricted on 7rp x Tf. Denote these two restrictions by LU and V respectively. From the above discussion, we see the Y components of WU(Tf ) and WS(Tf ) have a transversal intersection at Yf near y. Therefore, for conciseness, we may as well assume that m=O. For q E (WU(pd n W S(P2)) n 7rp , we denote Tl = (TqWU(pd n TqWS(P2)) n 7r p , T2 = (TqWU(pd)~ n TqWS(P2), T3 = TqWU(Pl) n (TqWS(p2))~' T4 = (TqWU(pl))~ n (TqWS(p2))~' These spaces may be parametrized by a, vf, vI and () respectively. Let = (a,vl), V U = (a,v l ). Then, by a similar discussion made in the previous section, we have the following expressions:
VS
v
=
{(z,x!)},
L U = {(z, x~)},
= (a,vf,mHz,a,vf), x~ = (a, mf(z, a, vf), vf, C r , i = u, s; j = 1,2. x:
m2(z,a,vf)), m~(z, a, vn),
where mj is At point Pf = (Xf' Zf) E "if' the tangent spaces to V and LU are spanned by column vectors in the following matrices:
D(z,VS)V(Pf)
=
I 0 0
0 I 0
0 0 I
~~~ 8z 8n 8vi ~~~ 8z 8n 8vi
, D(z,vu)LU(Pf)
=
I 0 0 0 0 I 8m) ~ 8m l 8z 8n 8v1 0 0 I 8m2 ~ 8m2 8z 8n 8v1
It should be clear that D(z,VS)V(Pf) has kl + £ - 1 columns and D(z,vu)LU(Pf) has k2 + £ - 1 columns, and the dimension of 7rp x Tf is n + £ - 1. Then, by the proofs of Theorems 6.2.7 and 6.2.9, and Remarks 5 and 7 in Sec. 6.2, we see that LS and LU intersect transversally at Pf. D
408
Chapter 6.
Melnikov Vector, Homoclinic and Heteroclinic Orbits
If y E T m , then M f is an (m + .e)-dimensional normally hyperbolic invariant torus. Hence, averaging is unnecessary in this case, and the projections of WU(Mf ) and WS(Mf ) on y space are all m-dimensional, which in turn means that the condition y~ = y: can be trivially satisfied. Then, by an analogous proof for the transversality as above, we obtain the following proposition. Theorem 6.3.11. Suppose that hypothesis (HI) holds, y E T m , m> 0, and there exists q = (t), za,ci, fl) such that M(q) = 0,
rank (DM(q)) = d,
rank (D(y,zop)M(q)) = b,
andp-d+b> O. Then, there is an El ::; Ea such that, forO < lEI < El, there exists a CT (p - d + b) -dimensional hypersurface H f C V near J..l = fl, and system (6.3.1) has an orbit If with x component close to {x: x = x(t,o.,y), t E m} andhomoclinic to aCT (m+.e)-dimensional normally hyperbolic invariant torus when J..l E H f • MGJ:f~ver, WU(Mf ) and WS(Mf ) intersect transversally near (x(t, a, y), y, za) for all t in some bounded interval if there are d = n + c - kl - k2 column vectors of the matrix D(y,zo,a)M(q) which are all nonzero.
Remark 3. The transversality claimed in Theorem 6.3.8 is valid along a sufficiently long segment of the orbit If if El is small enough, and is valid along the whole orbit If if v == 0 in (6.3.1). Remark 4. For system (6.3.1), the conclusions similar to those given in Corollary 6.2.8, Theorems 6.2.9 and 6.2.10 are still valid when m = O. We leave the concrete statements and the details of the proof to the readers. Remark 5. If (6.3.1) is a Hamiltonian system (i.e., f(x,y) JDxH(x,y), x E IR2n) and there exist n independent integrals Kl H, K 2 , ..• ,Kn with (DxKi, J DxH) = 0,
= =
then c = d = kl = k2 = n. By a proof similar to that of Proposition 6.3.6, we can take 'lj;i = DxKi(X, y), F;(x, y) = Ki(x, y). Therefore,
6.4.
Heterociinic Orbits in Singular Perturbation Problems
409
Theorems 6.3.8 and 6.3.11 extend and include the theorems 4.1.9, 4.1.10,4.1.13 and 4.1.14 contained in [180J.
Remark 6. When m = 0, we can use the orthogonality condition and transversality theory developed in Sec. 6.2 to obtain results similar to those given in Theorems 6.3.8 and 6.3.11. For details one may refer to [200J.
6.4.
Heteroclinic Orbits in Singular Perturbation Problems
Singular perturbation problems have extensive applications in engineering. The existence of heteroclinic and homo clinic orbits and their persistence under singular perturbations are of very important significance in dealing with traveling wave problems for reactiondiffusion equations or for viscous approximations of hyperbolic conservation laws (e.g., the existence of viscous profiles for all magnetohydrodynamic shock waves). For details one may see [159J and the references therein. In this section, we introduce the results given in [159,202J. By using Fenichel's geometric singular perturbation theory ([46]), we can show that the transversal intersection of stable and unstable manifolds of the reduced problem implies the existence of transversal heteroclinic or homoclinic orbits of the singularly perturbed problem. We derive some analytical conditions for transversality, and illustrate how these results can be used to prove the existence of heteroclinic or homoclinic orbits in singularly perturbed problems which depend on additional parameters. Consider the following singularly perturbed system,
where
E
E (-Eo, Eo), Eo
>
x
= f(x,
EY
=
°
y, fJ, E),
g(x, y, fJ, E),
(6.4.1)
small, (x, y) E M, fJ E U, M is an open
410
Chapter 6.
Melnikov Vector, Homociinic and Heterociinic Orbits
subset of lR n + k , U is a neighborhood of the origin in lRm , f, 9 E CT, r 2: 2. If we rescale the time t by T = t / E, then system (6.4.1) has the form x' = Ef(x, y, j.1, E), (6.4.2) y' = g(x, y, j.1, E). (6.4.1) and (6.4.2) are usually called slow system and fast system respectively. For conciseness, we use E- 1X f and Xf to denote systems (6.4.1) and (6.4.2). By setting E = 0 in ( 6.4.1) and (6.4.2), we obtain two different problems, the reduced problem defined by
x = f(x, y, j.1, 0), 0= g(x, y, j.1, 0),
(6.4.3)
and the layer problem defined by
x'= 0, y' = g(x, y, j.1, 0).
(6.4.4)
Let G(j.1) be a CT manifold of solutions of the equation
g(x, y, j.1, 0) = O.
(6.4.5)
Obviously, the reduced problem (6.4.3) defines a dynamical system on G(j.1), while G(j.1) is a manifold of equilibria for the layer problem (6.4.4). If we call x the slow variable and y the fast variable, then we see that the reduced problem essentially captures the slow dynamics and the layer problem, the fast dynamics. An appropriate combination of the results on the dynamics of these two limiting problems will give us a clear geometric construction of the dynamics of singularly perturbed problem (6.4.1) for small E. A good understanding of this relation depends on the theory of invariant manifolds for singularly perturbed problems developed in [46]. 6.4.1.
Geometric singular perturbation theory
In this subsection, we introduce briefly Fenichel's invariant manifolds theory. To some extent, it says that the regular singular perturbations (i.e., rank Dyg(x, y, f-L, 0) = k for (x, y) E G(j.1)) are not "all
6.4.
Heterociinic Orbits in Singular Perturbation Problems
411
that singular" which makes it reasonable to decouple problem (6.4.1) into two lower dimensional problems (6.4.3) and (6.4.4) for E = 0, and to apply the methods from dynamical system theory to singularly perturbed problems. Since the dependence on the parameter /.L is not discussed explicitly in this subsection, we drop it in all our notations for the moment. As a starting point, we introduce a dummy variable E in the phase space and consider the following equivalent system X f x 0 of (6.4.2),
x' y'
= Ef(x, y, E), = g(x, y, E),
E'
= 0
(6.4.6)
defined on M x (-Eo, Eo). The flow induced by X f x 0 is denoted by "·r". Let DX be the linearization of the vector field X. Since Xo vanishes identically on G, TmG is an invariant subspace of DXo(m) for any mEG. Consequently, DXo(m) induces a linear map
on the quotient space. Let KeG be a compact subset such that QXo(m) has k S eigenvalues in the left half-plane, k C eigenvalues on the imaginary axis, and k U eigenvalues in the right half-plane, for all m E K. Then, D(Xf x O)(m, 0) has k S eigenvalues in the left half-plane, k C + n + 1 eigenvalues on the imaginary axis, and k U eigenvalues in the right half-plane, for all m E K. It should be clear that k S + k C + k U = k. For each m E K, let E:n, E~ and E~ be the corresponding stable, center, and unstable eigenspaces associated with D(Xf x O)(m,O). We call the manifolds BS, BC and BU a center-stable, a center, and a center-unstable manifold for X f x 0 near K x {O} if they all contain K x {O}, and are all locally invariant under the flow of X f x 0 and tangent to E:n EB E~, E~, and E~ EB E~ at (m,O) respectively, for all (m,O) E K x {O}. Obviously, the dimensions of these manifolds are k S + k C + n + 1, k C + n + 1, and k U + k C + n + 1 respectively.
412
Chapter 6.
Melnikov Vector, Homociinic and Heterociinic Orbits
We define two submanifolds of G as follows: G R = {(x,y) E G: rankDyg(x,y,O) G H = {(X,Y) E G R : F = O}.
= k},
By the implicit function theorem, we can parametrize G R locally by solving the equation g(x, y, 0) = 0 locally for y = y(x). Notice that G R may be the union of several connected manifolds separated by submanifolds of singular points where some of the k eigenvalues may be zero. Moreover, for any compact set KeG H, K is a normally hyperbolic invariant manifold of the layer problem (6.4.4). Let N be the complement of the tangent bundle TG R. Then we have the splitting T Mlc R = TG R EO N, and t~rojection 7fc TMlc R --t TG R . Define -
The system XR(m) is called the reduced system of (6.4.1). It is easy to see that XR(m) is equivalent to system (6.4.3). Before stating the main result of Fenichel's invariant manifold theory, we introduce the following concept. Definition 6.4.1. Let BS be a center-stable manifold for X€ x 0 near K x {O}. We say that a family {PS(p) : p E B S} is a CT2 family of CT! stable manifolds for BS near K if (1) ps (p) is a CT! manifold for each p E BS. (2) p E PS(p) for each p E BS. (3) ps (p) and ps (q) are disjoint or identical for each p and q E BS. (4) pS(m,O) is tangent to E:n at (m,O) for each m E K. (5) {PS (p) : p E BS} is a positively invariant CT2 family of manifolds. Here, positive invariance means that
FS(p) . T C FS(p. T) for all p E BS and all T 2: 0 such that p. [0, T] E BS.
6.4.
HeterocJinic Orbits in Singular Perturbation Problems
413
The family of unstable manifolds {FU(p) : p E BU} can be defined similarly.
In order to help the reader keep track of the above definition, we make some explanation. The family of stable (resp. unstable) manifolds FS (resp. FU) provides a foliation of the center-stable (resp. center-unstable) manifolds BS (resp. BU). It means that
and the fibers F S( m, 0) are roughly parallel to E:n and FU( m, 0) are roughly parallel to E~. The following invariant manifold theorem ([46]) describes the geometric structure and its variations with t for the flow induced by (6.4.1) near G x {O} when t is small enough. Theorem 6.4.1. Let M be a C r + 1 manifold, 1 :::; r < 00. Let XEJ t E (-to, to), be a r family of vector fields on M, and let G be a C r submanifold of M consisting entirely of equilibrium points of Xo' Let kS, k C, and k U be fixed integers, and let KeG be a compact subset such that QXo( m) has k S eigenvalues in the left half-plane, k C eigenvalues on the imaginary axis, and P eigenvalues in the right half-plane, for all m E K. Then: (i) There are a C r center-stable manifold BS, a C r center-unstable manifold BU, and a C r center manifold BC for X E x 0 near K. (ii) There is a C r - 1 family {FS(p) : p E BS} of r stable manifolds for BS near K. If p E M x {t} then FS(p) c M x {t}. Each manifold FS (p) intersects BC transversally at exactly one point. There is a C r - 1 family {FU(p) : p E BU} of r unstable manifolds for BU near K. If p EM x {t} then FU(p) eM x {t}. Each manifold FU(p) intersects B C transversally at exactly one point. (iii) Let Ks < 0 be greater than the real parts of the eigenvalues of QXo( m) in the left half-plane for all m E K. Then, there is a constant C s such that if p E BS and q E FS (p), then
c
c
c
d(p· T, q . T) :::; Cse KsT d(p, q)
414
for the for qE
Chapter 6.
Melnikov Vector, Homoclinic and Heteroclinic Orbits
all T :2: 0 such that p . [0, T] C BS. Let Ku > 0 be smaller than real parts of the eigenvalues of QXo(m) in the right half-plane all m E K. Then there is a constant C u such that if p E BU and PU(p), then d(p . T, q . T) ::; Cue KuT d(p, q)
for all T ::; 0 such that p . [T,O] c BU. (iv) If K C G H , define for (m, E) E Be, if E # 0, if E=~ Then Xc is a
cr-l
vector field on Be near K x {O}.
In the case K C GH, the assertion (iv) above says that the vector field c 1Xf can be cr-l extended to E = 0 in Be near K x {O}, and so reduce the singular perturbation problems to regular perturbation problems. More explicitly, any structure in G H which persists under regular perturbations, also persists under singular perturbations when confined to the center manifold. In other words, normally hyperbolic invariant manifolds of the reduced problem persist under singular perturbations. This idea has been carried out in [46]. In the following, we quote the corresponding results obtained in [46] with a slightly modified version given in [159]. Theorem 6.4.2. Let M, Xu and G be the same as in Theorem 6.4.1, and 2 ::; r < 00. Let N C G H be a j-dimensional compact normally hyperbolic invariant manifold of the reduced vector field X R with a (j + p)-dimensional local stable manifold WS and a (j + jU)_ dimensional local unstable manifold WU. Then there exists El > 0 such that: (i) There exists a cr-l family of manifolds {Nf : E E (-EI' EI)} such that No = Nand Nf is a normally hyperbolic invariant manifold of X f • (ii) There are C r - l families of (j + jS + kS)-dimensional and (j + jU + kU)-dimensional manifolds {N: : E E (-EI' Ed} and {N: :
6.4.
HeterocJinic Orbits in Singular Perturbation Problems
415
E E (-El' Ed} such that for E > 0 the manifolds N: and N: are local stable and unstable manifolds of Ne (iii) For E > 0 the local stable and unstable manifolds N: and N: are given by N:
=
{F€S(p) : p E Wn,
N:
=
{F€U(p) : p E W€U},
where F;(p) (resp. F€U(p)) are the projections of FS(p) (resp. FU(p)) from M x (-El,El) onto M, and W€s (resp. W€U) are the local stable (resp. unstable) manifolds of N€ for the flow restricted into the center manifold Be for fixed E. For E < 0, the same conclusion is valid by interchanging FS and FU.
Clearly, wg = W S and W~ = W U. Theorem 6.4.2 is essentially a direct consequence of Theorem 6.4.1 and the invariant manifold theorem. 6.4.2.
Transversal heteroclinic orbits
We now consider the existence of transversal homoclinic and heteroclinic orbits of the singularly perturbed system (6.4.1). Assume that G H has several connected branches, two of which are given by G i = {(x, Yi(X, J.£)) : x E U1 C lRn} for i = 1,2, where U1 is a non-empty open set. Let Ni(J.£) C G i be an invariant manifold of the reduced problem X R . Definition 6.4.2. A connected set r is called a singular heteroclinic orbit of (6.4.1), if r consists of the orbits of Xo and XR, and connects N1(J.£) and N 2 (J.£).
In Definition 6.4.2, there may be three special cases: (1) Nl = N 2 ; (2) The orbit set of Xo is empty; (3) The orbit set of XR is empty. In the first case, r is a singular homo clinic orbit. In the second case,
r
is a heteroclinic orbit of XR.
416
Chapter 6.
Melnikov Vector, Homoclinic and Heteroclinic Orbits
In the following, we assume Ni(J.L) C G i is a ji-dimensional compact normally hyperbolic (or overflowing, or inflowing) invariant manifold of XR. Let W~(J.L) and W1(J.L) be the (jl + H)-dimensional unstable manifold and (h + jn-dimensional stable manifold of NI (J.L) and N 2(J.L) , respectively. Denote by YI(X, J.L) and Y2(X, J.L) the hyperbolic equilibria of the system Y' = g(x, y, J.L, 0), such that PI = (x, YI(X, J.L)) E W~(J.L) ~nd p2. = (x, Y2(X, J.L)) E W1.(J.L), an'
intersect transversally along the singular heteroclinic orbit r. Then there exist an €l > 0 and a transversal heteroclinic orbit of the singularly perturbed system (6.4.1) connecting the manifolds NI,E and N 2,E for 0 < € < €l. Theorem 6.4.4. Assume that NI (J.L) (resp. N 2(J.L)) is a compact manifold with boundary overflowing (resp. inflowing) invariant for the reduced vector field X R and satisfies the assumptions of the unstable (resp. stable) manifold theorem for overflowing (resp. inflowing) invariant manifolds (Th.6.3.2). Then, Theorem 6.4.3 is still valid. By Theorems 6.4.1 and 6.4.2, Nf(J.L) and N~(J.L) are C r - l and we have dimN~(J.L)
= jl + jf + kf,
dimN2(J.L) = j2 + ji
+ ki·
lt follows from Theorems 6.4.3 and 6.4.4 that, to show (6.4.1) has a transversal heteroclinic orbit near r, it suffices to show Nf(J.L) and N~(J.L) intersect transversally along r. To formulate the problem more
6.4.
Heteroc1inic Orbits in Singular Perturbation Problems
417
precisely we need the following two hypotheses. Denote POI = (Xo(J.t), YI(Xo, J.t)) E Wr(J.t),
P02 = (xo(J.t), Y2(Xo, J.t))
,= {(Xo,Yo(T)): T
Yo(-OO) = YI(Xo,J.t),
E
W 2(J.t),
E lR}
c
E,
Yo(oo) = Y2(X o,J.t),
and assume that (HI) dim(TqF;(poI) n T q F;(P02)) = C for any q E " and d == k c - k1 - k~ ~ 1, (H2) NI(J.t) and N 2(J.t) are hyperbolic equilibria of X R,
dim(TqWr n Tq Wn
do == n
= Co for any
+ Co - H -
+
q E L,
j~ ~ 1.
It should be clear that d ~ 0, do ~ 0, and that either d = 0 or do = 0 implies the transversality, a trivial case. Assume that (HI) is valid. Then, the linear variational system
(6.4.7) has exponential dichotomies in both lR+ and lR-. By the fact that a solution Y(T) of (6.4.7) is bounded on lR if and only if
it follows that (6.4.7) has exactly c linearly independent bounded solutions
'TlI(T) = Y~(T), 'Tl2(T), ... ,'Tlc(T) on lR. Due to Proposition 6.1.7, the adjoint equation of (6.4.7) has exactly d linearly independent bounded solutions
418
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
on 1R and
(1/Ji(T), 1Jj(T)) = 0,
for i
= 1, ... , d,
j
=
1, ... , c.
For i = 1, ... , d, denote
\
(6.4.8)
and let 7r(Wf) and 7r(W{) be the x-coordinates of the manifolds Wf(J.l) and W~ (J.l), respectively.
Theorem 6.4.5. Let Nl(J.l) C G l (resp. N 2(J.l) C G 2) be a jldimensional (resp. j2-dimensional) normally hyperbolic invariant manifold of X R . Suppose that (HI) holds. Then the manifolds Nf(J.l) and N~(J.l) intersect transversally at the point (xo(J.l), Yo(T)) E I if and only if there exist exactly (e - c) linearly independent vectors ~j E T xo 7r(Wf) n TXo7r(W{) such that Mi(J.l)~j
where e
= 0,
i
= 1, ... , d,
j
= jl + i1 + kl + i2 + j~ + k2 -
= 1, ... , e -
c,
(6.4.9)
n - k ~ c.
Proof. The intersection of N1(J.l) and N 2(J.l) at a point q (xo(J.l),Yo(T)) is transverse if and only if dim(TqN~
+ TqNn
= n
+ k.
By
it suffices to show that
(6.4.10) holds if and only if the conditions of the theorem are satisfied. Obviously, e ~ c. For conciseness, we may as well assume that the time T = 0 at point q. Let cl>r(q) be the flow defined by the layer
6.4.
Heterociinic Orbits in Singular Perturbation Problems
419
problem (6.4.4). The linearization of the flow 3, (6.5.6) and (6.5.8). 0 Remark 1. If a, /3 are the control parameters of the saddle-nodes p and q respectively, then, in (6.5.3), we have C = and
°
Vi(O,O,z,t,a,/3,E,1])
=
O(z2),
v
=
j,g, i
= 1,2,3,
(6.5.11)
hex == bp hl(P, 0, 0, 0, 0, 1]) > 0, h{3 == bq h2(q, 0, 0, 0, 0, 1]) > 0, (6.5.12) hi(O, 0, z, t, a, /3, E, 1])
=
O(z2), i
= 2,3 in
Up, i = 1,3 in Uq. (6.5.13)
For definition, we assume further (H4) bp = bq = 1. The persistence of heteroclinic orbit r is equivalent to there being a non-empty intersection of the stable and unstable manifolds bifurcated from W~s and W~u as a, /3 and fL vary. Before giving a measure of the separation of these manifolds as we did in Sec. 6.2, we need to establish the principal normal coordinates along the orbit r. Now
432
Chapter 6.
Melnikov Vector, Homoclinic and Heteroclinic Orbits
we apply the exponential trichotomy theory developed in Sec. 6.1 to attack this problem. Let A(t) = DF(r(t), 0, 0, 0, 0). Consider the linear system
x=
A(t)x
(6.5.14)
-A*(t)x.
(6.5.15)
and its adjoint system
x=
If (H1)-(H3) hold, then in the admissible variables, we have r(t) (x(t), 0, 0) for r(t) E Up and
A(t) = (
A + f(r(t))
+ fx(r(t))x(t)
fy(r(t))x(t) B + g(r(t))
fz(r(t))x(t))
0
o
o
o
o
=
.
(6.5.16) Since x( t) tends to 0 exponentially as t ---t +00, we get A( +(0) = diag (A, B, 0). The roughness of the exponential dichotomy means that (6.5.14) has an exponential trichotomy in IR+. Similarly, when r(t) E Uq , r(t) = (0,0, z(t)),
A(t) = diag (A + f(r(t)), B + g(r(t)), -2z(t) + (}'(z)).
(6.5.17)
Also, by the fact that z(t) tends to zero as t ---t -00 and A( -00) = diag (A, B, 0), it follows that (6.5.14) has an exponential trichotomy in IR-, and the corresponding constants K 2: 1 and a » (J" > 0 can be taken the same as in IR+ . Then, by Propositions 6.1.1 and 6.1.7-6.1.9, we have the following two propositions. Proposition 6.5.2. Suppose that (H1)-(H3) hold, then (6.5.14) and (6.5.15) have exponential trichotomies in both IR+ and IR- with the same constants K, a, (J", and the corresponding projections are P:(t), P~(t), P~(t) and P~*(t), P~*(t), P:*(t), for i = +, -, respec-
tively, such that
6.5.
Heteroc1inic to Nonhyperbolic Equilibria
433
Proposition 6.5.3. Suppose that (H1)-(H3) are true, and d = n - s+ - u-. Then (6.5.15) has exactly (d - 1) linearly independent
bounded solutions 'l/Jl(t), ... ,'l/Jd-l(t) in E(a,l,m), and exactly one linearly independent bounded solution 'l/Jd( t) in (E( -(]", 1, m+) - E(a, 1, m+)) When t
~
n E(a, 1, m-).
0,
'l/Ji(t) E ~(Ps+(t) 'l/Jd(t) E ~(Ps+(t)
+ Pc+(t))-L, + P:(t))-L,
i
= 1, ... ,d -1,
and when t :S 0,
Moreover, if we extend Ps+(t), Pc-(t) and Pu-(t) along r for t E (-00, +00), then Ed(t) == span {'l/Jl(t), ... , 'l/Jd(t)}
= (~P/(t) + ~(Pu-(t) + Pc-(t))-L = (TrCt) W: + Tr(t) W~U)-L C (TrCt)r)-L.
Definition 6.5.2. 'l/Jl(t), 'l/J2(t), . .. , 'l/Jd(t) are called the principal normals of r. 6.5.2.
Bifurcation equations
Now we consider the persistence problem of the heteroclinic orbit accompanied by saddle-node bifurcations. This will be accomplished by consideration of local bifurcations near equilibria and measurement of the separation of stable and unstable (or center-unstable) manifolds along the principal normals.
434
Chapter 6.
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
Proposition 6.5.4. Suppose that (Hl), (H2) and (H4) are valid, -Jaha, 8 = ..j{3h(3. Then, for 1{31, IILI and a > small enough, system (6.5.1) has exactly two hyperbolic T-periodic orbits pi = P~(3€ near p satisfying P~(3€ = p:
>.
°
=
pi = P + (0,0, i>.)*
+ O(a) + O(>'E) + 0(>'{3), i = +,-j
°
and for lal, IILI and {3 > small enough, (6.5.1) has exactly two hyperbolic T-periodic orbits qi = q~(3€ near q with q~O€ = q: qi = q + (0,0, i8)*
+ 0({3) + 0(&) + 0(8a),
i=
+,-.
Proof. We consider system (6.5.3) in Up. Applying the theory of linear periodic system to the system defined by the first two equations of (6.5.3), we see that there exists a unique and hyperbolic T-periodic solution (x(t),y(t)) = 0(z 2H) with H = O(a) + 0({3) + O(E). Substitute it into the third equation, we have
i = _z2 + O(z)
+ aha + O(az) + 0(z2 H).
Clearly, the autonomous system
i = _z2 + O(z)
+ aha has exactly two hyperbolic equilibria z = z+ = >.+O(a) and z = z->. + O(a) near z = 0. Let z = Ui + Zi, i = +, -, then Ui satisfies U = -2i>.u + Gi(u, t, b),
=
where b = (>., (3, IL),
Gi(u, t, b) = _u 2 + 0(u 3 )
+ (0(>') + 0(H))(u 2 + >.u + a),
and G i is T-periodic with respect to t. By [62] Th.lV.l.l, the system
U = -2i>.u + Gi(O, t, b) has exactly one T-periodic solution
u
= Ui(t) = J~oo
s
e 2i -X Gi (0, t
+ s, b) ds = O(a) + O(>'H).
6.5.
Heteroc1inic to Nonhyperbolic Equilibria
435
+ Ui(t), we get V = -2i'\v + Hi(v, t, b),
By the change of variables u = v
(6.5.18)
where
Hi ( v, ., b) is T-periodic in t. Let TJ(p, 0") be the Lipschitz constant of Hi(·,t,b) for Ivl::; p, Ibl::; 0". Then it is easy to see that TJ(p, 0") = O(p)
+ 0(.\2) + O('\H).
Denote K = I J:!.ioo e 2iAS dsi = (2.\t I ; then an application of the uniform contraction mapping theorem (cf. the proof of [62] Th.lV.2.1 and 3.1) shows that, for P = 0(0:) and 0" so small that KTJ(p,0") < 1, (6.5.18) has a unique stable (resp. unstable) T-periodic solution v = Vi(t) for i = + (resp. -) with IVi(t)1 ::; PI = 0(0:). This completes the proof of the proposition. 0 Now fix the points on the orbits pi and qi corresponding to t = to, and still denote them by pi and qi respectively. Let Wis (resp. w~) be the stable (resp. unstable) manifold of pi (resp. q-) under the solution map of time T, where w~ = w~s for 0: = 0 and w~ = W~u for f3 = o. Obviously, dim w~ Take Po
=
= r(O) =
dicula~ to
s+
+ 1,
dim w~
=
(xo, 0, 0) E Up" Let
s+, 7r
C
dim w~
= u- + 1.
Up be the section perpen-
TPor(t), LOu = W qCU L:- = W ~s n 7r,
where
W~s
W~s ---+
W; as 0:
L -; = W ~ n 7r,
n 7r ,
Lu = W~ n 7r,
is the strong stable manifold of p+. Clearly, we have ---+ O. From (6.5.4) we have L~ = {(x, 0, 0): Ixl« I}. By Proposition 6.5.3, 7r
= span{Ed(O), L~, TPoL~} n Up,
436
Chapter 6.
Melnikov Vector, Homociinic and Heterociinic Orbits
and L~ intersects L~ transversally in (Ed(O) )1.. Thus, for a, {3, E small enough, there is a unique x = x* near x = Xo such that PU= (* x ,YU,zU) E
Lu,
Pis
S S) = (* X 'Yi' Zi
E
LiS' (6.5.19)
Now the separation between L~ and Lu is equal to the separation between the points pf and pU. In the following, we want to determine the vector pU - pf. From Proposition 6.5.4, we see
pi
= P + (0,0, iA)* + O(a) + O(A{3) + O(AE).
If we translate the origin of (6.5.3) to pi, then the perturbation terms have order O(A) + O(H). By the fact that pU - pt --t which in turn means that pt --t Po, pU --t Po as A, {3, E --t 0, and (6.5.3) is C r- 2 with respect to the parameters, we obtain
°
pi = Po + O(A) + 0({3) + O(E).
(6.5.20)
Similarly, we can show
pU
=
Po + O(A) + 0(8) + O(E).
(6.5.21)
Let qU(t, to), t :S to and qt(t, to), t 2: to with qU(to, to) = pU, qt(to, to) = be solutions of (6.5.1). Then, by (6.5.20) and (6.5.21), we have
pf
qU(t, to) qt(t, to) where r
= =
r(r) + AqX(t) + aq~(t) + 8q8(t) + {3q~(t) + Eq~(t) + o(H), r( r) + Aqi(t) + aq~(t) + {3qb(t) + Eq!(t) + o(H),
= t - to.
On account of Proposition 6.5.4 we get
qt( +00)
=
(0,0,1)*,
q,\"(+oo)
=
(0,0,-1)*,
= (0,0, -1)*, q~(+oo) = q!(+oo) = q>:( -00) = q~( -00) = q~( -00) = 0.
qlf( -00)
0,
6.5.
Heteroc1inic to Nonhyperbolic Equilibria
= A(t - to).
Set D(t)
437
Then
it = D(t)u,
= qt, qA' qY,
where u
+ Fa(r(t - to), t, 0, 0, 0), , + Ff3(r(t - to), t, 0, 0, 0), q~ = D(t)q~ + FIL(r(t - to), t, 0, 0, 0)7], +, -; v = u, +, -. It follows from (6.5.19) that =
q~ D(t)q~ q~ = D(t)q~
where i =
d
pU - pi
= 2: l'l/Ji(0)1- 2'l/J;(0)(pU - Pi)'l/Ji(O).
(6.5.22)
i=l
Let
dij(t, a, (3, J.L)
= 'l/J;(t - to)(qU(t, to) - qj(t, to))
(6.5.23)
== A(Af(t) - Ai (t)) + a(Af(t) - Ai{t)) + 8b.f(t) + (3(Bf(t) -Bf(t)) where j =
+, -;
+ E(Ef(t) -
EI(t))
+ o(H),
(6.5.24)
i = 1, .... ,d;
Ai(t) = W(t Ay(t) = W(t Bi(t) = 'l/J;(t Ei(t) = W(t b.i(t) = W(t
- to)qX(t) , - to)q~(t), - to)q~(t), - to)q~(t), - to)qY(t).
= U,], v = U,],
v
v=
v
U,],
= U,],
Since
Ay(t) = Ai(t) = 0, Ay(t) = W(t - to)Fa(r(t - to), t, 0, 0, 0), Bi(t) = W(t - t o)Ff3(r(t - to), t, 0, 0, 0), Ei(t) = W(t - to)FIL(r(t - to), t, 0, 0, 0)7], and 'l/Ji(t) tends to we have
°exponentially as t
AY(to ) = Af(-oo) = 0,
-t
±oo for i
= 1""
A{(to ) = AH+oo) = 0,
b.f(to) = b.f( -(0) = 0,
,d - 1, (6.5.25) (6.5.26)
438
Chapter 6.
L: = L: = L:
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
Mt(to) =
1/J;(t)Fa(f(t) , t + to, 0, 0, 0) dt,
(6.5.27)
Mf(to)
1/J;(t)F{3(f(t), t + to, 0, 0, 0) dt,
(6.5.28)
1/J;(t)F/L(f(t) , t + to, 0, 0, 0) dt,
(6.5.29)
Mt(to)
where Mia(t o) == Ay(to) -A{(t o), Mf(t o) == Bi(to) - BI (to), Mt(to)'r} == Ei(t o) - Ef(to); i = 1"" ,d - 1; j = +, -. Clearly, the improper integrals (6.5.27)-(6.5.29) are absolutely convergent. Proposition 6.5.5. Let l1/Jd(t) I = 1, then 1/Jd(t) ~ 0.
(x(t), y(t),
z(t))* == (0,0,1)* for t Proof.
By Propositions 6.5.1 and 6.5.3,
for t ~ 0. Then, by the fact that (6.5.16) implies -J;d(t) = we get 1/Jd(t) = (0,0,1)* for t ~ 0.
°
for t ~
° 0
By Propositions 6.5.3 and 6.5.5, we can take 1/Jd(t) == (0,0,1)* for 0, and 1/Jd( t) --t exponentially fast as t --t -(X). Consequently, we have
t
°
~
A!(to) = A!(+oo) = 1,
A;;(to) = A;;(+oo) = -1,
(6.5.30)
Ad(to) = Ad( -(X)) = 0,
(6.5.31 )
~d(to) = ~d( -(X)) =
(6.5.32)
0.
Define Mg(to) = B~(to) - B~(to), M%(to)TJ = E~(to) - E~(to). Then since x( t) --t exponentially as t --t +00 and along f( t)
°
6.5.
Heteroclinic to Nonbyperbolic Equilibria
439
we see that the improper integrals
L: = L:
Mf(to) =
~d(t)Ffi(r(t), t + to, 0, 0, 0) dt,
(6.5.33)
M%(to)
~d(t)FJL(r(t), t + to, 0, 0, 0) dt
(6.5.34)
converge absolutely. Now, owing to (6.5.22)-(6.5.34), pU -
pf
exMt(to)+{3Mf(to)+Mt(to)/-L+o(H) = 0, -iA + (3Mf(to) + M%(to)/-L + O(ex)
=
°
if and only if
i = 1"" ,d-1, (6.5.35)
+ o(H) = 0,
i
= +, -. (6.5.36)
Definition 6.5.3. We call (6.5.35) and (6.5.36) bifurcation equations associated with a heteroclinic orbit r. Let M(t o) = (Mi(to),"" M%_l(t o)). If the rank of M(to) is d - 1, then there exist C r - 3 functions 1; and Ai such that the bifurcation equations (6.5.35) and (6.5.36) have solutions il = 1;(/-L*, (3, to), A = Ai(/-L*, (3, to), where {3 ;::: 0, /-L* and il are an (m - d + 1 )-dimensional vector and a (d - 1 )-dimensional vector consisting of different components of /-L, respectively. Thus, we have proved our main result stated as follows.
Theorem 6.5.6. Suppose that (H1)-(H4) hold, d = n - s+ - u-, m ;::: d. Then (6.5.1) has no T-periodic orbit near p (resp. q) which in turn means that there is no heteroclinic orbit near r for ex < (resp. (3 < 0). Moreover, if the rank of M(to) is (d - 1) for some to and ex ;::: 0, (3 ;::: 0, then:
°
(i) (6.5.1) has no heteroclinic orbit near r when il i= 1;(/-L*,{3,to); (ii) there exist two (m-d+2)-dimensional C r - 3 hypersurfaces Li = {(A, (3, /-L*): A ;::: 0, {3 ;::: 0, A = Ai(/-L*, (3, to) = i({3Mf(t o) + M%(to)/-L) + 0({3) + o(t:) for il = 1;(/-L*, {3, to), i = +,-} in the neighborhood of
the origin in (A,{3,/-L*) space such that, in different regions of the parameter space with {3 > (resp. (3 = 0), system (6.5.1) has 8 topologically different structures near r with q+ i= q- (resp. q+ =
°
q-
= q).
Chapter 6.
440
Melnikov Vector, Homoc1inic and Heteroc1inic Orbits
Remark 2. The bifurcation diagrams are shown in [203J. 6.5.3.
An example
To close of this section, we give an example to show the application of Theorem 6.5.6. Consider the system
+ XlX3 + ax~(x4 + 1)2 + ,Bx5, = -X2 + X2X3 + J.LX5 + ,Bx~(X4 + 1)2, (6.5.37) = -x5 - 4x3g(X4) + ax~ + (a + ,B)x5 + (,13 + J.L)X~(X4 + 1)2, = -X~ - X~ + XlX2X4 + ,B(X5 + x4 + 1) + aX5 + ,BX~(X4 + 1)2,
Xl = X2 X3 X4
Xl
°
where g(y) = for y ::; -1/2, g(y) = (y + 1/2)2 for y > -1/2. When a = ,13 = J.L = 0, system (6.5.37) has equilibria p(O, 0, 0, -1) and q(O,O,O,O) with eignvalues 1,-1,0,-1 and 1,-1,-1,0 respectively, and a heteroclinic orbit r = r(t) = (0,0,0, X4(t)) satisfying
It should be clear that u+ = c+ = u- = c- = 1, s+ = s- = 2, d = 1, and that, in some neighborhoods Up and Uq , (6.5.37) has the same form as (6.5.3) (in the case of Up, we need make the change X4 + 1 ---t X3, X3 ---t X4), and a, ,13 are control parameters of the saddlenodes p and q respectively. Here, the C l smoothness of g is enough. We now have
A(t) = diag (1, -1, -4g(X4), -2X4 - 3x~),
'l/Jd(t) = 'l/Jl(t) = (0,0, v(t), 0)*,
t> - 0', t ::; 0,
6.5.
Heteroclinic to Nonhyperbolic Equilibria
441
M 1i3 -- MJ.L1
= lXJ X~(X4 + 1)2dt + 1:00 X~(X4 + 1)2 exp{4 l(X4 + 1/2)2ds}dt =j-1/\x+1)dx+jO (x+1)exp{-4jX -1
-1/2
t+
(X2 1/ 2))2 dx}dx
-1/2 X
X
1
=~8 + jO-1/2 (x+1)exp{.!.-ln(x+1)!x!3+2-41n2}dx X 2 = (2 + e L~oo ue du)/16 = 5/16. U
Then we get the bifurcation equation and bifurcation surfaces as follows:
-iA
Li
=
+
156 (,6 + JL)
+ O(a) + 0(,6) + o(JL) =
{(A,,6, JL): A 2: 0, ,6 2: 0, A = i5(,6 + JL + 0(,6)
0,
+ 0(JL))/16},
where A =...;0., i = +, - and Li is C 1 . Thus, in the (A,,6,JL) space, L+ and L_ divide the half-space ,6 > (resp. subspace ,6 = 0) into eight different regions such that, in each different region, system (6.5.37) has a different type of heteroclinic orbit near r.
°
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+ x 3 = 0, J..
f32x2
=
+ f33x3 = 0,
