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O.
suitably, we may assume that le(x) 1 < g x ElRn. Since N(x) = O(lxl q ) as x'" 0,
for all
e
(2.5.5) where
C 1
is a constant.
Now
IQ(x,z) 1 < IQ(x,O) 1 IN(x) 1
+
IQ(x,z) - Q(x,O) 1
We can estimate
IQ(x,z) - Q(x,O) 1
constants of
and
function
F
keg)
with
G. k(O)
(2.5.6)
IQ(x,z) - Q(x,O) I·
+
in terms of the Lipschitz
Using (2.3.5), there is a continuous =
0, such that (2.5.7)
IQ(x, z) - Q(x, 0) 1 oS. keg) 1z 1 for and
Izl oS. g. Using (2.5.5), (2.5.6), (2.5.7), for x E lR n , we have that IQ(x,z) 1 oS. C11xlQ
+
z E Y
keg) Iz(x) 1
(2.5.8)
oS. (C 1 + Kk ( g)) 1 x 1 Q •
Using the same calculations as in the proof of Theorem 1, if x(t,x O)
is the solution of (2.5.2), then for each
there is a constant
M(r)
r
>
0,
such that (2.5.9)
where
y = r
+
2M(r)k(g).
Using (2.3.6), (2.5.8) and (2.5.9), i f
provided
£
and
r
are small enough so that
z E Y,
a -
qy >
o.
28
2.
By choosing ~
K large enough and
small enough, we have
£
that
C2
2.6.
Properties of Centre Manifolds
K and this completes the proof of the theorem.
(1) manifold.
PROOFS OF THEOREMS
In general (2.3.1) does not have a unique centre For example, the system
x
=
-x 3 ,
two parameter family of centre manifolds
y
y= =
-y, has the
h(x,c ,c 2) l
where c
1 -2 l exp (- 2" x ),
0
1 -2 c 2 exp( - 2" x ), However, if
hI
and
g
are
0
1. (2)
[441.
and
> 0
are two centre manifolds for q (2.3.1), then by Theorem 3, hex) - hl(x) = O(lxl ) as x ~ 0 for all
h
x x x
If
If
f
f and
g
Ck , (k ~ 2), then
h
is
Ck
are analytic, then in general (2.3.1)
does not have an analytic centre manifold, for example, it is easy to show that the system 3
-x,
x
Y
=
-y + x
2
(2.6.1)
does not have an analytic centre manifold (see exercise (1)). (3)
Centre manifolds need not be unique but there are
some points which must always be on any centre manifold. example, suppose that of (2.3.1) and let (2.3.1). if
r
(xO'YO)
y
hex)
For
is a small equilibrium point be any centre manifold for
Then by Lemma 1 we must have
Similarly, O is a small periodic orbit of (2.3.1), then r must lie
on all centre manifolds.
Yo = h(x ).
2.6.
Properties of Centre Manifolds
(4)
Suppose that
29
(x(t),y(t))
is a solution of (2.3.1)
which remains in a neighborhood of the origin for all
t
>
O.
An examination of the proof of Theorem 2, shows that there is a solution
u(t)
of (2.4.1) such that the representation
(2.4.5) holds. (5)
In many problems the initial data is not arbit-
rary, for example, some of the components might always be nonnegative. Suppose S cffi n +m with 0 € S and that (2.3.1) defines a local dynamical system on
S.
It is easy to check,
that with the obvious modifications, Theorem 2 is valid when (2.3.1) is studied on Exercise 1.
S.
Consider x
3 -x,
Y=
2
-y + x .
(2.6.1)
Suppose that (2.6.1) has a centre manifold h
is analytic at
x = O.
a n+2
= na n
x.
2 n=2
anx
n = 2,4, ... , with
for
n
a 2n + l = 0
Show that
h(x), where
Then
hex) for small
y
for all a2
=
1.
n
and that
Deduce that
(2.6.1) does not have an analytic centre manifold. Exercise 2 (Modification of an example due to S. J. van Strien [63]). each
If
f
and Cr
r, (2.3.1) has a
g
are
COO
functions, then for
centre manifold.
However, the
size of the neighborhood on which the centre manifold is defined depends on
r.
The following example shows that in
~eneral (2.3.1) does not have a
rand
g
are analytic.
COO
centre manifold, even if
2.
30
PROOFS OF THEOREMS
Consider 3 x = - e:X - x ,
-y +
y
Suppose that (2.6.2) has a for
o.
Ixl < 0, le:I
0- 1 . Then since
COO
Choose
in
2
x ,
x, there exist constants
such that hex, (2n)
-1
2n )
=
L
a.x i 1
i=l for
Ixl
that i f
small enough.
Show that
o
a.
1
for odd
i
and
n > 1, (2i) (2n)
(1
-1
)a 2i
(2i-2)a
_ , i 2i 2
2, ... ,n
(2.6.3)
Obtain a contradiction from (2.6.3) and deduce that (2.6.2) does not have a Exercise 3. odd, that is
COO
centre manifold.
Suppose that the nonlinearities in (2.3.1) are f(x,y)
=
-f(-x,-y), g(x,y)
that (2.3.1) has a centre manifold [The example
-he-x) .
•
x
y
=
-g(-x,-y).
= hex)
with
Prove
hex)
3
= -x , y = -y, shows that if h is
any centre manifold for (2.3.1) then
hex)
~
-he-x)
in
general.) 2.7.
Global Invariant Manifolds for Singular Perturbation Problems To motivate the results in this section we reconsider
Example 3 in Chapter 1.
In that example we applied centre
manifold theory to a system of the form
2.7.
31
Global Invariant Manifolds y'
e:f(y,w)
w'
-w + y2 - yw + Ef(y,w)
E'
where
0
-
= O.
f(O,O)
(2.7.1)
Because of the local nature of our results
on centre manifolds, we only obtained a result concerning small initial data.
= SeT), where
v = -w(l+y) + y2, s
Let
= 1 + yeT); then we obtain a system of the form
S'(T)
(2.7.2)
where
gi(O,O)
(y,O,O)
=
E'
0
0, i
= 1,2.
Note that i f
is always an equilibrium point for (2.7.2) so we ex-
pect that (2.7.2) has an invariant manifold fined for
-1
Theorem 4.
where
7
3.
38
EXAMPLES
By Theorem 1 of Chapter 2, (3.1.3) has a centre manifold y
= h(x).
By Theorem 2 of Chapter 2, the equation which
determines the asymptotic behavior of small solutions of (3.1.3)
is -feu - h(u)).
Using (3.1.2) and
(3.1.4)
h(u) u
(3.1.S)
Without loss of generality we can suppose that the solution u(t)
of (3.1.S) is positive for a11
t > O.
Using L'Hopital's
rule, . u -1 -1 = 1 1m ~ = lim t t~oo
Hence, if
wet)
u
t~oo
u(t)
=
1
is the solution of _w 3 ,
then
fU (t) s - 3 ds.
w(t+o(t)).
w(O)
1,
(3.1.6)
Since (3.1.7)
where
C
is a constant, we have that (3.1. 8)
To obtain further terms in the asymptotic expansion of u(t), we need an approximation to (Mcp)(x) If
cp(x)
=
h(u).
To do this, set
-cp'(x)f(x-cp(x)) + cp(x) + f(x-cp(x)).
= -x 3 then
Theorem 3 of Chapter 2, hex) = _x 3 + O(x S). into (3.1.4) we obtain
so that by Substituting this
3.2.
Hopf Bifurcation
39
(3.1.9)
u
Choose
T
so that
grating over
[T, t)
1 w- (u(t)) where
w
u(T)
=
= 1.
Dividing (3.1.9) by
u 3 , inte-
and using (3.1. 8), we obtain
t + constant + (3+a)
is the solution of (3.1.6).
f:
u 2 (s)ds
(3.1.10)
Using (3.1.8) and
(3.1.9), t
fTU
2
-ft
(s)ds
T
=
U(()~s u s
+ ft O(u 4 (s))ds T
(3.1.11)
-In t- 1 / 2 + constant + 0(1).
Substituting (3.1.11) into (3.1.10) and using (3.1.7), -3/2 1 t- 1 / 2 __ t _ _ [(a+3)ln t + C) + 0(t- 3 / 2 ) (3.1.12)
u(t)
12 where
412
C
is a constant.
If
(x (t) ,y(t))
is a solution of (3.1.3), it follows
from Theotem 2 of Chapter 2 that either
(x(t) ,yet))
tends to
zero exponentially fast or x (t) where 3.2.
u(t)
= ±u(t),
yet)
is given by (3.1.12).
Hopf Bifurcation There is an extensive literature on Hopf Bifurcation
[1,17,19,20,31,34,39,47,48,51,57,59) so we give only an outline of the theory.
Our treatment is based on [19).
Consider the one-parameter family of ordinary differential equations on ~2,
3.
40
EXAMPLES
f(x,a), such that
f(O,a)
for all
0
=
sufficiently small
that the linearized equation about Yea)
±
iw(a)
where
yeO)
= 0,
x
=
0
Assume
has eigenvalues
Wo + O.
w(O)
a.
We also assume
that the eigenvalues cross the imaginary axis with nonzero speed so that
+ O.
y'(O)
Since
y'(O)
+ 0,
by the implicit
function theorem we can assume without loss of generality that
yea)
=
a.
By means of a change of basis the differen-
tial equation takes the form
:it
A(a)x
+
(3.2.1)
F(x,a),
where A(a)
F(x,a)
=
2
O(lxl ).
Under the above conditions, there are periodic solutions of (3.2.1) bifurcating from the zero solution. precisely, for
a
More
small there exists a unique one parameter
family of small amplitude periodic solutions of (3.2.1) in exactly one of the cases (i)
a < 0, (ii) a
=
0, (iii) a
>
However, further conditions on the nonlinear terms are required to determine the specific type of bifurcation. Exercise
1.
Use polar co-ordinates for 2
aX l - wX 2
+
Kxl(x l
wX
+
Kx (x 2 2 l
l
+
aX
2
to show that case (i) applies if plies i f
K < O.
K
>
0
and case (iii) ap-
O.
3.2.
Hopf Bifurcation
41
To find periodic solutions of (3.2.1) we make the substitution Xl where
E
Er sin 8, a
Er cos 8, x 2
=
is a function of
Ea,
+
(3.2.2)
After substituting (3.2.2)
a.
into (3.2.1) we obtain a system of the form r
dar
+
2
r C3 (8,aE)1
+
2 3
E r C4 (8,aE)
+
3
O(E )
We now look for periodic solutions of (3.2.3) with and of
r 8
E
(3.2.3)
+
0
C3 and C4 are independent and the higher order terms are zero then the first near a constant
rOo
If
equation in (3.2.3) takes the form r
E [ar
+
2
Br 1
+
2
3
E Kr .
(3.2.4)
r = r O' where rO is a zero of the right hand side of (3.2.4). We reduce the
Periodic solutions are then the circles
first equation in (3.2.3) to the form (3.2.4) modulo higher order terms by means of a certain transformation. out that the constant
B is zero.
It turns
Under the hypothesis
K
is non-zero, it is straightforward to prove the existence of periodic solutions by means of the implicit function theorem. The specific type of bifurcation depends on the sign of so it is necessary to obtain a formula for
K. and let
Let
where
B~1
(xl'x 2 ).
K
is a homogeneous polynomial of degree
i
in
Substituting (3.2.2) into (3.2.1) and using (3.2.5)
we obtain (3.2.3) where for
• 3,4,
42
EXAMPLES
3.
1
(cos e)Bi_l(cOs e, sin e,a) +
Lemma 1.
(3.2.6)
2
(sin e)Bi_l(cos e, sin e,a).
There exists a coordinate change -
r
=
r
+
£ul(r,e,a,£)
+
£ 2u 2 (r,e,a,£)
which transforms (3.2.3) into the system
-r
Ear
Wo where the constant
+
£ 2-3 r K
+
0(£ 3 ) (3.2.7)
+
0(£)
K is given by (3.2.8)
where
C3
D3 (e,0)
C4
and =
are given by (3.2.6) and
(cos e)B~(cOS e, sin e,O) - (sin e)B~(COS e, sin e,O).
The coordinate change is constructed via averaging.
We
refer to [19) for a proof of the lemma. If
K
= 0 then we must make further coordinate changes.
We assume that
K
+0
from now on.
Recall that we are looking for periodic solutions of (3.2.7) with
£
+
gests that we set
0 a
and
-r
near a constant
rOo
This sug-
= -sgn(K)£ and rO = \K\-1/2.
The next
result gives the existence of periodic solutions of
(3.2.9)
with
r - rO
Lemma 2. £
small.
Equation (3.2.9) has a unique periodic solution for
small and
r
in a compact region either for
£
>
0
(whrn
3.2.
Hopf Bifurcation
K < 0) or
E < 0
43
(when
K > 0).
and the period of the solution
Also
T(E)
The periodic solution is stable if
is given by
K< 0
and unstable if
o.
K >
Lemma 2 is proved by a simple application of the implicit function theorem.
We again refer to [19) for a proof.
Lemma 2 also proves the existence of a one parameter family of periodic solutions of (3.2.1).
We cannot immedi-
ately assert that this family is unique however, since we may have lost some periodic solutions by the choice of scaling, i.e. by scaling
a
+
aE
and by choosing
E
=
-sgn(K)a.
In order to justify the scaling suppose that Xl
= R cos e, x 2
bifurcating from
e
R sin xl
= x2
O.
aR
R When
is a periodic solution of (3.2.1)
+
a
+
aE.
R satisfies
O(R 2 ) .
R o
R attains its maximum,
justifies the scaling
Then
so that
a
= O(R).
This
A similar argument applied to
periodic solutions of (3.2.7) justifies the choice of
E.
Theorem.
Suppose that the constant
nonzero.
Then (3.2.1) has a unique periodic solution bifur-
cating from the origin, either for a < 0
(when
K > 0).
If
x
K defined by (3.2.8) is
a > 0
(when
K < 0) or
= R cos e, y = R sin e, then the
periodic solution has the form
44
3.
with period
R(t,a)
laK- l l l / 2 + O(lal)
6(t,a)
l 2 Olot + 0(lal / ) 2 + 0(lal l / ).
= (2~/OlO)
,(a)
tion is stable if
K< 0
EXAMPLES
The periodic solu-
and unstable if
K > O.
Finally, we note that since the value of only on the nonlinear terms evaluated at
a
=
K depends
0, when apply-
ing the above theorem to (3.2.1) we only need assume that the eigenvalues of
A(a)
cross the imaginary axis with non-zero
speed and that A(O)
[0
-OlO] 0
Olo
with 3.3.
non-zero. Hopf Bifurcation in a Singular Perturbation Problem In this section we study a singular perturbation prob-
lem which arises from a mathematical model of the immune response to antigen [52].
1) b - ~] [ 3 + ( a-Zx+ -x ~
EX
where a
and
E,Ii'Yl'Y2 b
The equations are
1
a
2" 1i(1-x) - a - ylab
b
-ylab + Y2b
are positive parameters.
(3.3.1)
In the above model
represent certain concentrations so they must be
non-negative.
Also, x
measures the stimulation of the system
and it is scaled so that
Ixl
~
1.
The stimulation is assumed
to take place on a much faster time scale than the response so that
E
is very small.
3.3. Hopf Bifurcation in a Singular Perturbation Problem
45
The above problem was studied in [52] and we briefly outline the method used by Merrill to prove the existence of periodic solutions of (3.3.1).
=
e:
Putting
in the first
0
equation in (3.3.1) we obtain x Solving (3.3.2) for tain
x
=
F(a,b)
3
x
1 Z)x + b
-
+ (a
-
1 Z
as a function of
O. a
(3.3.2) and
b
we ob-
and substituting this into the second equa-
tion in (3.3.1) we obtain 1
2"
a
o(l-F(a,b)) - a - ylab (3.3.3)
b
Using
0
as the parameter, it was shown that relative to a
certain equilibrium point place in (3.3.3).
(aO,b ) O
a Hopf bifurcation takes
By appealing to a result in singular per-
e:
turbation theory, it was concluded that for
sufficiently
small, (3.3.1) also has a periodic solution. We use the theory given in Chapter 2 to obtain a similar result. Let bO
+0
then
(xO,aO,b O) aO
= x
be a fixed point of (3.3.1).
Y2!Yl
3
o
+
and
Y2 Yl
(-- -
1 -2)x
xO,b O 0
+
b
O
1 Y2 - 0 (l-x ) - - Y b 2 0 Yl 2 0
If
satisfy 1 - -2
=
0, (3.3.4)
=
O.
Recall also that for the biological problem, we must have bO > 0 b
O
and
IXol
~ 1.
We assume for the moment that
satisfy (3.3.4) and these restrictions.
these solutions are considered later. for
th~
rest of this section.
We let
Xo
The reality of aO
=
Y2!Yl
and
3.
46
Let where
1jJ
=
EXAMPLES
y = a - a O' z = b - b O' w = -1jJ (x-x O) - xoY - z 2 1 3x O + a O - 2· Then assuming 1jJ is non-zero, e:w
g(w,y,z,e:) f 2 (w,y,z,e:)
y
(3.3.5)
where g(w,y,z,e:)
fl(w,y,z,e:) - e:x Of 2 (w,y,z,e:) - e:f 3 (w,y,z,e:)
=
-1jJw + N(w+xOY+z,y) Ii
f 2 (w,y,z,e:)
(21jJ
-1
Ii
Xo - 1 - ylbO)y + (2 1jJ
-1
- Y2)z
+
f 3 (w,y,z,e:) N(e,y)
=
-1jJ
= -ylbOY - YlYz -2 3
e
+ 31jJ
-1
xOe
2
- yeo
In order to apply centre manifold theory we change the time scale by setting respect to
s
by
t
= e:s.
We denote differentiation with
and differentiation with respect to
t
Equation (3.3.5) can now be written in the form
by
w,
g(w,y,z,e:)
y'
e:f 2 (w,y,z,e:)
z,
e:f (w,y,z,e:) 3 O.
e:' Suppose that
1jJ
>
O.
(3.3.6)
Then the linearized system corresponding
to (3.3.6) has one negative eigenvalue and three zero eigenvalues. fold
w
By Theorem 1 of Chapter 2, (3.3.6) has a centre manih(y,z,e:).
By Theorem 2 of Chapter 2, the local be-
havior of solutions of (3.3.6) is determined by the equation
3.3. Hopf Bifurcation in a Singular Perturbation Problem
y'
e:fZ(h(y,z,e:) ,y,z,e:)
z'
e:f 3 (h(y,z,e:) ,y,z,e:)
47
(3.3.7)
or in terms of the original time scale y
fZ(h(y,z,e:) ,y,z,e:)
Z
f 3 (h(y,z,e:) ,y,z,e:).
(3.3.8)
We now apply the theory given in the previous section to show that (3.3.8) has a periodic solution bifurcating from the origin for certain values of the parameters. The linearization of the vector field in (3.3.8) about y
z
=0
is given by cS
2"
1jJ
-1
-
yz ]
J (e:)
+ 0
(e:) .
o If (3.3.8) is to have a Hopf bifurcation then we must have cS -1 trace(J(e:)) = 0 and 2" 1jJ - yz > O. From the previous ana1ysis, we must also have that with
Xo ,b O are solutions of (3.3.4)
1, b O > 0 and 1jJ > O. We do not attempt to obtain the general conditions under which the above conditions
IXol
(y,z),y,z,O)
Then by Theorem 2 of Chapter 2, if we can find (M<j>)(y,z)
= O(y4+z4)
then
(3.3.10) <j>
such that
= <j>(y,z) + O(y4+z4).
h(y,z,O)
Suppose that (3.3.11) where
<j>. J
is a homogeneous polynomial of degree
j.
Substi-
tuting (3.3.11) into (3.3.10) we obtain (M<j» (y,z)
= 1jJ<j>2(Y'z) - 31jJ -1 xO(xoY+z) 2 + y(xoY+z) + 0 ( 1y 1 3 +
3
1z 1 ) .
Hence, if (3.3.12) then
(M<j>)(y,z)
(3.3.10) with (M<j»(y,z)
= o(
3 lyl 3 + IzI ).
<j>2(Y'z)
given by (3.3.12), we obtain
1jJ<j>3(Y'z) + 1jJ
-2
(xoY+z)
4
4
+ y<j>2(Y'z) + O(y +z ). lienee,
Substituting (3.3.11) into
3
3.
50
CP2(y,z) - ljI
h (Y. z)
-3
(xoy+z)
3
+ 6lj1
-2
EXAMPLES
x O(x Oy+z)CP2(y,z)
- ljI-lYCP2(Y'Z) + O(y4+Z4) where
CP2(y,z)
is defined by (3.3.12).
culated as in the previous section. depend on
and
can now be cal-
The sign of
K(O)
will
is non-zero then we can
K (0)
If
K(O)
apply Theorem 1 of Section 2 to prove the existence of periodic solutions of (3.3.1).
The stability of the periodic
solutions is determined by the sign of
If
is
K(O)
K(E).
zero then we have to calculate 3.4.
K(O).
Bifurcation of Maps In this section we give a brief indication of some re-
sults on the bifurcation of maps.
For more details and
references the reader is referred to the book by Iooss [40). Let let
F~:
q ~ Em
V be a neighborhood of the origin in mm and V + mm be a smooth map depending on a parameter
with
F (0) ~
0
~.
for all
If bifurcation is to
take place then the linearized problem must be critical. assume that with Fa
Fb
A(O), X(O)
has simple complex eigenvalues I, A(O)
IA(O)I
We
±l, while all other eigenvalues of
~
are inside the unit circle.
By centre manifold theory
all bifurcation phenomena take place on a two-dimensional manifold so we can assume
m
2
without loss of generality.
We first consider the case when sional parameter and we assume that
~
A(~)
is a one-dimencrosses the unit
circle, that is
0 and By redefining
.
3.
52
EXAMPLES
small enough, the equation
~2
Re(h(r,e,~))
has a unique (small) solution
= 0
r = r(~I'~2,e)
with (3.4.1)
Substituting this into the equation Im(h(r,e,~))
=0
(3.4.2)
we obtain
I Im(ak(~))r2k + Im(b(~)e-ine)r2p+1 + o(r2p +2). ~2(~I'~2,e) = I Im(ak(~))r2k. We now make the change k=2
~2 Let
=
k=2
of parameter the map
E
= EI
+ iE2
=
~l + i(~2 - ~2)'
E is one-to-one for
~ +
~
By (3.4.1),
small enough.
Equation
(3.4.2) now takes the form
Since
b(O)
equation has
~
0, by the implicit function theorem the above n
IE I < CE(2P+I)/2 2
1
fixed points for If A3 (0)
er(E), r = 1, ... ,n, for is a constant. Thus Fn has
distinct solutions '
where
~
C
in a region of width
~
O(~~n-2)/2).
= 1, then for one parameter families of maps,
in general, there are bifurcating fixed points of order 3 1 and no invariant closed curves [40,41]. The case A4 (0) is more complicated. the normal form
In this case the map can be put into
3.4.
Bifurcation of Maps
=
a(~)
where
O(~).
i +
fixed points of order cate from c(~)
F~
53
~
If 4
Em
then in general either
or an invariant closed curve bifur-
0; the actual details depend on
[40,41,68].
If
~
a(~),
b(~)
and
is a two-dimensional parameter then
is studied by investigating bifurcations of the differen-
tial equation (3.4.3) A(~),
where one map
B(~)
T(~)
and
C(~)
are chosen such that the time
of (3.4.3) satisfies
The bifurcation diagrams given in [4] for equation (3.4.3) seem only to be
conjec~ures.
studied in [54]
in the case
parameters
Re(~)
and
Bifurcations of (3.4.3) are A(~)
= ~
E
¢
with bifurcation
Re(B(~)).
We note that the above bifurcation results do not apply if the map has
special properties.
For example, F
~
could represent the phase flow of a periodic Hamiltonian system so that
F
~
is a symplectic map.
is due to Moser [53]: F(O)
The following result
if
F
is a smooth symplectic map with
= 0, then in general
F
has infinitely many periodic
points in every neighborhood of the origin if the linearized map
F'(O)
has an eigenvalue on the unit circle.
For an ap-
plication of this result to the existence of closed geodesics on a manifold see [46].
CHAPTER 4 BIFURCATIONS WITH TWO PARAMETERS IN TWO DIMENSIONS
4.1.
Introduction In this chapter we consider an autonomous ordinary dif-
ferential equation in the plane depending on a two-dimensional parameter
E.
We suppose that the origin
point for all
E.
X
x
=
0
is a fixed
More precisely, we consider f(x,E),
x E1R
z,
E
=
(El,E Z) E1R
z,
(4.1.1)
f(O,E) - O. The linearized equation about
x
x and we suppose that
A(O)
=
0
is
A(E)x,
has two zero eigenvalues.
ject is to study small solutions of (4.1.1) for
The ob-
(El'E Z)
in
a full neighborhood of the origin.
More specifically, we wish
to divide a neighborhood of
into distinct components,
such that if
E,E
E
=
0
are in the same component, then the phase
portraits of (4.1.1)E and (4.1.1)E are topologically equivalent.
We also want to describe the behavior of solutions for
each component.
The boundaries of the components correspono
4.1.
Introduction
55
to bifurcation points. Since the eigenvalues of that either (i)
A(O)
A(O)
are both zero we have
is the zero matrix, or Cii)
A(O)
has
a Jordan block,
A(O)
~
= [:
l
There is a distinction between (i) and (ii) even for the study of fixed points.
Under generic assumptions, in case (ii),
equation (4.1.1) has exactly 2 fixed points in a neighborhood of the origin.
For case (i)
the situation is much more com-
plicated [21,29). Another distinction arises when we consider the eigenvalues of
A(E).
values of
A(E)
bifurcation.
We would expect the nature of the eigento determine (in part) the possible type of
If
I El l
A(E)
0 E2
0
then the eigenvalues of
A(E)
J
are always real so we do not
expect to obtain periodic orbits surrounding the origin.
On
the other hand if A(E) =
[0
1
El
E2
then the range of the eigenvalues of of the origin in ber then
A(E)
~,that
is, if
has an eigenvalue
z
j A(E)
is a neighborhood
is a small complex numz
for some
We shall assume from now on that (4.1. 2) •
(4.1.2)
A(E)
E. is given by
56
4.
BIFURCATIONS WITH TWO PARAMETERS
Takens [64,65) and Bodganov [see 3) have studied normal forms for local singularities.
Takens shows, for example,
that any perturbation of the equation
is topologically equivalent to
for some
E ,E 2 . There are certain difficulties in applying l these results since we must transform our equation into nor-
mal form, modulo higher order terms. In [47, p. 333-348), under the assumptions that
where
f2
Kopell and Howard study (4.1.1) A(E)
is given by (4.1.2) and that
is the second component of
f.
Their approach con-
sists of a systematic use of scaling and applications of the implicit function theorem. In this chapter, we use the same techniques as Kopell and Howard to study (4.1.1) when the nonlinearities are cubic. Our results confirm the conjecture made by Takens [64) on the bifurcation set of (4.1.1). Similar results are given in [4) together with a brief outline of their derivation. The results on quadratic nonlinearities are given in Section 9 in the form of exercises. be found in Kopell and Howard [47).
Most of these results can
4.Z.
Preliminaries
4.Z.
Preliminaries
57
Consider equation (4.1.1) where (4.1.Z). is
A(E)
is given by
We also suppose that the linearization of
f(X,E)
A(E)X, and that f(O,E)
=0
0,
f(X,E)
=0
-f(-X,E).
(4.Z.1)
The object is to study the behavior of all small solutions of (4.1.1) for
E
in a full neighborhood of the origin.
Equa-
tion (4.1.1) is still too general, however, so we shall make some additional hypotheses on the nonlinear terms. f
=
(f1'fZ) , 3
ex
= -a 3 fZ(O,O),
3
8
aX l
(HI)
ex
f
0
(HZ)
8
f
0
(H3)
3
a -::z-
fZ (0,0) •
axlx Z
O.
(HI) implies that for small or
Set
T
fixed points.
E, (4.1.1) has either
1
Under (HI), it is easy to show that by
a change of co-ordinates in (4.1.1), we can assume (H3) (see Remark 1).
We assume (H3) in order to simplify the computa-
tions. Under (HI) we can prove the existence of families of periodic orbits and homoclinic orbits.
Under (Hl)-(H3) we can
say how many periodic orbits of (4.1.1) exist for fixed The sign of
8
E.
will determine the direction of bifurcation
and the stability of the periodic orbits, among other things. From now on we assume (Hl)-(H3).
4.
58
BIFURCATIONS WITH TWO PARAMETERS
Level Curves of Figure 1
2
1
Bifurcation Set for the Case
a < 0,
e
0
are obtained by using the change of and
t
+
-to
The
4.2.
Preliminaries
59
3
2 ----------------~----~------------ £1
Bifurcation Set for the Case
a > 0, 8 < O.
Figure 3 The cases ferent.
a > 0
and
a < 0
are geometrically dif-
The techniques involved in each case are the same
and we only do the more difficult case a > 0
a < O.
The case
is left for the reader as an exercise. From now on we assume
a < 0
in addition to (Hl)-(H3).
Note that this implies that locally, (4.1.1) has point for
£1 < 0
and
3
fixed points for
1
£1 > O.
fixed
60
4.
BIFURCATIONS WITH TWO PARAMETERS
REGION 1
REGION 2
------------~.---------
REGION 3
--.....-------------------~,-------
REGION 4
Phase Portraits for the Case Figure 4
a
0
and
El < 0
are treated separately.
El~
El > 0, equation (4.3.1) becomes YZ Yl
+
a
Z
gl(~,a,y)
+ ~YZ
H(Y I ,Y Z)
3 Yl
+
Z . (YZ/2)
Z ayylyz
+
a
- (y/Z)
+
Z
(4.4.1)
Z
gz(~,a,y).
4
(y/4) .
Then alon).!
4.4.
The Case
El > 0
65
solutions of (4.4.1), (4.4.Z) ~
Note that for
= 0 = 0, H
The level curves of of eight i f Figure 1) .
is a first integral of (4.4.1). H(yl,yZ) = b
consist of a figure
b = 0, and a single closed curve i f
b > 0
(see
b > 0, the curve
H(yl'YZ) = b passes through the point Yl = 0, YZ = (Zb)l/Z For b > 0 and sufficiently small, we prove the existence of a function ~
For
~l(b,o)
=
with
~
point
=
-
= -yP(b)o
~l(b,o)
+
Z
0(0)
such that for
0
b > 0, (4.4.1)
has a periodic solution passing through the
Yl = 0, YZ = (Zb)l/Z, and with
~ = ~1(0,0), (4.4.1)
has a figure of eight solutions. For fixed
the number of periodic solutions of
~,o,
(4.4.1), surrounding all three fixed points, depends upon the number of solutions of (4.4.3) We prove that b
l for
> 0
PCb)
such that
~
as
00
b
P'(b) < 0
~
for
b > b l . These properties of of solutions of (4.4.3).
00
and that there exists b < b PCb)
l
and
P' (b) > 0
determine the number
Suppose, for simplicity of exposition, that y
bl
such that
~
=~l(bZ'o),
o.
~l(b,o)
=
0 < b Z < b l , then there exists ~l(bZ'o) - ~1(b3'0). Hence, if If
then (4.4.1) has two periodic solutions, one
0, YZ = (Zbz)l/Z, the other passing (Zb ) l/Z. I f ~ >~1(0,0), then (4.4.1) through YI = 0, YZ 3 has one periodic solution surrounding all three fixed points.
passing through
Yl
4.
66
~ =
Finally, if
~(bl'o),
BIFURCATIONS WITH TWO PARAMETERS
then the periodic solutions coincide.
In Figure 4, the periodic solutions
surrounding all
three fixed points in regions 3-5 correspond to the periodic ~
solutions of (4.4.1) which are parametrized by
b > b l . Similarly the "inner" periodic solutions in region 5 are parametrized by ~ = ~l(b,o), 0 < b < b l . The curve Ll in (El'E Z) space corresponds to the curve ~ = ~1(0,0). Similarly the curve
LZ
~ =
corresponds to the curve
~l(bl'o)
(see Figure Z). ~l
In general
(b,o)
is not identically equal to
-yP(b)o, but the results are qualitatively the same. example, we prove the existence of a function ~,o
0(0), such that if
satisfy
~
=
~ = ~l(bl(o)
,0)
bl(o)
~l(bl(o)
tion (4.4.3) has exactly one solution.
For bl
=
+
,0), then equa-
The curve
which is mapped into the curve
LZ
in
(El,E ) space corresponds to the points where the two Z periodic solutions coincide. If H(Yl'YZ) points
b
O.
Similarly,
S(~,O)
when
be the stable and unstable mani-
hits
yz
H(yl'YZ)
U(~,O)
when
is the value of
H(~,o,-)
=
is a saddle
0, Yl > O.
H(~,o,±)
are
well defined since stable and unstable manifolds depend continuously on parameters. Let
I(~,o,+) U(~,O)
the portion of YZ
=
a to YZ
denote the integral of
=
with
0, Yl > O.
Yl > 0, YZ > Then
H(~,o,+)
Similarly, I(~,o,-) the portion of to
=
with
over
a from Yl
(4.4.5)
I(~,o,+).
denotes the integral of
S(~,O)
H(Yl'YZ)
H(Yl'Y Z)
Yl > 0, YZ < a from H(~,o,-) = I(~,o,-).
Yz = 0, Yl > 0, so that Equation (4.4.1) has a homoclinic orbit (with
over
Yl = Yz = Yl > 0)
if and only if H(~,o,+)
-
H(~,o,-)
=
O.
(4.4.6)
We solve (4.4.6) by the implicit function theorem.
a
4.
68
BIFURCATIONS WITH TWO PARAMETERS
Using (4.4.2) and (4.4.5), 2
f
22
= + (~Y2 + YOYIY2)dt +
H(~,o,+)
O(~
2
2 + 0 ),
(4.4.7)
where the above integral is taken over the portion of from
Yl
=
Y2 = 0
H(~,o,-)
=
to
L(~Y~
Similarly,
0, Yl > O.
Y2
2 + YOY 2Y2)dt + I
O(~
2 + 0 2) ,
where the integral is taken over the portion of Yl
= Y2
=
0
to
Y2 = 0, Yl > O.
U(O,O)
(4.4.8) 5(0,0)
from
Using (4.4.7), (4.4.8) and
U(O,O) = -5(0,0), we obtain (4.4.9) Using (4.4.7) and (4.4.9),
so that by the implicit function theorem, we can solve (4.4.6) for
~
as a function of
0, say
to get an approximate formula for
=
~
~(o).
~(o).
We now show how
We can write equa-
tion (4.4.6) in the form
Hence, using (4.4.1) and
Y2
-YOf+
J+
Y2dy 1
This completes the proof of Lemma 1. Lemma 1 proves the existence of a homoclinic orbit of (4.1.1) when
(£1'£2)
lies on the curve
Ll
given by
4.4.
The Case
Using
El > 0
69
= -f(-x,E), when
f(x,E)
(El,E Z)
lies on
Ll , equa-
tion (4.1.1) has a figure of eight solutions. We now prove the existence of periodic solutions of (4.4.1) surrounding all three fixed points.
In the introduc-
tion to this section, we stated that (4.4.1) has a periodic 0, Y = (Zb)l/Z for any Z In Lemma Z, we only prove this for "moderate" values of
solution passing through
=
Yl
The reason for this is that in (4.3.1) the only for
Y
in a bounded set.
gi
b >
o.
b.
are bounded
In Section 6, we show that by
a simple modification of the scaling, we can extend these resuIts to all Lemma Z.
b > O.
Fix
b >
o.
Then for
0 < b < band
ently small, there exists a function O(oZ)
such that if
]..I
=
]..11 (b, 0)
b
~
=
]..11 (b,o)
suffici=
-yP(b)o +
in (4.4.1), then (4.4.1) has
a periodic solution passing through As
]..I
0
0, Y = (Zb)l/Z. Z the periodic solution tends to the figure of
0
Yl
=
eight solutions obtained in Lemma 1. Proof:
Let
H(]..I,o,b,+)
be the value of
orbit of (4.4.1) which starts at sects
yz = O.
y
1 = Similarly, H(]..I,o,b,-)
H(yl,y ) when the Z 0, y = (Zb)l/Z interZ is the value of
H(yl'YZ) when the orbit of (4.4.1) which starts at Yl = 0, YZ = _(Zb)l/Z is integrated backwards in time until it intersects YZ = o. Then (4.4.1) has a periodic solution passing through
Yl
=
0, YZ
=
(Zb)l/Z
H(]..I,o,b,+) Let
I(]..I,o,b,+)
-
i f and only i f
H(]..I,o,b,-)
and finishing at
(4.4.10)
O.
denote the integral of
the portion of the orbit of (4.4.1) with Yl • 0, Y L - (2b)1/Z
=
H(yl,y Z)
over
YZ > 0, starting at YZ • 0, Yl
> O.
4.
70
Similarly, in time.
BIFURCATIONS WITH TWO PARAMETERS
is defined by integrating backwards
I(~,o,b,-)
Thus, b +
H(~,o,b,±)
I(~,o,b,±).
(4.4.11)
Using (4.4.Z) and (4.4.11),
where the above integral is taken over the portion of the orbit of (4.4.1) with to
Yl = c, YZ
=
0
~
0 = 0
=
from
Yl
0, Yz
=
=
where 4b
Similarly,
(Zb) l/Z
(4.4.13)
I(~,o,b,-)
=
+
-I(~,o,b,+)
O(~
Z + 0 Z), so
that equation (4.4.10) may be written in the form (4.4.14) Hence by the implicit function theorem, we can solve (4.4.14) to obtain
~ = -yP(b)o + O(oZ)
where
f yiYZdYl
P (b)
(4.4.15)
J YZdYl In order to prove that the periodic solution tends to the figure of eight solutions as H(~,o,b,±)
+
b
0, we prove that
+
H(~,o,±)
as
b
+
O.
(4.4.16)
This does not follow from continuous dependence of solutions on initial conditions, since as
b
+
periodic solution tends to infinity.
0
the period of the
The same problem occurs
in Kopell and Howard [47, p. 339] and we outline their method. For
Yl
and
YZ
small, solutions of (4.4.1) behave like
4.4.
The Case
El > 0
71
solutions of the linearized equations.
The proof of (4.4.16)
follows from the fact that the periodic solution stays close to the solution of the linearized equation for the part of the solution with
(yl,yZ)
small and continuous dependence on
initial data for the rest of the solution. Lemma 3.
PCb)
+
as
00
b
+
00
Proof:
The integrals in (4.4.lS) are taken over the curve Z Zb]l/Z, from Yl = 0 to Yl = c where YZ = [Yl - (yilZ) + c is defined by (4.4.13) . Thus, JO(b)P(b) = J 1 (b) where
w = cz
Substituting
in (4.4.17) we obtain J. (b)
=
c Z fl (cz) Zig(z)dz 0
1
[(zZ_l)
where g(z) g(c- l ) for
o
0
for
such that
pI (b) < 0
for
b > b . l
It is easy to show that
pI (b)
+
-00
by Lemma 3 it is sufficient to show that if
as
b
+
O.
pI (b l ) = 0
Hence, then
pit (b ) > O.
l
Let
r(w)
=
[w Z - (w 4 /Z)
(4.4.17) with respect to
b
in~:
hy pans in
Zi ~ dw.
fo r
1
.10
Zb]l/Z.
Differentiating
we obtain c
J!
Intcgrat
+
l
w)
we obtain
(4.4.18)
4.
72
J Also J
-r
r2 (w) 0 rCw) dw
O-
=
BIFURCATIONS WITH TWO PARAMETERS
O
f:
f:
[w 2
4 2 (w -w 2 dw. rCw)
(4.4.19)
-r tW4w) / 22
(4.4.20)
+ 2b] dw.
Similarly,
Using (4.4.18) - (4.4.21) we can express JI
terms of
J O and
Jl
in
A straightforward calculation yields
o
(4.4.22) l5J l Suppose that
4bJ O + (4+l2b)Ji'
P'(bl)
J],(b l ) - P(bl)J'O(b l ).
=
O.
Then
JO(bl)PIt(b l )
Using (4.4.22) we obtain
4b l (4b l +l) [J],(b l ) - P(bl)J'O(b l )] =
Hence
PIt(b l )
(4.4.23)
JO(b l ) [p2(b l ) + 8b l P(b l ) - 4b ]· l
has the same sign as (4.4.24)
Since
pI (b l ) = 0 we have that (4.4.22) we obtain
Ji (b l )
=
P(bl)JO(b l ).
Using
(4.4.25) Using P(b l )
b l > 0, it is easy to show that (4.4.25) implies that Using (4.4.24) and (4.4.25), PIt(b ) has the same l P (b l ) - P2 (b ). This proves that pit (b l ) > 0 as l
< l.
sign as required.
4.4.
The Case
Lemma S.
For
El > 0
0
73
sufficiently small there exist
b l + 0(0), bZ(o)
b Z + 0(0), where
=
prO)
=
bl(o)
P(b Z)' with the
following properties: Let
0 > O.
(i)
if
\l
=
\ll(bl(o),o), then the equation (4.4.Z6)
has exactly one solution. (ii)
If
y
0
fo r
(-1,0).
0 < c < 1.
Proof:
We write Q,J and J l as functions of b where O 4 2 4b = c - 2c . Since (db/dc) < 0 for 0 < C < 1, we must
prove that
Q' (b) < 0
for
-1 < 4b < O.
procedure as in Lemma 4, we find that equation (4.4.22).
Thus, i f
Following the same
JO,J O' Jl,Ji
Q'(b l ) • 0
then
satisfy
4.5.
The Case
77
El < 0
(4.4.36) -1 < 4b l < 0, the roots of (4.4.36) are less than 1. Hence, i f Q'(b l ) = 0 then Q(b l ) < 1. Also, from (4.4.Z3), Since
Q' (b l ) = 0 then Q"(b l ) has the same sign as 4b l 8b l Q(b l ) - QZ (b l )· Using (4.4.36) and Q(b l ) < 1, this im-
if
plies that
Q" (b l ) < O.
shows that
Q' (b) < 0
Since for
Q(-1/4)
1, Q(0)
-1 < 4b < O.
=
4/5, this
This completes the
proof of the lemma. 4.5.
El~
The Case With
El < 0, equation (4.3.1) becomes YZ
Yl
°Zgl(\l,o,y)
+
-
3 Yl
Let
+
(yilZ)
tions of (4.5.1),
Z YZ
-Yl
Yz
\lYZ
+
HI
=
+
+
Z oyylyz +
Z Z oYYlYZ
+
Z g Z(\l , ,y) .
°
(4.5.1)
°
(yi/ 4). Then along soluZ + 0(0 ). Using the same
methods as in the previous lemmas, we prove that (4.5.1) has a periodic solution passing through and only i f J O(c)R(c)
\l
= \l3(c,0)
=
-yR(c)o
+
Yl = c > 0, YZ O(OZ) , where
0, i f
= J 1 (c) , Ji(c) = r(w)
c z' 0 w lr(w)dw,
I
[Zb - (w 4 /Z) - WZ]l/Z, 4b
In order to prove that for fixed
=
monotonic. R' (c) > 0
for
c > O.
c4
+
Zc Z.
\l,0, equation (4.5.1)
has at most one periodic solution we prove that
Lemma 8.
=
R
is strictly
BIFURCATIONS WITH TWO PARAMETERS
4.
78
Proof:
We write
R,JO,J l
ent to prove that
R'(b)
as functions of 0
>
for
b
b.
It is suffici-
O.
>
Using the same methods as before, we show that
lSJ l
(12b+4)J
(4.S.2)
1 - 4bJb
c 3w4dw
fo rtWJ Now
R'
(4.S.3)
has the same sign as
S
where
S
By (4.S.2), S
=
(8b-4)J'J'
o
By (4. S. 3) , bJ b > J
1·
1
=
S(J,)2 + 4b(J,)2 1
Hence, for
0
b
S > (J )2 b -l(8b-4-Sb+4)
1
Similarly, i f
0 < b < 2, then
~
=
2, 3(J,)2 > O. 1
S > 3b 2 (Jb)2 > O.
This com-
pletes the proof of Lemma 8. 4.6.
More Scaling In Section 4 we proved the existence of periodic solu-
tions of (4.4.1) which pass through the point Lemma 3 indicates that we may take please. are
b
(O,(2b)1/2).
to be as large as we
However our analysis relied on the fact that the
0(1)
and this is true only for
y
gi
in a bounded set.
Also, our analysis in Sections 4 and S restricts
El ,E 2 to a region of the form {(E l ,E 2 ): [E l [ ~ EO' El ~ (constant) E~}. To remedy this we modify the scaling by setting I) [E a- l [1/2 h -l, where h is a new parameter with, say, l o < 4h ~ 1. The other changes of variables remain the same.
Note that if
~.I)
lie in a full
n~ighborhood
of the
4.6.
More Scaling
origin then
79
(El,E Z)
and
(xl'x Z)
lie in a full neighbor-
hood of the origin. After scaling, (4.1.1) becomes
YZ For
0 < 4h < 1
0(1)
for
Y
Let
and
sufficiently small, the
~,o
gi
are
in a bounded set.
El
>
O.
Let
HZ(Yl'YZ)
=
Z
(YZ/Z)
-
~Y~
Then along solutions of (4.6.1), HZ
O(~Z
oZ).
+
Z Z (h yl/Z)
+
+
Following the same procedure as in Sec-
tion 4, we find that (4.6.1) has a periodic solution passing through
Yl
= 0, YZ = I, if and only if h
~
where
Zb
hZP(b)
=K+
=
Z
~l
(b,o)
= -yh ZP(b)o
+
Z 0(0 ),
h- 4 .
An easy computation shows that as Z O(h ), where JOK = 12 J l , J. 1
Thus, for small
h ~ 0,
= fl WZi(1_w4)1/Zdw. 0
h, (4.6.1) has a periodic solution passing
-yKo + Yl = 0, YZ = 1 i f and only if ~ O(hZ[o[ + o Z) . In particular, when El = 0, E > 0, 8 < 0, Z (4.1.1) has a periodic solution passing through xl = 0, through
X
z=
[a[(-8K)-lE For
z
+
O(E~/Z).
El < 0, a similar analysis shows that (4.6.1) has
a periodic solution passing through
Yl
= I,
YZ
=
0, if and
only if
~ = hZ~Z(h-l.o) = _yhZQ(h-l)o and t hat
II
s
h
• (), 1/ Q(h - 1) -
( KI fl)
+
+ 0 ( h Z) .
O(oZ),
4.
80
4.7.
BIFURCATIONS WITH TWO PARAMETERS
Completion of the Phase Portraits Our analysis in the previous sections proves the exist-
ence of periodic and homoclinic orbits of (4.1.1) for certain (El,E Z)
regions in
space.
We now show how to complete the
pictures. Obtaining the complete phase portrait in the different regions involves many calculations.
However, the method is
the same in each case so we only give one representative example.
We prove that if
y < 0, then the "outer" periodic
We use the scaling given in
in region 5 is stable.
orbit
Section 3. Since we are in region 5, \ll(O,c) > \l > \ll (b (c) ,c). l Fix
and
\l
with
c
and let
b Z > bl(c).
solution
r
bZ
\l = \ll(b,c)
be the solution of
Then we have to prove that the periodic Yl = 0, YZ = (Zb Z) l/Z
of (4.4.1) passing through
is stable. Let b3
b 3 · Let c (b, +) the orbi t of (4.4.1) starting at
be the value of
hits
is the value of
YZ = O.
Similarly, c (b , - )
the orbit of (4.1.1) starting at integrated backwards until it hits
Yl
0, YZ
YZ
-f(-X,E), the solution passing through
=
O.
Yl when (Zb)l/Z is
Using
f(X,E)
(O,(Zb)l/Z), spirals
inwards or outwards according to the sign of Using the calculation in Lemma Z, c(b,+) +
c(b,+) + c(b,-).
~(b,-)
has the
4.8.
Remarks and Exercises
H(~,o,b,+)
same sign as
-
81
H(~,o,b,-)
which in turn has the
same sign as
~l(b,o)
Using the properties of positive if
b < b
2
given in Lemma 5, S
and negative if
b > b . 2
is
The result now
follows. 4.8.
Remarks and Exercises
Remark 1.
Consider equation (4.1.1) under the hypothesis
that the linearization is given by is defined by (4.1.2). hold.
r
The map
x
~
-x
A(E)
x
=
B(E)X, where
and
a =
=
will be one-to-one for
Using the fact that
where
Suppose also that (4.2.1) and (HI)
Make the change of variables
B(E) = I - ra -lA(E)
x = A(E)X
A(E)
and
B(E)
E
sufficiently small.
commute, it is easy to
show that the transformed equation satisfies all the above hypotheses and that in addition it enjoys (H3). lar, if the transformed equation is
x = F (x, E) then
o
In particu-
BIFURCATIONS WITH TWO PARAMETERS
4.
8Z
3 3a fl(O,O)
3
+
Remark Z.
We have assumed that
aX l
f(x,E)
=
-f(-x,E).
If we
assume that this is true only for the low order terms then we would obtain similar results. get a homoclinic orbit with curve
L'1
For example, on xl > O.
and
L'1
we would
Similarly on another
we would get a homoclinic orbit with
general
xl < O.
In
would be different although they would ~ =
have the same linear approximation Remark 3.
Ll
-(4/S)oy.
Suppose that we only assume (HI) and (H3).
Then
we can still obtain partial results about the local behavior of solutions.
For example, in Lemma Z we did not use the
hypothesis
nonzero.
8
a periodic solution through for some
and
b > 0, (4.1.1) has l Z 0, X = I a I- / (Zb) l/ZE z I However, we cannot say
Hence, for each xl
with
anything about the stability of the periodic orbits and we cannot say how many periodic orbits (4.1.1) has for fixed
El
and Exercises (1) that for ing orbit.
Suppose that (El,E Z) (Hint:
a
and
8
are negative.
Prove
in regions 5 and 6, (4.1.1) has a connectUse the calculations in Lemma 1 and
Lemma 6.) (Z)
Suppose that
a
(E , E ) is in region 3. Let l Z the point (0,0) in (4.1.1).
and
8
are negative and that
U be the unstable manifold of Prove that
gion of attraction of the periodic orhit.
U
is in the re-
4.9.
Quadratic Nonlinearities
(3)
Suppose that
a
83
is positive and
8
is negative.
Show that the bifurcation set and the corresponding phase portraits are as given in Figures 3 and 4.9.
s.
Quadratic Nonlinearities In this section we discuss the local behavior of solu-
tions of (4.1.1) when the nonlinearities are quadratic.
Most
of the material in this section can be found in (47). Suppose that the linearization of (4.1.1) is where
A(E)
x
A(E)
is given by (4.1.Z) and that f(O,E)
We also assume that
0,
0,
8 > 0, a < 0; the results for the other
cases are obtained by making use of the change of variables, t
~
-t, EZ ~ -E Z ' xl ~ ±xl' Introduce parameters
and a new time
t 1
We assume that the case
X
z~ ~,o,
+x Z. scaled variables
Yl'YZ
by the relations Ela -Ill/Z ,
El > 0
in what follows, see Exercise 8 for
El < O.
After scaling, (4.1.1) becomes
(4.9.1)
X
BIFURCATIONS WITH TWO PARAMETERS
4.
84
where the dot means differentiation with respect to t, the Yi lie in a bounded set and Y = Blal- l/Z . Note that by making a change of variable we can assume that fixed point of (4.9.1) for
(1,0)
is a
sufficiently small.
~,o
The object of the following exercises is to show that the bifurcation set is given by Figure 6 and that the associated phase portraits are given by Figure 7. Exercises. (4)
Let
Show
that along solutions of (4.9.1), . H (5)
o
0
0
(4.9.1) has at P' (c)
use the same techniques as in the proof of Lemma 4. ternative method of proving
>
>
0
we can
An al-
is given in [47] .)
4.9.
Quadratic Non1inearities
Put
E:1 < 0, then after scaling (4.1.1) becomes
If
(8)
z
-
85
1,
Y1
Y2
Y2
-Y1
II
II
z
Y2
Y2
z
2 0(0 )
+
- yo. + +
-
llY2
+
2 Y1
+
YOY1 Y2
+
2 0(0 ).
Then
2 0(0 )
-
llY2
z2
+
oyzY2
+
0(0 2 )
which has the same form as equation (4.9.1) and hence transforms the results for
E:l
>
0
into results for the case
E:1 < O. (9)
Show that the bifurcation set and the correspond-
ing phase portraits are as given in Figures 6-7.
4
3
5 6
L
Bifurcation Set for the Case Figure 6
a < 0, B
>
O.
86
4.
BIFURCATIONS WITH TWO PARAMETERS
REGION 1
REGION 2
Phase Portraits for the Case a < 0, B > O. (For the phase portraits in regions 4-6 use the transformations in Exercise 8.) Figure 7
4.9.
Quadratic Nonlinearities
87
ON
L
REGION 3
Figure 7 (cont.)
CHAPTER 5 APPLICATION TO A PANEL FLUTTER PROBLEM
5.1.
Introduction In this chapter we apply the results of Chapter 4 to a
particular two parameter problem.
ic = Ax
+
The equations are
f(x)
(5.1.1)
where T [flex) ,fZ(x) ,f 3 (x) ,f 4 (x)] ,
f(x)
o A
-c
fl (x)
f 3 (x) = 0,
fZ(x)
Xlg(x),
f4 (x)
4x 3g(x) ,
Zg(x)
-'J[ 4 (kx Z l
c = b.
J
8p
3'
+
1
0
0
bl
c
0
0
0
1
0
aZ
bZ
axlx Z
4kx;
_'J[ZjZ['J[ZjZ
a.
J
_ [Cl'J[4 j 4
+
+
IP
0];
+
+
4ax 3x 4 ) ,
fl,
Cl • 0.005,
/)
0.1,
5.2.
k
>
Reduction to a Second Order Equation
0, a
>
0
are fixed and
p,r
89
are parameters.
The above
system results from a two mode approximation to a certain partial differential equation which describes the motion of a thin panel. Holmes and Marsden [36,38] have studied the above equation and first we briefly describe their work. calculations, they find that for 2
-2.23n , the matrix
p
=
Po
108, r
=
~
rO
A has two zero eigenvalues and two
eigenvalues with negative real parts. !r-rO!
~
By numerical
Then for
!p-PO!
and
small, by centre manifold theory, the local behavior
of solutions of (5.1.1) is determined by a second order equation depending on two parameters.
They then use some results
of Takens [64] on generic models to conjecture that the local behavior of solutions of (5.1.1) for
!p-PO!
and
!r-rO!
small can be modelled by the equation
for
a
and
b
small.
Recently, this conjecture [37], in the case
a
has been proved by Holmes
= 0, by reducing the equation on the
centre manifold to Takens' normal form.
We use centre mani-
fold theory and the results of Chapter 4 to obtain a similar result. 5.2.
Reduction to a Second Order Equation The eigenvalues of
A
are the roots of the equation (5.2.1)
where the
ui
are functions of
zero eiIlCHlvlI!U(,s then
u3
rand
• d 4 • O.
p.
If
A
has two
A calculation shows that
5.
90
d
d3
if
APPLICATION TO A PANEL FLUTTER PROBLEM
0, then
4
2 a 1a 2 + c
a b
1 2
or in terms of
and
r
0 (5.2.2)
+ b a 1 2
0
p,
4 2 2 4'1f ('If +r) (4'1f +r) + 64 p2
9
=
4 1/2 2 2 1/2 2 (16o'lf +t5P ) ('If +r) + 4 (O'lf +t5p ) (4'1f +r)
0
(5.2.3)
=
We prove that (5.2.3), (5.2.4) has a solution From (5.2.3) we can express
P
(5.2.4)
O. r = r
in terms of
' O
P = PO'
r.
Sub-
stituting this relation into (5.2.4) we obtain an equation H(r) r
1
=
some
=
Calculations show that
O.
-(2.225)'If
2
and
r
2
=
H (r 1) < 0, H (r 2) > 0 where 2 -(2.23)'If , so that H (r 0) = 0 for
Further calculations show that (5.2.3) ,
rO E (r 2 ,r 1 ) •
(5.2.4) has a solution
rO'PO
with
107.7 < Po < 107.8.
In the subsequent analysis, we have to determine the sign of various functions of know
and
rO
rO
and
PO'
Since we do not
exactly we have to determine the sign of
Po
these functions for
rO
and
Po
in the above numerical
ranges.
r
When
Po, the remaining eigenvalues of
are given by
and a calculation shows that they have negative real parts and non-zero imaginary parts. We now find a basis for the appropriate eigenspaces when
r
=
r
O
'
P
=
PO'
Solving
Av 1 • 0
we find that
A
5.2.
Reduction to a Second Order Equation
v
The null space of onical form of 2
A v2
0
(5.2.5)
=
1
A
91
is in fact one-dimensional so the can-
A must contain a Jordan block.
Solving
we obtain (5.2.6)
The vectors
vI
eigenspace of
and A
v2
form a basis for the generalized
corresponding to the zero eigenvalues.
Similarly, we find a (real) basis for the by the eigenvectors corresponding to Az
A3 z , we find that
=
z
Let P
=
PO'
AO
Let
z
+
z,
=
v3
V
spanned
A4 • Solving and v 4 where
= i(z-z),
2v 4
(5.2.7)
=
denote the matrix S
and
A3
is spanned by
V
2v 3
space
[vl,v2,v3,v41
A when where the
by (5.2.5), (5.2.6) and (5.2.7) and set
y
r
= rO
vi
and
are defined
= S-lx.
Then
(5.1.1) can be written in the form
Y where
F(y,r,p)
By
=
S
-1
B
and where
A3 • PI
+
S
+
-1
(A-AO)Sy
+
F(y,r,p)
f(Sy) , 0
1
0
0
0
0
0
0
0
0
PI
P2
0
0
-P2
PI
=
iP2' PI
0
is easy to verify conditions (i),
Thus
(ii), (iii).
and it e Ct
defines a semigroup. Example 2.
Let
Z
be a Banac!l space of uniformly continuous [O,~)
bounded functions on (T(t)f) (8)
=
with the supremum norm.
f(8+t),
f
€
Z, 8
~
Define
0, t > O.
Conditions (i) and (ii) are obviously satisfied and since IIT(t)f - fll
=
sup{lf(8+t) - f(8) I: 8 ~ O} ... 0
(iii) is satisfied. Definition. T (t)
Hence
T(t)
as
t ... 0+,
forms a semigroup.
The infinitesimal generator
C
of the semigroup
is defined by T(t)z - z , I Im+ t t ... O
Cz whenever the limit exists. of all elements also say that
z € Z C
The domain of
C, D(C), is the set
for which the above limit exists.
generates
T(t).
We
6.
100
INFINITE DIMENSIONAL PROBLEMS
In Example 1 the infinitesimal generator is
C E L(Z)
while in Example 2 the infinitesimal generator is
= f' (8),
(Cf) (8) Exercise 1.
If
C
E
L(Z)
D(C)
=
{f: f'
prove that
II e
Ct
E
-
Z}. as
I II ... 0
t ... O. Exercise 2.
Prove that generator
Does
R,2
Z
Let
T(t)
is a semigroup on
T (t) : Z ... Z by
Z with infinitesimal
given by
IIT(t) - I II'" 0
Exercise 3.
and define
as
t ... O+?
Prove that if IIT(t) 112. Me
for some constants
w
~
T(t) wt
,
is a semigroup then t > 0,
0, M ~ 1.
(6.2.4)
(Hint:
Use the uniform
boundedness theorem and property (iii) to show that is bounded on some interval
[0,£1.
IIT(t) II
Then use the semigroup
property. ) Exercise 4. Find a 2 x 2 matrix C such that if Ct II e II 2. M for a11 t > 0 then M > 1. Exercise 5.
Let
which satisfies
C be the generator of a semigroup T(t) wt IIT(t) 112. Me , t > O. Prove that C - wI
is the generator of the semigroup IIS(t)II2.M
fora11
t>O.
Set)
K
e-wtT(t)
and that
6.2.
Semigroup Theory
Exercise 6.
If
w E Z, T(t)w Let
101
T(t)
is a semigroup, prove that for each
is a continuous function from T(t)
[0,-)
be a semigroup with generator
into
C.
Now
h- 1 [T(h) - I]T(t)w
h- 1 [T(t+h)w - T(t)wl
(6.2.5)
T(t)h- 1 [T(h)w - wl. If
w E D(C)
T(t)Cw
then the right side of (6.2.5) converges to h ~ 0+.
as
Z.
Thus the middle term in (6.2.5) conand so
verges as
T (t)
maps
D (C)
into
D (C) •
Also
the right derivative satisfies
dTJ~)W
=
CT(t)w
Similarly, the h
-1
= T(t)Cw,
t
>
0, w E D(C).
(6.2.6)
ide~tity
[T(t)w -
T(t-h)h
T(t~h)lw
-1
[T(h)w - wl
shows that (6.2.6) holds for the two-sided derivative. Exercise 7.
For
w E Z, prove that wet)
and that dense in
~
w
wet) E D(C)
where
J: T(s)w ds,
as
Deduce that
D(C)
is
and
C
Z.
Exercise 8.
If
wn E D(C), then from (6.2.6) T(t)w
n
- w = Jt T(s)Cw ds. nOn
Use this identity to prove that
C
is closed.
Equation (6.2.6) shows that if the generator of a semigroup
Wo E D(C)
T(t), then
wet)
is
= T(t)w O is a
solution of the Cauchy problem (6.2.2) - (6.2.3).
In applica-
102
6.
INFIMITE DIMENSIONAL PROBLEMS
tions to partial differential equations, it is important to know if a given operator is the generator of a semigroup. see the problems involved, suppose that traction semigroup
T(t), that is
The Laplace transform R(A)w exists for T(t)
Re(A) > 0
and
is in some sense
resolvent of
IS,
and is easy to prove.
generates the con-
~
IIT(t) II
1
for a11
e-AtT(t)w dt,
IIR(A) II < A-I
(AI - C)
-1
.
t > O.
(6.2.7)
for
exp (Ct), we expect .
C, that
I:
C
A > O.
RCA)
Since
to be the
This is indeed the case
The main problem is in finding the in-
verse Laplace transform, that is, given
C
A > 0, say, is
exists for
semigroup?
The basic result is the Hille-Yosida Theorem.
Theorem 1 (Hille-Yosida Theorem).
C
such that
(AI - C) -1
the generator of a
A necessary and sufficient
condition for a closed linear opera tor
C
with dense domain
to generate a semigroup of contractions is that each is in the resolvent set of
= II
IIRCA)II
To
A> 0
and that
C
(AI-C)-lll ~ A-I
A > O.
for
(6.2.8)
The reader is led through the proof of the above theorem in the following exercises. Exercise 9. A
>
0
define
Let
T(t)
R(A)
h-l(T(h)-I)R(A)w
be a contraction semigroup and for
as in (6.2.7) for Ah l h- [ (e - 1) - e
to prove that
R(A)
maps
Z
Ah
into
I:
w E Z.
Use the identity
e-AtT(t)wdt
J: e-AtT(t)w dt] D(C)
and that
6.2.
Semigroup Theory
103
(AI - C)RCA) = I.
w E D(C). R(A) (AI - C)w = wand deduce
Prove also that for that
R(A)
is the resolvent of
Exercise 10.
Suppose that
C
C. is a closed linear operator on
Z with dense domain such that set of
(O.~)
C with (6.2.8) satisfied.
C (A) E L(Z)
is in the resolvent
For
A
0
>
define
by AC (AI - C)-l
C CA)
Prove that (i)
(ii)
CCA)w
-+
Cw
as
A
Ilexp(tC(A)) II ~ 1
-+
~
for
w E D(C).
and
Ilexp(tCCA))w - exp(tC(ll))wll ~ tlICCA)w - C(ll)wll t
for
A.ll
> O.
> 0
and
w E Z.
Use the above results to define T(t)w
T (t)
by
lim exp (C CA) t)w A-+~
and verify that with generator Exercise 11.
T(t)
is a semigroup of contractions on
C. Let
Z =.Q,2
complex numbers such that Define
Z
and let
{an}
be a sequence of
a = sup{Re(a ): n n
1.2 .... }
O.
generates
104
6.
Remark.
INFINITE DIMENSIONAL PROBLEMS
The above version of the Hille-Yosida Theorem char-
acterizes the generators of semigroups which satisfy wt IIT(t) II.::. e (see Exercise 5). A similar argument gives the characterization of the generators of all semigroups. Remark.
Let
C be a closed operator with dense domain such
that all real nonzero A are in the resolvent set of C and II (AI - C)-III < IAI- l for such A. Then C is the generator of a group all
t,s
and
€ JR
Example 3. space
T(t), t
Let
H with
€ JR,
T(t)z
that is, T(O) = I, T(t+s) ~
z
as
t
~
0
T(t)T(s) ,
for each
z
€
z.
C be a selfadjoint operator on a Hilbert C < O.
Then
II(AI - c)-III < A-I
for
A> 0,
so by the Hille-Yosida Theorem, C generates a contraction semigroup. Example 4.
Let
A
on a Hilbert space
iC
=
H.
_A)-III < IAI- l
liP.!
Let semigroup
f
€
where
so that
on
Z.
A,
A generates a unitary group. and let
C be the generator of a
Then formally the solution of
Cw + f ( t),
w
is a selfadjoint operator
Then for all nonzero real
C([O,T] ;Z)
T(t)
C
0 < t < T,
w( 0 )
=
w0 '
(6.2.9)
is given by the variation of constants formula wet) If
f
=
T(t)w O +
f~
is continuously differentiable and
is easy to check that
wet)
(6.2.10)
T(t-s)f(s)ds.
Wo
€
D(C)
defined by (6.2.10) satisfies
(i)
wet)
is continuous for
(ii)
wet)
is continuously differentiable for
o
d
dt<w(t),v> where
=
w(O)
C*
w € C([O,T] ;Z)
Wo
and if for every
Z
v € D(C*), the
is absolutely continuous on =
<w(t),C*v> +
is the adjoint of
ing between
is a weak solution
C
and
[O,T]
and
a.e.
denotes the pair-
and its dual space.
The proof of the following theorem can be found in 19]. Theorem 2.
The unique weak solution of (6.2.9) is given by
(6.2.10). A semigroup
T(t)
if in addition the map for each
z € Z.
properties.
o.
t
~
T(t)z
T(t), then
Thus if
is real analytic on
(0,00)
Analytic semigroups have many additional
In particular, if
tic semigroup t >
is said to be an analytic semigroup
f
T(t)
C
is the generator of an analy-
maps
Z
into
D(C)
for each
is continuously differentiable, then
wet)
defined by (6.2.10) is a strong solution of (6.2.9) for each
Wo
€ Z
if
T(t)
is an analytic semigroup.
An example of a
generator of an analytic semigroup is a nonpositive selfadjoint operator and in general, solutions of parabolic equations
6.
106
INFINITE DIMENSIONAL PROBLEMS
are usually associated with analytic semigroups.
Hyperbolic
problems, however, do not in general give rise to analytic semigroups. of a group t
>
C
O.
To see this, suppose that T(t)
and that
T(t)
maps
C
is the generator
Z
into
D(C)
By the closed graph theorem, CT(l) E L(Z)
CT(l)T(-l)
for
and hence
is a bounded linear operator.
In applications, it is sometimes difficult to prove directly that
C + A is a generator, although
erator and
is in some sense small relative to
A
C is a genC.
The
simplest such result is the following. Theorem 3. T(t)
and let
(group) IIU(t) II
Let
U(t).
O.
Outline of the Proof of Theorem 3: tor of
If
A + C
is the genera-
U(t), then by the variation of constants formula, U(t)
must be the solution of the functional equation U(t) = T(t) +
I
to T(s)AU(t-s)ds.
(6.2.11)
Equation (6.2.11) is solved by the following scheme: co
U(t)
L Un(t)
(6.2.12)
n=O where
UO(t) = T(t)
and
Un+l(t) =
I:
T(s)AUn(t-s)ds.
A straightforward argument shows that (6.2.12) is a semigroup with generator Exercise 12.
U(t)
defined by
A + C.
Prove Theorem 3 in the case
Mal, w • 0
in
6.2.
Semigroup Theory
the following way:
107
let
B
=A
C
+
operator with dense domain
D(B)
;\ > IIAII, IIA(AI - C) -111 < 1
so that
exists for (;\ - IIAID -1
;\ > IIAII. for
so that D(C).
K
is a closed
Show that for
(I - A(;\I - C)-l)-l II (AI - B) -111 ~
Hence show that
;\ > IIAII
B
and use the Hi1le-Yosida Theorem.
Example 5 (Abstract Wave Equation).
Let
selfadjoint operator on a Hilbert space
A H
be a positive B E L(H).
and let
Consider the equation
v
+
Bv
Define a new Hilbert space duct
< , >
+
Av
o.
=
Z = D(AI/2) x H with inner pro-
defined by l 2 l 2 ) <w,w> - (A / wI' A / wI
where
(,)
W= [wl ,w2 ].
(6.2.13)
+
is the inner product in
( w ,w -) 2 2
Hand
w
= [w l ,w 2 ],
Equation (6.2.13) can now be written as a first
order system on
Z, Cw
(6.2.14)
where
D(C)
= D(A)
l 2 x D(A / ).
If
B
=
0
then
C
joint so by Example 4 it generates a group.
is skew-selfadFor
B
r
0, C
is the sum of a generator of a group and a bounded operator so by Theorem 3 it generates a group. As a concrete example of the above, consider the wave equation
is
6.
108
Vtt + v t - /::'v = 0,
x
INFINITE DIMENSIONAL PROBLEMS
n,
€
v(x,t) = 0,
x
an,
€
is a bounded open subset of :rn. n
with boundary an. 2 This equation has the form (6.2.13) if we set H = L (n)' where
n
D(A) = {w this case
H, w = 0 on an}, A = - /::, , B = I. In H~(n) x L2 (n) (see [2] for a definition of
H: -/::'w
€
Z =
the Sobolev space Let
T(t)
H with generator
€
1
HO(n)). be a contraction semigroup on Hilbert space C.
Then for
11T(s+t)zll so that
II T(t) z II
II ret) zl12
IIT(t)T(s)zll ~ IIT(t)zll
is nonincreasing.
is differentiable in
V'(O)
t
For
z
D(C), vet)
€
and
+ .
v I (t)
Since
=
s > 0,
~
0, we have that
+ A linear operator
~
0,
z
€
D(C).
(6.2.15)
C with dense domain is said to be dissi-
pative if (6.2.15) holds. Exercise 13.
If
C
is dissipative and is the generator of a
semigroup, prove that the semigroup is a contraction. Theorem 4 (Lumer-Phillips Theorem). operator on the Hilbert space
B> 0
the range of
BI - C
H. is
Let
C
be a dissipative
Suppose that for some H.
Then
C
is the generator
of a contraction semigroup. Exercise 14. (6.2.15) that
Prove the above theorem. II(AI-C)zll ~ ~Izll
for
(Hint: A> 0
Use this and the fact that the range of
and
B1 - C
deduce from z is
D(C).
€
If
10
6.2.
Semigroup Theory
deduce that
C
109
is closed.
{A > 0: range of
AI - C
Show that the non-empty set is
H}
is open and closed and use
the Hille-Yosida Theorem.) Remark.
There is also a Banach space version of the above
result which uses the duality map in place of the inner product. Exercise 15.
Let
C be the generator of a contraction semi-
group on a Hilbert space and let D(B)
:::>
D(C)
be dissipative with
and IIBwl1 < allCwl1
for some
B
a
with
IIB(AI-C)-lwll < (2a
+
bllwll,
0 < 2a < 1. +
Use the inequality
bA-l)llwll
invertible for large enough
w € D(C)
to show that A.
Deduce that
AI - B - C B
C
+
is
generates
a contraction semigroup. Exercise 16. bert space
Let H
C be a closed linear operator on a Hil-
and suppose that
Prove (i) the range of (ii)
=
0
for all
duce that the range of C
I - C
C
and
is closed in h € D(C)
I - C
C*
is
H
are dissipative. H,
implies
Z =
O.
De-
so that by Theorem 4,
generates a contraction semigroup on
H.
If all the eigenvalues of a matrix C have negative -wt real parts then II exp (Ct) II .::. Me , t ~ 0, for some M > 1 and
w > O.
sions is:
The corresponding question in infinite dimenif
Re(cr(C)) < 0 w > 0
and
C
is the generator of a semigroup T(t), does imply that IIT(t)II'::'Me- wt , t ~ 0, for some
M > I?
(cr(C)
general, the answer is no.
denotes the spectrum of
C).
In
Indeed, it can be shown that if
6.
110
a < b
INFINITE DIMENSIONAL PROBLEMS
then there is a semigroup
such that
sup Re(a(C))
=
a
and
T(t)
on a Hilbert space bt IIT(t) II = e [691. Thus to
obtain results on the asymptotic behavior of the semigroup in terms of the spectrum of the generator. we need additional hypotheses. is the spectral radius of
If
T(t). that is
lim IIT(t)k I11/k.
k+ao
then a standard argument shows that for w > Wo there exists wt M(w) such that IIT(t) II ~ M(w)e . Hence we could determine the asymptotic behavior of of
T(t).
However the spectrum of
the spectrum of
from the spectral radius T(t)
is not faithful to
C. that is. in general the mapping relation a(T(t))
is false.
T(t)
=
exp(ta(C))
(6.2.16)
While (6.2.16) is true for the point and residual
spectrum [551 (with the possibility that the point
0
must
be added to the right hand side of (6.2.16)) in general we only have continuous spectrum of T(t) Example 6.
Let
Z = 9,2
and
=exp(t(continuous
spectrum of C)).
C(xn ) = (inx n ), D(C) = C generates the semigroup
{(xn ) E Z: (nx n ) E Z}. Then T(t) given by T(t) (x ) = (eintx ). The spectrum of C is n n {in: n = 1.2 •...• } while the spectrum of T(l) is the unit circle so (6.2.16) is false. For special semigroups. e.g .• analytic semigroups. (6.2.16) is true (the point zero must be added to the right hand side of (6.2.16) when the generator is unbounded). but
6.2.
Semigroup Theory
111
this is not applicable to hyperbolic problems.
For the prob-
lem studied in this chapter, the spectrum of the generator consists of eigenvalues and the associated eigenfunctions form a complete orthonormal set.
Thus the asymptotic be-
havior of the semigroup could be determined by direct computation.
It seems worthwhile however to give a more abstract
approach which will apply to more general problems. Let
be the generator of a semigroup
C
wt
II T (t) II < Me . I A I ~ exp(wt) . group
U(t) .
values of e
At
A
Hence the spectrum of A E L(Z).
Let
Q=
in
consists only of points
e
,
A E
be compact.
such that
Proof:
A
Then T (t)
U(t)
II T (t) II
with
T (t)
From Theorem 3, A
+
Then i f
C
+
{/.:
A E
A
I AI
Q,
is comwt } > e
Q.
Theorem 5 (Vidav [67], Shizuta [62] ) . tor of a semigroup
isolated eigen-
in the set
U(t) At
generates a semi -
We prove that i f
U(t) .
pact then the spectrum,of
U(t)
C
Re (A) > w}.
{A:
is an eigenvalue of
A E L(Z)
+
Suppose that there are only + C
lies in the disc
T (t) A
Then
with
T(t)
Let
C
wt < Me
be the generaand let
generates a semigroup
is compact. C
generates a semigroup
U(t)
which is a solution of the functional equation U(t)
T(t)
We prove that the map in norm on in norm.
[0, t]. Since
s
+
+ {
to T(s)AU(t-s)ds.
f(s)
= T(s)AU(t-s)
(6.2.17) is continuous
Thus the integral in (6.2.17) converges
f(s)
is a compact linear operator and the
set of compact operators in
L(Z)
operator topology, U(t) - T(t)
is closed in the uniform
is compact.
l1Z
6.
INFINITE DIMENSIONAL PROBLEMS
To prove the claim on the continuity of prove that t
0
> S >
s
AU(t-s)
+
and
hI
is continuous in norm.
0
>
f
with
hI
we first Let
sufficiently small.
Then
the set {A(U(t-s-h) - U(t-s))w: Ilwll has compact closure.
For
£
>
0
=
Let
g(s)
Ihl < hI}
it can be covered with a
finite number of balls with radius wl,wZ, ... ,w n .
1,
and centres at
£
= AU(t-s).
Then for
Ilwll
= 1,
Ilg(s+h)w - g(s)wll ~ IIA(U(t-s-h) - U(t-s)) (w-w k ) II + IIA(U(t-s-h) - U(t-s))wkll . Hence if
0
is chosen such that
IIA(U(t-s-h) - U(t-s))wkll < for
Ihl < 0
(constant)
£.
and
k
=
1,Z, ... ,n
The continuity of
£
then f
Ilg(s+h) - g(s)11
~
fol1ows from similar
reasoning applied to f(s+h) - f(s)
=
T(s+h) [g(s+h) -g(s)] + (T(s+h) -T(s))AU(t-s).
This completes the proof of the theorem. Since of
U(t)
and
U(t) T(t)
T(t)
is compact, the essential spectrum
coincide [43].
(It is important to note
that we are using Kato's definition of essential spectrum in this claim.)
Hence we have proved:
Corollary to Theorem S. The spectrum of U(t) lying outside the circle IAI = e wt consists of isolated eigenvalues with finite algebraic multiplicity.
6.2.
Semigroup Theory
113
The above result is also useful for decomposing the space
Z.
Suppose that the assumptions of Theorem 5 hold.
Since there are only isolated points in the spectrum of U(t) wt outside the circle IAI = e , there is only a finite number of points in the spectrum of ~
Re(A)
w +
£
for each
£
sum
such that
£
>
in the halfplane
Let 0
AI, ... ,A n denote these and let P be the corres-
Then we can decompose
PU(t) + (I-P)U(t)
M(a)
O.
>
eigenvalues for some fixed ponding projection.
A + C
and for any
a >
£
II (I-P)U(t) II ~ M(a)e(w+a)t.
U(t)
into the
there exists Finally, PZ
is
finite dimensional and so it is trivial to make further decompositions of
PU(t).
We now show how the above theory can be applied to hyperbolic problems. product on
H
pact.
, ). such that
Let
Let
H
be a Hilbert space with inner
A be a positive selfadjoint operator
A-I
is defined on all of
H
and is com-
Consider the equation
v
+ 2av + (A+B)v
0
where B € L(H) and a > O. Let Z be the Hilbert space D(A I / 2 ) x H with inner product < > as defined in Example
s.
We can recast the above equation in the form
w
Cw
C
As in Example 5, C
is a bounded perturbation of a skew-
selfadjoint operator and so
C
generates a group
Set).
Let
114
6.
u (t) Then
U(t)
o
[ a:
I
INFINITE DIMENSIONAL PROBLEMS
J
S (t) [
;l
I -aI
is a group with generator
[ -aI
+
and the compactness of the injection C2 : Z + Z is compact. Since C1 generates a group that
the asymptotic behavior of
:J
[ a 2 I:B
-a: ]
-A
D(A 1 / 2 )
+
H
implies
Thus the above theory applies. U1 (t) with IIU 1 (t)ll.::.. e- at , U(t) (and hence Set)) depends
on a finite number of eigenvalues.
H
=
D(A)
The above theory applies, for example, to the case 2 L (n), where n is a bounded domain in mn, A = -~, {v E H:
=
~v
E H and -1
elliptic theory, A Example 7. U
tt
v
0
=
an}
on
since by standard
is compact.
Consider the coupled set of wave equations +
2au t - u xx
+
Bf:
g(x,s)u(s,t)ds -
Bv
=
0
(6.2.18)
for
0 < x
0 and B small enough the
n real parts of aU the eigenvalues are negative so we expect that liT (t) II < Me -wt , t > 0, for some w > O. This formal
argument can be rigorized by applying the above theory. a = O.
We now suppose that
In this case, for
sufficiently small all the eigenvalues of ginary and they are simple.
M.
are purely ima-
By analogy with the finite di-
mensional situation we expect that some constant
C
B
IIT(t) II.::. M, t ~ 0, for
We show that this is false.
Consider the following solution of (6.Z.18): -1
u(x, t)
Zrr
v(x,t)
m- 1 sin mx cos mt
where
m(cos mt - cos amt)sin mx
m is an integer and
(6.Z.19)
a~
the initial data corresponding to (6.Z.19) is bounded independently of m. Let tm Zm 3rr. Then for large m, Z
1 - cos(rrzB) so that
Z lux(x,t m) I ~ (constant)m . Z
+
Thus
O(m- 4 )
>
IIT(t m)
constant,
112.
(constant)m . The above instability mechanism is associated with the fact that for
a
= 0, B f 0, the eigenfunctions of C do not
form a Riesz Basis.
For further examples of the relationship
between the asymptotic behavior of solutions of the spectrum of
C see [15,16].
We now consider the nonlinear problem
w = Cw
and
116
6.
w where
C
= Cw
+
INFINITE DIMENSIONAL PROBLEMS
N(w),
= Wo
w(O)
is the generator of a semigroup
N: Z + Z.
(6.2.20)
E Z, T(t)
on
Z and
Formally, w satisfies the variation of constants
formula wet) = T(t)w O Definition.
A function
of (6.2.20) on and if for each
[O,T]
w E C([O,T];Z) if
C*
ing between
is a weak solution 1
w(O)
wOI f(w(')) E L ([O,T];Z)
the function
[O,T]
is the adjoint of
<wet) ,v>
is ab-
and satisfies
d dt <wet) ,v> = <wet) ,C*v>
where
(6.2.21)
to
v E D(C*)
solute1y continuous on
ft T(t-s)N(w(s))ds.
+
+
C and
< ,>
a.e.
denotes the pair-
Z and its dual space.
As in the linear case, weak solutions of (6.2.20) are given by (6.2.21).
More precisely:
Theorem 6 [9].
A function
of (6.2.20) on
[O,T]
and
w
w: [0, T] + Z
if and only if
is a weak solution 1 f(w(·)) E L ([O,T];Z)
is given by (6.2.21). As in the finite dimensional case, it is easy to solve
(6.2.20) using Picard iteration techniques. Theorem 7 [61].
Let
N: Z + Z
be locally Lipschitz.
Then
there exists a unique maximally defined weak solution wE C([O,T);Z)
of (6.2.20). Ilw(t)II+ex>
Furthermore, if as
T < ex>
then (6.2.22)
t+T
As in the finite dimensional case, (6.2.22) is used as a continuation technique.
Thus if for some
Wo
E Z, the
6.3.
Centre Manifolds
solution
wet)
of (6.2.20) remains in a bounded set then
wet)
exists for all
6.3.
Centre Manifolds Let
Z
117
t > O.
11'11.
be a Banach space with norm
We consider
ordinary differential equations of the form
w = Cw where
C
semigroup
N(w),
+
w(O) E Z,
(6.3.1)
is the generator of a strongly continuous linear Set)
N: Z
and
second derivative with
Z
+
0, N' (0) = 0 [N'
N(O)
Frechet derivative of
has a uniformly continuous is the
NJ.
We recall from the previous section that there is a unique weak solution of (6.3.1) defined on some maximal interval
[O,T)
and that if
T
O.
X along
x E X, Y E Y, let
(6.3.2)
Y.
Let
118
6.
= PN(x+y),
f(x,y)
INFINITE DIMENSIONAL PROBLEMS
g(x,y)
= (I-P)N(x+y).
(6.3.3)
Equation (6.3.1) can be written
X
Ax + f(x,y)
y
= By + g(x,y).
(6.3.4)
An invariant manifold for (6.3.4) which is tangent to X space at the origin is called a centre manifold. Theorem 8. y
=
There exists a centre manifold for (6.3.4), 2 hex), Ixl < 0, where h is C . The proof of Theorem 8 is exactly the same as the
proof given in Chapter 2 for the corresponding finite dimensional problem. The equation on the centre manifold is given by
u = Au In general if
yeO)
is not in the domain of
will not be differentiable. yet) = h(x(t)), and since and consequently Theorem 9.
+ f(u,h(u)).
(6.3.5) B
then
yet)
However, on the centre manifold X is finite dimensional
x(t),
yet), are differentiable.
(a) Suppose that the zero solution of (6.3.5) is
stable (asymptotically stable) (unstable).
Then the zero
solution of (6.3.4) is stable (asymptotically stable) (unstable) . (b) stable.
Suppose that the zero solution of (6.3.5) is
Let
(x(t),y(t))
be a solution of (6.3.4) with
\I (x(O) ,y(O))
\I
tion
of (6.3.5) such that as
u(t)
sufficiently small.
Then there exists a solut
+ .. ,
6.3.
Centre Manifolds
119
x(t) yet) where
y
>
= u(t) + O(e -yt )
(6.3.6)
h(u(t)) + O(e- yt )
O.
The proof of the above theorem is exactly the same as the proof given for the corresponding finite dimensional result. Using the invariance of
h
and proceeding formally
we have that h' (x) [Ax + f (x, h (x) ) 1
= Bh (x) + g (x, h (x) ) •
( 6 . 3 . 7)
To prove that equation (6.3.7) holds we must show that is in the domain of Let
E
Xo
the domain of
B. be small.
X
Let
(6.3.4) with
To prove that
h (x O)
is in
it is sufficient to prove that
B
lim+ t+O exists.
hex)
= xO.
h(x O)
h(x(t))
x(t), yet) x(O)
-
U(t)h(x O) t
be the solution of
As we remarked earlier, yet)
differentiable.
From (6.3.4)
yet)
= U(t)h(x o) +
J:
is
U(t-T)g(X(T) ,Y(T))dT,
so it is sufficient to prove that lim+ t1 t+O exists.
Jt
0
U(t-T)g(X(T) ,y(T))dT
This easily follows from the fact that
strongly continuous semigroup and h(xO)
is in tlie domain of
B.
g
is smooth.
U(t) Hence
is a
120
6.
Theorem 10. origin in Hx) q
>
X into
D(B).
E
~
Let
be a
INFINITE DIMENSIONAL PROBLEMS
1 C
map from a neighborhood of the
Y such that
~(O)
= 0,
~'(O)
x'" 0, (M~)(x)
Suppose that as
= 0 and
= O(lxlq),
1, where (M~)
Then as
=
(x)
~'(x)
[Ax
+ f(x,~)l
- BHx) - g(x,Hx)).
= O(lxlq).
x'" O,llh(x) - Hx)11
The proof of Theorem 10 is the same as that given for the finite dimensional case except that the extension
s: X ... Y
of
domain of 6.4.
~
must be defined so that
is in the
B.
Examples
Example 8. V
tt
+
Consider the semi1inear wave equation v t - vxx - v v
where as
Sex)
f
v'" O.
C3
is a
+
fey)
=
0
0, (x,t)
=
at
E
(O,rr)
(0,"') (6.4.1)
x
= O,rr
x
function satisfying
fey)
=
v3
+
O(v 4)
We first formulate (6.4.1) as an equation on a
Hilbert space. Let
Q
=
(d/dx)
2
+
I, D(Q)
Q is a self-adjoint operator.
=
2
1
H (O,rr) n HO(O,rr).
Let
1
Then
2
Z = HO(O,rr) x L (O,rr) ,
then (6.4.1) can be rewritten as
w = Cw
+
N(w)
(6.4.2)
where N(w)
Since
C
is the sum of a skew-selfadjoint operator and a
6.4.
Examples
121
bounded operator, C generates a strongly continuous group. 3 Clearly N is a C map from Z into Z. The eigenvalues of +
A1
and all the other eigenvalues have real part less
0
o.
than
Care
The eigenspace corresponding to the zero eigen-
value is spanned by
q1
where
~ ] s in x.
[
To apply the theory of Section 3, we must put (6.4.2) into canonical form.
We first note that
Cq2
= -q2
where
and that all the other eigenspaces are spanned by elements of the form
an sin nx, n
2
~
2, an E1R.
other eigenvectors are orthogonal to X
span(q1) , V
Z
X III Y.
= span(q1,q2) ,
The projection
In particular, all q1
and
q2·
Let
span(Q2) III V.l , then
Y
P: Z ... X
is given by
(6.4.3) where w.
J
Let
w
= -2 TT
JTT
0
w.(e)sin e de. J
= sQ1 + y, s E 1R, Y E Y and B
(I-P)C.
Then we can
write (6.4.2) in the form
(6.4.4) y
= By + (I-P)N(sQ1+Y).
122
6.
INFINITE DIMENSIONAL PROBLEMS
By Theorem 8, (6.4.4) has a centre manifold h(O)
=
=
0, h' (0)
0, h:
y
=
h(s),
By Theorem 9, the equa-
(-eS,eS) + Y.
tion which determines the asymptotic behavior of solutions of (6.4.4) is the one-dimensional equation sq1
PN( sq 1 + h(s)).
=
(6.4.5)
Since the non1inearities in (6.3.4) are cubic, h(s) so that
or (6.4.6)
s
Hence, by Theorem 9, the zero solution of (6.4.4) is asymptotically stable.
Using the same calculations as in Section 1 of
Chapter 3, if
s(O)
Hence, if vt(x,O)
v(x,t)
> 0
then as
set)
=
t
(4 t) -1/2
+
00,
+ oCt -1/2).
is a solution of (6.4.1) with
small, then either
v(x,t)
(6.4.7) v(x,O),
tends to zero exponen-
tially fast or v(x,t) where
set)
=
±s(t)sin x + 0(s3)
(6.4.8)
is given by (6.4.7).
Further terms in the above asymptotic expansion can be calculated if we have more information about f. Suppose that fey) = v 3 + av 5 + 0(v 7 ) as v + O. In order to calculate an approximation to (M4» (s)
=
h(s)
set
4>' (s)PN(sq1+4>Cs)) - B4>Cs) - (I-P)N(sQ1+4>Cs)) (6.4.9)
6.4.
Examples
where
4>: IR
(M4» (s)
+
123
Y.
= 0 (s 5)
•
To apply Theorem 10 we choose 4>(s)
If
= 0(s3)
4>(s)
so that
then
(6.4.10)
where
sin 3x.
q(x)
If
4>(s)
(6.4.11)
then substituting (6.4.11) into (6.4.10) we obtain
(M4» (s)
Hence, i f
a
= 3/4,
B1
1/32, BZ
0, then
M4> (s)
so by Theorem 10 h(s)
=
43 qzs 3
+
1[1] 32 0 qs 3
+
O(s 5 ).
(6.4.12)
Substituting (6.4.1Z) into (6.4.5) we obtain
The asymptotic behavior of solutions can now be found using the calculations given in Section 1 of Chapter 3. Example 9.
In this example we apply our theory to the equa-
tion -
[B +
4 (Z/rr)
f1 (v o
s
(s,t)) Zds]v
xx
= 0, (6.4.13)
• 0 at x = 0,1 and given initial conditions xx V(x,O), vt(x,O), 0 < x < 1. Equation (6.4.13) is a model for
with
v • v
124
INFINITE DIMENSIONAL PROBLEMS
6.
the transverse motion of an elastic rod with hinged ends, v
B a constant.
being the transverse deflection and
The
above equation has been studied by Ball [7,81 and in particu-
B = _~2
lar he showed that when
the zero solution of
(6.4.13) is asymptotically stable.
However, in this case the
linearized equations have a zero eigenvalue and so the rate of decay of solutions depends on the non-linear terms.
In
[131 the rate of decay of solutions of (6.4.13) was found using centre manifold theory.
Here we discuss the behavior
B + ~2
of small solutions of (6.4.13) when
is small.
As in the previous example we formulate (6.4.13) as an ordinary differential equation. vatives and
z
We write
for time derivatives.
2
1
2
H (0,1) n HO (0,1) x L (0,1),
w
[
Ww
1
2
1'
Cw
=
[
Let
-v''''
Qv
2 w ] ' Qw -w l 2
for space deri-
+
Bv",
N(w)
Then we can write (6.4.13) as w
It is easy to check that group on If
Z and that
N
Cw
+
N(w).
C
generates a strongly continuous
is a
COO
A is an eigenvalue of
trivial solution
u(x)
(6.4.14)
map from
Z
u"" - Bu" + (A+A 2)u
0
u(O)
u" (1)
u" (0)
=
Z.
C then we must have a non-
of
=
into
u(1)
o.
6.4.
Examples
lZ5
An easy computation then shows that C
A is an eigenvalue of
if and only if
Let
£
=
~ZB + ~4.
Then the eigenvalues of
Care
An(£)'
where
2A (£) = -1 + [1_4£]1/Z, AZ(£) = -1 - A (£), and all l l the rest of the An(£) have real parts less than zero for £
sufficiently small.
The eigenspaces corresponding to
are spanned by
and
and
.[
~x,
then
Z= X
(jj
Y
~x
] sin
Z
span(ql)' y = span(ql ,qZ)' Y = span (qZ)
(jj
are spanned by elements orthogonal to
X
1 A (£) Z
>
while the eigenspaces corresponding to
Let
where
and the projection
and
ql
n
for
An(£)
qZ'
P: Z ... X
yol,
is given by
where 1
w. = Z J J
Let
w
=
0
w. (e)sin J
sql + y, where
s E ffi
~e
de.
and
y E Y.
Then we
can write (6.4.14) in the form Al(£)sql + PN(sql+Y) (6.4.15)
By + (I-P)N(sql+Y) £ =
O.
By Theorem 8, (6.4.15) has a centre manifold lsi