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Lecture Notes in
Physics
Monographs Editorial Board
Beig, Wien, Austria J. Ehlers, Potsdam, Germany U. Frisch, Nice, France K.'Hepp, Zfirich, Switzerland W. Hillebrandt, Garching, Germany D. Imboden, Zfirich, Switzerland R. 1 Jaffe, Cambridge, MA, USA R.
Kippenhahn, G6ttingen, Germany Lipowsky, Golm, Germany H. v. L6hneysen, Karlsruhe, Germany 1. Ojinla, Kyoto, Japan A. A. Weidenmiiller, Heidelberg, Germany R. R.
J. Wess, Mfinchen, Germany J. Zittartz, K61n, Germany
':
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http://wwwspringerde/phys/books/Inpm
Michel Henon
Generating Families in the Restricted
Three-Body Problem II.
Quantitative Study of Bifurcations
A-11
4 3
-ISpringer
Author Michel H6non CNRS
Observatoire de la C6te d'Azur B-P 4229
o6304 Nice C6dex 4, France
Library of Congress Cataloging-in-Publication Data applied for. Die Deutsche Bibliothek
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CIP-Einheitsaufnahme
H6non, Michel: Generating families in the restricted three-body problem / Michel H6non.
-
Berlin
Heidelberg
; New York ; Barcelona ;
London ; Milan Paris ; Singapore ; Tokyo (Physics and astronomy online library)
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Hong Kong
Springer
Quantitative study of bifurcations. 2001 (Lecture notes in physics: N.s. M, Monographs ; 65)
2.
-
ISBN 3-540-41733-8
ISSN 0940-7677
This work is
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SPIN:lo644571
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5 4 3
2 10
Preface
previous volume (H6non 1997, hereafter called volume I), the study of generating families in the restricted three-body problem was initiated. (We recall that generating families are defined as the limits of families of periodic orbits for p -4 0.) The main problem was found to lie in the determination of the junctions between the branches at a bifurcation orbit, where two or more families of generating orbits intersect. A partial solution to this problem was given by the use of invariants: symmetries and sides of passage. Many simple bifurcations can be solved in this way. In particular, the evolution of the nine natural families of periodic orbits can be described almost completely. However, as the bifurcations become more complex, i.e. when the number of families passing through the bifurcation orbit increases, the method fails. This volume describes another approach to the problem, consisting of a detailed, quantitative analysis of the families in the vicinity of a bifurcation orbit. This requires more work than the qualitative approach used in Vol. I. However, it has the advantage of allowing us, in principle at least, to determine in all cases how the branches are joined. In fact it gives more than that: we will see that, in almost all cases, the first-order asymptotic approximaIn
a
tion of the families in the
This
allows,
in
particular,
neighbourhood a
of the bifurcation
can
be derived.
quantitative comparison with numerically found
families.
Chapter 11 deals with the relevant definitions and general equations. The quantitative study of bifurcations of type 1 is described in Chaps. 12-16. The analysis of type 2 is more involved; it is described in Chaps. 17-23. Type 3 is even more complex; its analysis had not yet been completed at the time of writing. As was the case for the previous volume, this work is sometimes lacking in mathematical rigor; there is certainly much room for improvement. However, a
number of factors lead
me
to believe that the results
are
correct:
qualitative analysis of Vol. I; agreement computations; internal consistency; and, simple intuition.
with the results of the
agreement
with numerical
VI
Preface
My thanks go to Larry Perko, who read a draft version of this volume and made many helpful comments and suggestions. I also thank Alexander Bruno for many discussions by e-mail, and for sending an english translation of parts of his
new
Nice, March 2001
book in advance of
publication.
Michel H6non
Contents
.........................
1
.............................
1
........................................
1
11. Definitions and General
11.1 Introduction
................
11.2 The 0 Notation 11.2.1
Definitions
.....................................
.............................
.........
4
.....................................
4
Computation GeneralEquations
Rules
113.1
Definitions
11.3.2
Intermediate Arcs
11.3.3
Orders of
11.3.4
More Accurate Estimate of
Matching
6
...............
of
Magnitude Relations
Api
and
Api
.............
7
..................
9
.............................
11
Ajbi
The Case p = 0 11.4 General Method ..........................................
13
Quantitative Study of Type 12.1 Fundamental Equations
.............................
17
.....
17
..................................
17
11.3.6
12.
.......
.....................
11.3.5
1 2
11.2.2
11.3
Equations
.................................
I
.............................
12.1.1
Arc Relations
12.1.2
Additional Relation for Two Arcs
12.1.3
Encounter Relations
12.1.4
Recapitulation
I
.................
12.3 The Case 12.3.1
12.3.2,
v
0
........................................
25
1/2 1/2
26 27 28
............................
30
................................
34
Species Species Orbit Sides of Passage
First
25
............I..................
Orbit
12.4.1
21
23
Species Orbit ................................ Second Species Orbit ............................. v
19
20
...................
First
12.4 The Case 0
pi /2
I
.............
...................................
Equations
16.2 Method of Solution
126
..............................
...............................
132
.....................................
135
.....................................
135
.....................................
136
......................................
136
..........................................
138
16.7 Conclusions ............................................. 142 16.8
Appendix: 16.8.2
17.
No TT Node*
'
Total
17.1 New Notations
17.2 Fundamental
of
Tf,
Type
T9
2
,
Equations
Arc Relations
Separation
17.3 The Case
v
=
0
..............................
143 143 147 149
.................
149
.................................
150
..................
Encounter Relations
17.2.1 17.2.3
........
................................
Quantitative Study
17.2.2
.........................
T-Sequence ................................ T-Sequence
Partial
16-8.1
.............................
151
..................................
151
of the Case
n
=
1
......................
........................................
158 158
17.3.1
T-Arc
........................
......
158
17.3.2
S-Arc
- .......................................
159
17.4 The Case 0
0,
applied:
1. Weestimate the orders of
magnitude
of the variables
by extrapolating
from the previous case. 2. These orders of magnitude suggest an appropriate change of variables. Intuition plays a role here: the "good" change of variables is not always obvious. In most cases, the new variables happen that some of them are o(l).
are
0(l)
for IL
-+
0; but
it
can
3. At this stage it is generally convenient to make another change of variables which consists simply in a change of scale for each variable, so as
simplify the equations as much as possible. equation, the dominant terms are collected in the left-hand member, and the equation is divided by an appropriate factor to make these terms 0(l). The right-hand member then should contain only terms to
4. In each
which
are
o(l).
5. We determine under which conditions the
right-hand
member is indeed
This may determine an upper limit for v, which is the limit of the presently considered interval for v.
o(l).
The asymptotic equations member to zero.
are
then obtained
by equating the right-hand
asymptotic equations. If there are several solutions, the by continuity with the previous interval. 7. We compute the Jacobian JJJ of the system of equations, for the asymptotic solution. (Note: usage varies concerning the terms used to designate the Jacobian matrix, J, and the Jacobian determinant, JJJ. In the present work, only the determinant will be needed, and we will call it simply Jacobian.) If I JI is. non-zero, the procedure is successful: the change of variables selected at step 2 above was indeed the good one. The implicit function theorem guarantees then the existence of a solution in the vicinity of the asymptotic solution, for p 54 0. Moreover, the difference between this true solution and the asymptotic solution is given in order of magnitude by the maximum of the right-hand members.
6. We solve the
relevant solution is determined
8. Sometimes the
error
sidering separately
estimates
some
can
equations.
be refined for
some
variables
by
con-
11. Definitions and General
16
9. We
can
ables.
go
back
Equations
to the initial variables
by inverting the changes of
vari-
Quantitative Study of Type
12.
12.1 Fundamental
1
Equations
12.1.1 Are Relations
The intermediate orbit
i, defined
in Sect.
11.3.2,
is
a
keplerian'orbit.
Therefore
a definite relation exists.between the positions and velocities at the two ends, corresponding to the times ti-1 and ti. We compute now these relations
explicitly. Until define it
now some
precisely
arbitrariness has been left in the choice of ti. Now
we
the time of the intersection of the true orbit Q A with
as
the orbit of M2. This
was
found to
give the simplest calculations
in the
case
of type 1. We define yj as the oriented length of the arc from M2 to M3 at time tj (Fig. 12. 1). yj is also the lead of. M3 over M2 for the passage at the intersection
point.
Fig.
12.1. Definition of yi.
We consider
now
the intermediate orbit i
of M2 at a time t 71'. Since the radial of type 1, we have
41
=
Z
We call
ti +
Y'
0( A)
the
velocity
is
(i
A).
It intersects the orbit
for bifurcation orbits
(12.1)
-
corresponding
E
non-zero
lead. There is also
M. Hénon: LNPm 65, pp. 17 - 38, 2001 © Springer-Verlag Berlin Heidelberg 2001
Quantitative Study of Type
12.
18
yz f
orbit of M2 at
t'j
=
yz
a
=
time
O(I.L)
ti-1 +
i intersects the
previous encounter, the intermediate orbit tj, with a lead y'j, and there is
at the
Similarly,
(12.2)
OW
Yi +
=
1
(12.3)
,
(12.4)
O(A)
Yi-1 +
keplerian orbit and therefore can be represented by a point (A, Z) plane. (We recall that the (A, Z) coordinates elements (a, e) by a simple change of variables; orbital the are related to call that We see Chap. 4.) point (Ai, Zj). We call Ci the Jacobi constant of call (A, Z) the point representing the bifurcation We orbit. intermediate the We define Jacobi constant. its C and orbit, The intermediate orbit is
a
in the
Ai
=
A + AAj
Zi
,
=
Z + Azi
Ci
,
=
C + Aci
(12-5)
.
a sequence of alternating basic arcs PQ and We consider first the case where the basic arc i is a first
The bifurcation orbit consists of
QP (Sect. 6.2.1.2). arc PQ. In the particular case collision in P, i.e. the lead yi' is zero,
where the intermediate orbit i has
basic
and the lead at the end is, from
Y2 1 In the
yj"
=
t2
t4
-
t20
case
yi'
27r(Zi
=
+
y'
-
2
arc
y
+
(12.5)
note also that 6
we
a
In the
V'j
=
27r(Zi
+
a
-
can
yi'
arc
(6.8)
-
#Aj)
we
(12-6)
.
have then
for the bifurcation orbits; to (6.17). We obtain
according
obtain from
we
arc
i is
a
second basic a
collision
(4.20), (4.23), (4.28)), (12.9)
.
yi'l
+
27r(Zi
+
0
=
00
J for the second basic
Yj'1
=
-
a
where the intermediate orbit i has
yj"
There is
a
4.3.1,
(12.8)
where the final lead -
+
.
case
a
2-7r(Zi
is non-zero,
the relation
general =
in Sect.
(12.7)
is zero,
PAj)
--
be made when the basic
case
yj"
i.e. the final lead
y
PAi)
-
the first basic
#oAAi)
-
analysis
.
we use
#0 for
A similar computation QP. In the particular
P,
2-7r(a
#Aj)
-
and
=
27r(AZi
=
Z
t40 +
-
the
(4.20), (4.23), (4.28)),
where the initial lead
general
We substitute
in
=
we can use
flAi)
is non-zero,
we
have then
(12.10)
.
arc
according
to
(6.18),
and
we
obtain
Y'j
-
27r[AZi
-
(Oo
-
J)AAi]
.
(12.11)
We define Si
1
if the
arc
i is
a
+1
if the
arc
i is
a
first basic arc, second basic arc.
(12.12)
12.1 Fundamental
+1 if the encounter i is at P, -1 if words, si equations (12.8) and (12.11) can then be merged into In other
y'
=
y
-
21
=
z
I
27rSi
-Azi
(186
+
I
-
i-
Si
2
Equations
it is at
Q.
19
The two
J) AAil
(12.13)
2
There is
Azi
( 5C_ 09Z)
=
'9Z)
ACi+
We call AC the
AAi+O(ACi2)+O(AA?).
M
A
(12.14)
C
displacement of the Jacobi constant for the true
orbit with
respect to the bifurcation orbit. Since the true orbit and the intermediate orbit agree within O(M) (Sect. 11.3.3), we have
Ci From
=
C+
(4.10)
we
3
AAi
06U)
2
Aci
=
OW
AC +
(12.15)
.
have
v/aAai
+
O(Aa?)
(12.16)
2
where Aai is the variation of the semi-major axis of the intermediate orbit i with respect to the bifurcation orbit. Substituting (12.14) and (12.16) into (12.13), and using also (12.2), (12.4), and (12.15b), we obtain yi
-
(aC az) +37rsiv/a [_ (OA az) yi-i
=
AC
-27rsi
A
+
flo
J_
-
2
C
+O(P)
O(AC2)
+
O(Aa?)
+
Aai
i E A
,
71
il
si2
equation, linking the value ai of an intermediate ends, will be called an arc relation.
This
(12.17)
.
arc
and the values yi-I
and yi at its
12.1.2 Additional Relation for Two Arcs
We prove now a relation useful later on. If and
we
(12.4), yi+l
(12.13). for
add
-
as
well
yi-i
as
+
The values of Ci and
-
AZi
+O(p)
+
the
si+1
arcs
-si,
=
27rsi(AZi+1
=
-7rJ(AAi
AZi+1
involving
AAi+,) Ci+1
+
are
aZ) o9A
-
i and i + we
1, and take
into account
(12.2)
obtain
AZi)
-
2.7r Si
(00 -J) (AAi+l
-
2
0(p)
the
Till be
two consecutive basic arcs, which
(12.18)
same
(AAi+l
AAi)
-
within
O(p) (see (12.15)).
Therefore
A4i)
0
0[(AAi+l
-
AAi )2]
(12.19)
Quantitative Study of Type
12.
20
yj+j
yi-1
-
(12.18)
this into
Substituting
37rJ
37rsi
=
,Fa(Aaj
-
2
using (12.16),
and
[( ) 09Z
OA
+
1
-
Po
+
C
Aai+,)
+
O(M)
obtain
we
il \/a-(Aai+l
-
2
+
O(Aa,?)
-
Aaj) 2
O(Aai+
+
i E C
(112.20) This relation is stronger than the relation which would have been obtained by the arcs i and i + 1: there is no error term O(AC2).
simply adding (12.17)'for
12.1.3 Encounter Relations
We establish
relation for encounter i
now a
where this encounter is
case
near a
point
orbit i intersects the orbit of M2 at time
vi"
call
Oj'
and
polar
the
t ',
coordinates of the
coordinates, (see Fig. 8.2). There is
(i E C) We (si +1). -
P
=
consider first the The intermediate
vicinity of collision i. We velocity at that time, in rotating in the
(see (8.3))
V 12 =3-Ci
(12.21)
,j
(12.15)
and therefore from
Vi" where
V
=
is the
v
cos
WI
in fixed
velocity modulus for the true orbit. expressed as a function -of 0 and Vi", the velocity modulus
be
can
axes
(12-22)
O(P)
+
(Fig. 8.2): V!12
_
-
Cos
V 2112
(12.23)
2vill 71
and
we
have
V112 i so
(see (4.12a)) (12.24)
=2 ai
that Cos
V2
Oi'
-
IN
2v
+
(12.25)
O(P)
Similarly, the intermediate orbit i + 1 intersects the orbit of M2 at time in the vicinity of collision i, and we define v +j and W+1 as the polar coordinates of the velocity at that time. A similar relation is obtained:
t +,,
z
z
Cos
and
we
Oi+1 have
1
_
V2
-
-
2v
1/ai+l
+
O(P)
2
(12.26)
12.1 Fundamental
041
Cos
COS
-
Aai+l
0
Aai
-
[1
2va2 We
use now
project same
on
at
ti'
+
O(Aai)
+
the modified second
,bi (ti) gives
in
0(p),
+
projection
O(Aai+,)]
+
(12. 27)
0(p)
matching equation (11.79), which we (Thi& direction is not exactly the
0(p)
and therefore
t + 1 with an error 0 (p); v cos 0i'+1 + 0(p). In the same
or
0. /
v cos
21
OW
replace ti by
we can
vi'+, cos 0i'+1
is then
ai+1
ai
the tangent to the orbit of M2. and at t'i+,; but the difference is
i+ 1 (ti),
the term
+
2v
Equations
+ 0 (p). On the other
negligible.)
In
the proj ection way, the term
hand, the distance d
from M2 to the tangent to the orbit can be computed with an error 0(q2) with M2 replaced by a point M2* situated on the tangent to the circle, at the same distance yi from the intersection *point. We have thus: d yi I sin O ' + 0 (d). the of the the circle is sin W, x sign(yi). j on Finally projection tangent to
Therefore
YiV(cos M+1
-
Cos
21L
0'
+
V
0(/tyi)
+
0 /_t(cos
+1
-Cos
(12.28) We
can
for
use
v
the value of the bifurcation
Substituting (12.27),
yi(Aai+l
-
i E C
Aai)
orbit, with
an
error
0(6i).
obtain
we
4pa
2
i-
-
V
0(pyi)
+
0(/LAai)
0(pA.ai+1)
+
,
(12.29)
-
This equation, linking the value of yi at an encounter and the values Aai and Aai+l of the adjacent arcs, will be called an encounter relation. It is identical with the equation obtained by Guillaume (1971, p.83, equation for Aal). For an encounter near a point Q, exactly the same equation is obtained. For yo
a
=
12.1.4
partial bifurcation, the equations (11.71) give
OW
Yn
,
simplify 4a2
1 +
them
2
constitute
a
set of
equations for the
yi and the
Aai.
by writing
(OC [( TA 'Z)C-po] G,
27r
'Z
G2 A
=K
37rJ.,Fa 2
G3
(12.31)
reproduces (8.14)). The quantities G1, G2, G3, K are given bifurcation. G, and G3 are always positive. G2 and K functions of the partial derivatives of the function Z (A, C); in view of the
(the
last definition*
constants for are
(12.30)
OW
Recapitulation
(12.17), (12.29), (12.30) We
=
a
22
Quantitative Study of Type
12.
1
complicated expression of Z (see (4.28);
(Fig. 4.15),
surface
Therefore
we
Conjecture This
make the
is
(see
also
4.79)),
and the appearance of the real numbers.
arbitrary, ordinary following fundamental conjectures:
expect them
12.1.1. K
conjecture
bifurcations
we
takes
never
to be
integer values.
supported by the numerical computation of K for many 8.2.1). It will be frequently used in what follows.
Sect.
Conjecture 12.1'.2. G2
takes the value 0.
never
This will also be needed in what follows. We obtain the fundamental equations for bifurcations of type 1: yi
-
yi-i
=
G3(1
-G23i,
kC
Aai)
-Gip
-
+
Ksi)Aai
+
0(/-t)
+
O(AC2)
+
O(Aal?)
E A
yi(Aai+l YO
-
O(IA)
=
=
Yn
The additional relation yi+l
-
yi-i
+O(M) (12.32a)
is
a
=
+
+
O(A)
=
for
(12.20)
-G3(l
0(pyi)
O(Aa?)
+
relation between
-
as
shown
-
-
schematically by Fig.
i E C
(12-32)
Ksi)Aai+l
AC, Aai, given,
yi-1, and yi.
(12.33) (12.32b)
successive elements
is
can
a
rela-
thus be
(12-34)
,
12.2.
Aai 12.2.
-
Yi
Yi- i
Fig.
G3(1
i E C
Aai, yi, Aai+,. computed step by step, in the order -
O(ILAai+,)
2
For AC
Aai, yi, Aai+,, yi+,,
+
partial bifurcation.
O(Aa i+
tion between
...
a
O(fzAai)
becomes
Ksi)Aai
+
+
Yi+ i
>
Aai+l
Aai+2
Step by step computation.
1 We remark also that for a partial bifurcation, we have a system of 2n equations for the 2n variables AC, y, to yn-1, Aal to Aan (yo and y,, are given by (12.32c) and (12.32d)), and therefore a one-parameter family of solutions, which is the usual family of periodic orbits; these equations are therefore sufficient. For a total bifurcation, we have 2n relations for the 2n + 1 variables AC, y, to Yn, Aal to Aa, and again the equations are sufficient. -
In
The values of the si in (12.32) and (12.33) can be specified partial bifurcation, for the si to be determined according to
a
as
follows.
(12.12),
we
12.2 Exclusion of Successive Identical T-Arcs
whether the
specify
must
(12.32)
and
obtained
by
starting point
is in P
or
in
Q.
23
But the equations
show that for any solution, there exists another solution changing the signs of K and of the si. Therefore the study of a
(12.33)
starting point in P is identical with the study of the value starting point in Q (see Sect. 8.4.1), and we will consider only the case where the starting point is in P. In exchange for this simplification, we will have to consider all values of K, positive or negative. In a total bifurcation, by convention, the origin is in P; see Sect. 6.2.1.2. It follows from these conventions that, in all cases: value K with the -K with the
si
=
(-l)'
(12-35)
.
12.2 Exclusion of Successive Identical T-Arcs In this section
we
make
a
short excursion away from bifurcation orbits. It developed in the previous chap-
turns out that the formalism which has been
one can be applied not only to a bifurcation orbit, but also to ordinary generating orbit. We will use it to prove the fundamental Proposition 4.3.2, which has been already used many times in previous chapters: An ordinary generating orbit of the second species cannot contain two identical T-arcs of type 1 in succession. We recall that an ordinary generating orbit is defined as a generating orbit which is not a bifurcation orbit, i.e. which belongs to only one family. Assume that an ordinary generating orbit contains a sequence of two or more identical T-arcs of type 1. Each T-arc has collisions only in P or only in Q; otherwise the orbit would be a bifurcation orbit of type 1 (Sect. 6.2.1.2). If the T-arcs have intermediate collisions, they can be decomposed into smaller T-arcs. We assume that this decomposition has been carried out.
ter and in this an
restriction, because we still have a sequence of two or more identical T-arcs, to which the proposition applies.) Each T-arc is then a basic
(This
is not
arc, with
no
a
intermediate collisions.
All T-arcs of the sequence begin and end in the same point (P or Q). The deflection angles'between them vanish (all arcs have the same supporting ellipse). We distinguish two cases, by analogy with total and partial bifurcations
1) 2)
(Sect. 6.2): These T-arcs make up the whole orbit. We call this a total T-sequence. The T-arcs make up only part of the generating orbit. We call this a
partial T-sequence. The deflection angles
at the two ends of the sequence do
not -vanish.
We call A the number of T-arcsin the lisions and the T-arcs
as
Sect. 11.3.2. We obtain the tions:
(11.65), (11.70)
or
T-sequence. We number the col-
in Sect. 11.3.1. We define intermediate same
matching
(11.79), (11.71).
Sect. 12.1.1. The computations of the
arc
relations
as
in the
We define ti, yi,
case
arcs as
tY, y ', t , y ,
relation, however,
in
of bifurcaas
in
must be redone
24
Quantitative Study of Type
12.
since
basic
a
arc
is
from P to P
now
where the intermediate orbit i has
y'i
1
from
or
Q
Q.
to
particular case point, i.e. the lead
In the
collision in its initial
a
is zero, the lead at the end is
yi'
-27rJAAi
=
general
In the YZ
yi,
-
case
(12.36)
.
where the initial lead
-27rJAAi
=
yi-1
-
The collision relation tions
(12.30)
we
have then
(12-37)
-37rJ.\,FaAai
=
is non-zero,
.
Using (12.2), (12.4), and (12.16), yi
yi'
+
we
0(p)
(12.29)
is
obtain the +
O(Aa?)
unchanged.
arc
relation
,
So
i E A
(12-38)
the
boundary condi-
are
for
a partial T-sequence. multiply (12.38) by Aai:
We
yiAai
-
yi-i
Aai
=
-37rJVa-(Aai)2
O(pAai)
+
O(Aa )
+
E
A
(12-39) We consider first the
(12.39)
for i
=
case
1 to A and the
of a total T-sequence. Adding the equations 1 to A, we obtain equations (12.29) for i =
ft
0
-37rJV4a_J:(Aai )2
=
41a2h+O(pAa)+O(Aa')+O(/_Ly)
_
V
(12.40)
i=1
where
Aa For p
have written
we =
-+
max(lAail)
0, the last three
y
=
max(lyil)
terms become
(12.41)
.
negligible
in
comparison
to the first
of order Aa 2 and p, respectively.'On the other hand, the first term is negative or zero, and the second term is negative. We have reached
two, which an
of
are
impossibility, and therefore total T-sequence.
an
ordinary generating
orbit cannot be made
a
Next
(12.39) the
we
for i
consider the =
case
1 to h and the
of a partial T-sequence. We add the equations 1. Using also 1 to h equations (12.29) for i =
boundary conditions (12.30), ft
0
=
-37rJv a E(Aai) 2
we
4pa -
V
-
obtain
2
(h
-
1)
+ 0 (pAa) + 0 (Aa
3)
+
O(tly) (12.42)
orbit cannot
ordinary generating again impossible. Thus, partial T-sequence of two or more T-arcs. 1 is possible, because the second term vanishes. This correThe case A to a single T-arc, not adjacent to identical T-arcs. sponds The proposition 4.3.2 concerns only identical T-arcs. It is perfectly possible for a T-arc to be followed by another T-arc which differs in the'values of I and J, or in the superscript i or e (Sect. 4.3.4). In that case the deflection angle does not vanish. For h > contain
1, this is
a
=
an
12.3 The Case'v
12.3 The Case
v
=
25
0
0
=
study of the vicinity of bifurcations. We consider first 0. This case is simple and it 0, corresponding to v is not necessary to use the machinery described in Sect. 11.4. The solutions are known: they correspond to a displacement from the bifurcation orbit along one of the branches which emanate from it. For a total bifurcation, this can be either a first species elliptical orbit (particular case) or a second species orbit, formed of S- and T-arcs (general case) (Sects 6.2.1.2). For a partial bifurcation, we have a bifurcating arc, again formed of S- and T-arcs We return
now
the
=
case u
(Sect.
to the
0, AC
=
6.2.2. 1).
We determine first the orders of and yi. The fundamental yj
yi-1
-
i E
=
-G2SiAC- G3(1
-
Aaj)
=
0
i E C
,
0
for
The additional relation
(12.33)
yo
yj+j
0
-
+
Ksi)Aai
Y"
,
yi-1
Z
12.3.1 First
=
-GO
=
+O(Aa?)
+
+
a
yo
-
yo
=
yj
=
yo(Aal
=
(12.43)
Ksi)Aai
-
GO
-
Ksz-)Aai+l (12.44)
i E C
Species Orbit Using (12.35),
arcs.
equations for the
4
we
4 unknowns yo, yl,
find that the
Aal, Aa2:
O(AC2) + O(Aal2), -G2AC-G3(1 + K)Aa2 + O(AC2) + O(Aa22)
G2AC-GO
-
K)Aal
+
'
-
Aa2)
-G3(1
yo and yj
,
becomes
2
=
0
yj
)
and the additional relation 0
O(Aa?)
+
,
This orbit consists of the two basic
-
O(AC2)
partial bifurcation.
O(Aa i+ 1),
equations (12.43) give yj
+
quantities Aaj
to
A,
yi(Aai+l =
in AC of the
magnitude
equations (12.32) reduce
K)Aa2
+
are non-zero
-
for
(Aa2
-
Aal)
(12.44)
becomes
G3(1
K)Aa'l
a
-
first
+
=
0
(12.45)
,
O(Aa 2).+ O(Aa 22) 1
(12.46)
species orbit outside of the bifurcation;
therefore, from (12.45c,d) Aal Rom
Aa2
(12.46)
Aal We
=
=
recover
we
Aa2
(12.47)
-
have then
=
(12.48)
0
the invariance of the
period along
(This is an exact invariance, not simply (12.45a,b) reduce to a single equation for
an
a first species elliptical family. approximation for small AC.)
two unknowns:
'
yi
-
of
Quantitative Study
12.
26
yo
G2AC
--.,:
Type
1
O(AC2).
+
(12.49)
degenerate. This degeneracy can be removed by using the fact species elliptical orbits of interest here are symmetric. Therefore
The system is that the first
-yo, and
yj
G2 Y'
=
ou
2
12.3.2 Second
The orbit
or
O(AC2).
AC +
-
(12.50)
Species Orbit
bifurcating
arc, in
a
family emanating
Yj =0
(12.51)
As a consequence, we obtain for each which can be solved separately.
equations (12.43)
i + 2. The yj+j Yi+2
yj
-
G28iAC
=
Yi+1
-
=
yj+j (Aaj+2
-
Aai+,)
Yi
-
2
+O(Aai+,) Yi=0'
T-arc, running from
-
Ksi)Aai+l
i to
+
,
(12-52)
0
becomes
Ksi)Aai+l
-
GO
+
Ksi)Aai+2
2
(12-53)
O(Aai+2)
+
from
Aai+l From
Yi+2
non-zero
therefore,
=
(12.53)
Aai+l
an
-,
a
(12.51) gives
yj+j is
We
-
(12.44)
-GO
=
of
equations,
2 O(AC2) + O(Aai+,) 2 GO + Ksi)Aai+2 + O(AC2) + O(Aai+2)
GO
=
case
set of
become
-G28iAC
the additional relation Yi+2
-
independent
arc an
12.3.2.1 T-Arc. We consider first the
.
bifurcation,
a
which is
and
from the
a sequence of S- and/or T arcs.. We have, for every collision i node, and also for the two ends of a bifurcating arc
consists'in
=
recover
exact
yj+j
=
0
(12-54)
-
for the T-arc outside of the bifurcation
is
an
antinode);
(12.52c)
Aai+2 and
(12.55)
-
(12.54),
Aai+2
=
0
we
have then
(12.56)
-
the invariance of the
invariance.) Finally, =
(it
G2SiAC
+
period along
from
O(AC2)
(12.52a,b),
a
T
we
family. (Here again
this is
have
(12.57)
12.4 The Case 0
0, a solution close to the asympsolution, the distance being of the order of p 1/2 Step 8. does not apply here. Step 9. After an asymptotic solution (Yi, Xi, W) is found, we will be able come back to the physical variables with (12.112) and (12.111), obtaining
totic
to
Yi
SdGiG3 /11/2yi + 0(tt)
Aai
=
-'si
FG
A 1/2Xi +
OW
3
AC
G2
The dots in
N/G1 G3 A 1/2W. Fig.
12.3 represent the orders of
12.6 The Case
v
The'detailed
of the
study
(12.115)
>
yi,
Aai.
1/2 case
and 14 shows that all branches
outlined in Sect. 11.4 is
magnitude of AC,
v
=
are
1/2
which will be made in
joined
two
by
Chaps.
13
two. Thus the program
completed.
might ask whether there exist any solutions of the fundamental equacorresponding to values v > 1/2. In the variables Yi, Xi, W defined in Sect. 12.5, this range corresponds to the limit W -+ 0. Indeed an examination of the characteristics representing the families of solutions of (12.114) in a plane (W, Y1) or (W, YO) reveals that they intersect sometimes the axis W 0 (see for instance Figs. 13.1 to 13.7, 14.1 to 14.4). Any such point with One
tions
=
12.6 The Case
W
=
formally
0 must
be excluded in the
study of the
case v
>
v
=
1/2
1/2,
it violates 11.80). In order to study those points and their vicinity, once more th e method described in Sect. 11.4.
Step
=
t1i /2y* i
Aaj
=
Step Yj
+
p 112X* i
=
(1
-
X,)
Step equations
totic
+
+
Xi
(12.117)
.
+ 1
Yj_j
Yi(Xi+l Y0=0'
-
+
,
members do tend to
Xi)
obtained +
by equating
Ksi)Xi
+ 1
,
=
right-hand
(1
=
=
.
are
in the
i E A O(IL1/2) + 0(/_1-1/2AC) i C E 0(ttl/2) y for a partial bifurcation. (12.118) 0(111/2)
Ksi)Xi
0(,Ll/2) 5. The
Yj
FG3
Substituting in (12.32), collecting the dominant terms member, and dividing by appropriate factors, we obtain Yj_j
same
(12.116)
4.
y,(X,+, + y0
Aaj the
yj and
Yj*=sjsign(cos0)V1G_jG3Yj, Xj*=-sjsign(cos0) left-hand
apply
previous considerations suggest that the yj and Aaj should same order of magnitude as in the case v 1/2. On the other
hand, we assume AC < t11/2. Steps 2 and 3. This suggests that we use for the changes of variables as in (12.111) and (12.112): =
because
we
1. The
remain of the
Yi
37
=
0
=
0
1
for
Y"=0
i E A
,
i E C a
zero
them to
for IL
-+
0. The asymp-
zero:
,
,
(12.119)
partial bifurcation.
Step 6. Even though the system (12.119) is slightly simpler than (12.114), apparently cannot be solved explicitly in general. For a partial bifurcation, (12.119) is a system of 2n + 1 equations for 2n + 1 variables; for a total bifurcation, it is a system of 2n equations for 2n variables. Thus we expect it still
isolated solutions. .
Step
7. The results found in the next two
chapters suggest that
the
Jacobian does not vanish in
general. Step 8. does not apply here. Step 9. After an asymptotic solution (Yi, Xj) is found, we the physical variables with (12.117) and (12.116), obtaining Yi
8iNj1GjG3 P1/2yi
+
O(P)
Aaj
=
-si
FG3
A
112X,
come
+
back to
O(P) (12.120)
It would
seem
that
we
need to
study the solutions of (12.119). Actually
this is riot necessary. The solutions of that system are obviously a subset of the solutions of the system (12.114), which will be studied in great detail in
13 and 14. All that will be necessary is to select among the solutions (12.114) those for which W .0.
Chaps. of
=
12.
38
Quantitative Study
of
Type
1
'
necessary to consider these solutions
separately. fundamentally different from the other solutions of (12.114). The need to treat them separately is an artifact, resulting from the assumption (11.80), namely that AC should be strictly of the order of j?. This assumption is appropriate in general: it allows us to define V unambiguously. Here, however, it is in a sense too strong. If we replace it by In practice it is not
They
are
AC
=
even
not
O(Al/2),
(12.121)
then the points with W = 0 can be included in the previous Section 12.5, and the present section becomes unnecessary.
13. Partial Bifurcation of
Type
1
begin now the study of the system of equations (12.114), partial bifurcation. The equations are then We
Yj
Yj_j
+
Yj(Xj+j YO
-
+
Xj)
-
(1
+ 1
Yn
0,
=
W
+
=
Ksi)Xi
0
i
,
=
=
i
0
1,
.'n
=
-
1
in the
case
a
1'...'n 1
(13-1)
0-
=
of
equations form a system of 2n + 1 equations for the 2n + 2 variables W, YO to Yn, X, to Xn. Therefore we expect one-parameter families of solutions. For a given value of p, there is a one-to-one correspondence, given by (12.111) and (12.112), between the present variables W, Yj, Xiand the physical variables AC, yi, Aaj. Thus, the one-parameter families of solutions of (13.1) correspond simply to the ordinary one-parameter families of orbits These
(see
Sect.
2.3).
We recall that
(Sect. 12.1.4);
(-1)i
si
we
as a
consider
only
the
starting point
is in P
(13.2)
.
one
made in Sect. 8.4.1.
Properties
1. The
quantities Yj for
i
=
Therefore each Yj has
principle of
a
1,...,n-1 can never vanish, because of (13.1b). "constant sign along a family. We recover the
the invariance of the side of passage
(Broucke's principle),
Chap. 8. For any solution, by applying the fundamental symmetry stricted problem (Sect. 2.7), we obtain another solution:
which 2.
where the
consequence,
This convention is identical with the
13.1
case
was
(Yi, Xi, SO
F, 3. The
used in
equations(13.1)
I-+
(yn-i, Xn+l-i, sn+l-i)
E of the
re-
(13.3)
show that for any solution, there exists a symmietby changing the signs of all variables Yj, Xj, W.
rical solution obtained
We call this symmetry V:
EI
:
(Yi, Xi, W)
-+
(-Yi, -Xi, -W)
M. Hénon: LNPm 65, pp. 39 - 78, 2001 © Springer-Verlag Berlin Heidelberg 2001
.
(13.4)
13. Partial Bifurcation of
40
Therefore, for
1
family, there exists study one of them. We
any
be sufficient to
Y,
Type
symmetrical family, and it by postulating that
a
(13-5)
> 0
This convention is identical with the
made in Sect. 8.4.1. E,
one
in fact to, the second internal
responds
will
do this
symmetry which
cor-
used in
was
Sect. 8.4. 1. 4. We
can
(1
+
we
obtain
Ksi)Yi+l
us
define
G
=
Rom
relation
a
(1
+
1'...'n 5. Let
by extracting them from (13.1a) and substituting involving three successive values of Yi:
eliminate the Xi
(13.1b);
in
-
Y2k+1
Ksi)Yi-l
-
I
77k
(1
-
-
2Yi
2W
(1
-
+
1
-
K2
-Y
2W
0
(13.6)
=
(13.7)
Y2k+2 have then
we
K) k
I
277k
-
-
1 + K 77k+1
1 +
.
(13-6), using (12.35), 2W
+
K)?7k
77k -
K i
1 + K
26k+l
(13.8)
K
Thus, the problem is formally equivalent to the study of a plane mapping. As is easily shown, that mapping is area-preserving. Numerical explorations (H6non, unpublished) exhibit the mixture of regular and chaotic orbits characteristic of
6.
non-integrable systems.
This supports the
conjecture that the system of equations (13.1) is not explicitly solvable in general. By comparing (12.111), (12.112), (12.31), and the definition (8.22) of s, we
find s
13.1.1
=
sign(W)
Asymptotic
In the limit W
-+
(13.9)
.
Branches,for W
oo,
we
enter the
-+
oo
region JACI
> it 1/2 ,
or v
equations of Sect. 12.4 are applicable. The bifurcation arc again be decomposed into a sequence of T and S arcs (or, as a
the
first species orbit). By comparing the equations (12.112), we obtain the correspondence relations
a
Yi Rom totic
=
W Yi
I
Xi
=
or
1/2,
and
orbit
special
can
case,
(12.63), (12.64), (12.111), (13-10)
WXi
(12.75), (12.91), (12.92), (12.93), equations:
0, the stability index jumps then from large positive values to large negative values (or vice versa). There exists a-short interval of stable orbits.
Such
jump happens when, for one of the arcs, the following condition "perturbations in the direction of the departure velocity have no effect on the impact parameter at the next encounter" (Hitzl and H6non 1977b, p, 1029). These particular arcs, or critical arcs, have been computed in Hitzl and H6non 1977b; they correspond to extremums in C along arc a
is realized:
families
(see
Sect.
4.6).
The value of C is determined at which do not
a
partial bifurcation.
In
general, the
arcs for this belong bifurcating particular value of C (see Bruno 1973; 1994, Chap. IV). For the bifurcating arc itself, however, the above condition can be realized. We consider the system of equations (13.31) for a given value of W, with the last equation deleted, We have then a system of 2n 1 equations for 2n variables. Starting from a given value X, and using the equations one by one, we obtain successively Y,,. in the same way, starting from a variation dX1, we can Y1, X2, compute successively dY1, dX2, dYn. The direction of the initial velocity is a function of X1, and the final impact parameter is proportional to Yn. Therefore the above condition is simply written arcs
to the
arc are
-
...,
...,
not critical
13.1
d Y,, --':
dXj A
cating
(13.34)
0.
bifurcating
45'
Properties
which verifies this condition will be called
arc
a
critical
bifur-
arc.
We have the variational equations
of,
i x-i
dXj
+... +
19hn-i dXj ,gxl We
can
'9h 09yn
+... +
=
0,
'Mn-i dYn Igyn
=
1(13.35)
0.
solve this system to obtain
f2n-1) 10(h) Xn) o9(XI, Y1, X2,. f2n-1) 19(h Xn) Yn) O(Y1 X2) ...
dYn _
(13.36)
.'
7...
...
i
seen
diagonal
1
.
. -
dXj
As is
dYn
(13.31),
from
)
the denominator reduces to 1
0 and all elements in the
are
dYn
diagonal
are
(all
1)
elements above the
and
we
have
(13.37)
IjI
dXj,
We have thus shown
Proposition 13. 1. 1. In a partial bifurcation oftype 1, if and only if the bifurcating arc is critical.
(13.31)
We have from
dXi+l dXj from which
dYj dXj
dY2,
dYj
dXj
dYj
y2 i dXj
dXj
dXj
1 -
we can
-
K2
dXj dY3 dXj
=
(1
+
Ksj)
dXj dXj
dYj
-
dXj
1 ,
(13-38)
compute the successive derivatives
K
1 1
the Jacobian vanishes
(1
-
2,
K2)(1 y2 Y 2
-
K)
2(1
-
Y2
K)
2(1
-
Y2
K)
+3-K,
(13.39) given values of X, and W, one can compute successively Y1, X2, Yn. Therefore Yn is an analytical function of X, and W, which Y2, can be represented as a surface in a 3-dimensional space (WX1,Yn). The characteristics of the bifurcating arcs of order n, in the (WXl) plane, are 0. given by the intersection of that surface with the horizontal plane Yn For
...'
=
-
46
13. Partial Bifurcation of
We consider a
critical
now
Type
plane tangent
the
bifurcating
arc.
1
to the surface at
We have then
aY,,1,9X1
a
=
point A representing 0, and two cases are
possible: 1. The
tangent plane is
plane
not horizontal. It intersects then the horizontal
straight line parallel to characteristic; therefore: the
in
to the
a
the X, axis. This line characteristic has
an
is
the
tangent
extremum in W
at the
point A. tangent plane is horizontal. The surface Yn(W, X1) has then a maximum, a minimum, or a saddle at A. Since this point is on a characteristic,
2. The
only the
case
of
the horizontal
a
saddle is
The intersection of the surface with
possible.
consists then in
plane
general of two
curves
which intersect
at A: the characteristic intersects another characteristic at the
Thus
critical
corresponds to either an extremum in W family. Conversely, in an extremum in W or 0 and therefore a family, there is dYn/dX1
bifurcating
arc
intersection with another
or an
an
a
point A.
intersection with another
critical
bifurcating
=
arc.
All this has been shown to hold
more
generally for
the restricted
problem
(Sect. 2.8). Examples of both cases will be found below. The first case (extremum) correspond to a true singularity; we can obtain a non-zero Jacobian by changing the independent variable used as parameter of the characteristic, for instance by taking X, instead of W. The second case (intersection) corresponds to a true singularity: the implicit function theorem cannot be used, and the general method described in Sect. 11.4 fails. The reason is clear: for p 54 0, the four branches arriving at the intersection can be joined in different ways. This is the same situation as a bifurcation involving two families of generating orbits (Fig. 1.1). It will frequently be possible, however, to save the situation and establish the junctions by having recourse to symmetry considerations and to Restricdoes not
tion 7.3.1
(see
13.3.3.2).
Sect.
The intersections which
,
are
mentioned here should not be confused with
(see Chap. 1) corresponds to the intersection of generating orbits, i.e. to an intersection which exists
bifurcations. A bifurcation two
or more
families of
after the limit in the
IL
---
0 has been taken. On the contrary, here the characteristics families of periodic orbits, for tZ small but
(WYl) plane represent
not zero, i.e.
before
the limit has been taken. We
intersections between two such families. upper
cases
of
Fig.
Examples
concerned with the
are
can
be
seen
in the two
13.2.
If we go back from the W, Y, variables to the physical variables AC, y, and let tt -+ 0, the whole figure in the (AC, yi) plane shrinks to a point, which corresponds to the bifurcation orbit. We see thus that the present intersections take
orbits.
place
in
a
finer level of
description
than the bifurcation
13.2 Small Values of
13.1.5
Asymptotic
In that
case
dYO
=
07-
a
we
can use
N, i,,
=
dYn
UlU2
=
...
JWJ
--+ oo
computations of Sect. 13.1.2.
the
From
(13.29),
i
Thus, dYn/dX,
=
with
obtain
n we
UN
(13.40)
y2
2 y2 ily
dXj
Behaviour for
47
n
E)(W2N-2):
13.2 Small Values of
the Jacobian
never
vanishes in this limit.
n
simpler to use only the Yi variables, and the equations (13.6) (with boundary conditions (13.1c), (13.1d)). If desired, the Xi can then be deduced using (13. 1a). The characteristics will be represented in the (W, Yj) plane. It will be
the
13.2.1
n
We have
2Y,
a
single equation 1
+
Yj
=
-.K -
Y,
Depending resented
2
=
on
on
=
the value of
Fig.
0 and Yj.
2W
13-1. It is
=
W
(13.41)
0.
K, the characteristic has one of the shapes rephyperbola; its asymptotes are the straight lines
a
(dashed line).
In accordance with
half-plane
only the upper half-plane Yj > 0; the lower respect to the origin. +2
Y,
+2
Y1
(13.5), is
we
represent
symmetrical with
+2
Y1
+
-1
K< -1
Fig.
< K
13. Partial Bifurcation of
58
Type
1
13.3.2.5 Packets. The results of Sect. 1.3.3.2.2 show
dYi. From
For
(13.54)
dYj a
0
=
a
O(W-1)
=
=
that, for
and
>
0,
(13.76)
(13.56)
we
have then
()(WI-2a).
(13.77)
(Sect. 13.3.2.3), the difference between O(W). Therefore (13.77) covers this
branches is
the values of Yj for the two case
also, with the proviso
a small quantity anymore. equation shows that for JWJ large, the branches are organized hierarchically. The characteristics of two branches which differ in their first arc lie at a distance E) (W) from each other in the (W, Yj) plane; they diverge for JWJ -4 +oo. The characteristics of all branches having a given first arc in common are at a distance O(W-1) of each other; they tend towards each other for W -4 +oo. We will say that they form a first-order packet. Inside such a packet, the characteristics of all branches having in common their first two arcs are at a distance O(W-3) of each other; they tend towards each other even more rapidly, and form a second-order packet; and so on. This phenomenon is clearly seen for.instance on Figs. 13.3 to 13.7. Incidentally, the fact that the characteristics of two branches converge quickly towards one another for JWJ -+ oo as soon as a reaches a few units
that dYj is not This
is
one reason
13.3.2.6 An
ceding
for the numerical difficulties alluded to above.
Example.
sections allow
all branches
are
us
The rules which have been established in the precompletely how the characteristics of
to determine
ordered in Y1. As
an
example, consider the
case
1 < K
0 for 1< K
only two branches (see Fig. 13. 1).
0, there
therefore necessarily joined
are
+
2
13.3 Positional Method
-
-
-
-
+2 +21 - 3
-
+
12
1
K < -1
111 11 12
-
+1
Fig.,
-
-
-
-1
3 +2 - 21 -
-
-
-
-
-
-
-
< K-
1.
Polyhedra
15.3 Newton
first step consists thus in
The of
(15.4)
15.3.1 ]Encounter
n
n
....
equations are the equations f2i All these equations are identical except for
1.
therefore
consider
we
fi(X)
d)'
its faces IF ik
=
0
the
Equations
1 encounter
-
-
for each equation
polyhedron 1Fj, equations fj'A, (X), and their normal cones &d) ik
truncated
The
determining,
in turn, the associated Newton
=
one
of them for
a
a
0 in
(15.4),
shift of the
with i
1, subscripts; =
fixed i.
The support S2i is infinite, because the terms 0() hide a series expansion in the variables. However, S2i can be replaced by its minimal dominant subset
S12j, which is finite it contains the 3
Q21 1,
(Bruno 1.8):
points
(1, 0'...' 0)
=
=
El
,
2n
Q21 2
(01
=
...
301151)01
(0)
z
...
A 17 110)
2i+1
Here and in what
(For the
the
Q points,
manner
1
0)
=
E2i+1
+
E2i+2
=
E2i+2
+
E2i+3
i
2n-2i-1
2i
Q21 3
...
...
)
0)
-
(15.25)
2n-2i-2
follows, Ej' represents the we
of Bruno
unit vector in the direction qj. equation as an exponent, in
write the index 2i of the
1.9.)
going from the full support S2i to the minimal dominant equivalent to omitting the 0() terms in the equation, which
We note that subset
S'j 2
is
reduces to I
fj
=
X2i+2(X2i+3
-
X2i+l)
+
Gjxj
=
0
.
(15.26)
15.3 Newton
99
Polyhedra
polyhedron and its faces for a given set of complex operation (see Bruno 1.4, 1.5, 1.7). In our case, points contains only 3 points, which are not aligned. Thus the Newton however, S'j 2 2. It is therefore a is simply a triangle. Its dimension is d polyhedron is faces the such In a finding very simple: every case, simplex (Bruno 1.1). face. to one subset of S'j corresponds 2 the Newton
general, finding
In
S
be
can
a
-
=
(d) designate a face by r 2i,k where d is the dimension of the face, 2i identifies the equation, and k represents a numbering of the faces for each dimension (Bruno 2.8). We define a face by its associated boundary subset,
We
as
,
in Bruno 1.3. We obtain
]p(2)
2ij
D
one
face with dimension 2
(15.27)
Q2i, Q21 w2i, 2 3 1
and three faces with dimension 1 (1)
r2ij
D
j(2) 2ij
-
2
l) A2i,2
-
-
X21'+2
We remark that
Finally, a
f(2) NJ
d b
Q2i} f Q2i, 3 1
+
X2i+l ) +
Q2' QN, 31 2
D
G, x,
P9 i1)j
,
P9, i, 3
Gjxj J21*
-
-X2i+lX2i+2 +
X2z+2 (X2i+3
participate
in that
case.
cones are
P1
=
P2i+1 + P2i+2
U2(i,)l U2(1i,)2 U2(i, 3
JP*
P1
=
P2i+1 + P2i+2
P2i+1 > P2i+3
1p:
P1
=
P2i+2 + P2i+3
P2i+3 > P2i+1
fP:
P2i+l
some
to d
a
are
Gjxj
(15.29)
X2i+1
-
JP:
older work
(1 5.28)
are
F'.- all terms
-
rM 2i,3
U(2) 2i,
The letters order.
-
the normal
I
C
D
truncations
(X2i+3
X2z+2X2i+3
E'N,2
)
corresponding
The
from
(1)
2*
2'
M% Q221
=
P2i+3
,
P2i+2 + P2i+3
(15-30)
P2i+l + P2i+2 > P1
labels which will be needed below. They
(unpublished);
this is
why they
are
not in
are
inherited
alphabetical
,
15.3.2 Are Equations: General Case The
general
corresponds to an arc which is not one of the two end arcs 1. Their n 0 in (15.4), with i 2, f2j-1 only if n 2! 3. All these equations are identical shift of the subscripts; therefore we consider one of them for a
case
of the equations 2: they exist number is n and to
one
=
=
-
...,
-
except for
a
fixed i. The term
respect
=
the support S2i-1 'can be replaced by its minimal dominant which is finite: it contains the 4 points
again,
S'j 2 -1,
can be neglected, because it is negligible with yi-I and X2i+2 = yj as shown by (11.29).
0(p)
to the terms X2i
Here subset
O(xi)
15. The Newton
100
0-1 0-1 2
Approach
(01 11 0'..., 0) (0,
=
-
-
0, 1, 0'...' 0)
-,
2i-1
Q2i-1 3
E2 =
EV
=
E2i+1
=
E2i+2
2n-2i+l
(0'...' 0, 1, 0'... 0)
=
7
,
,
2n-2i
2i
-
Q2i-1 4
(0'...' 0, 1, 0'...' 0)
=
2i+1
This is
(15.31)
2n-2i-1
equivalent
to
omitting the 0()
terms in the
equation, which reduces
to
f2i-1 The 4
=
X2i+2
points
X2i +
-
not
are
Its dimension is d
=
2i-I'l
D
+
G3(1
a
Ksi)X2i+l
+
coplanar. again a simplex:
The Newton
3. It is
to a face. We obtain
r(3)
G2SiX2
=
(15.32)
0
polyhedron
every subset of
is
a
tetrahedron.
corresponds S'j-1 2
face with dimension 3:
0-1, 0-1 0-1, w2i-l, 1 2 3 4
(15.33)
4 faces with dimension 2:
r(2)
D
Q2i-1, IQ2i-1, Q2i-l} 1 2 3
][,(2) 2i-1,2
D
Q2i-1, IQ2i-1, Q2i-l} 1 2 4
r(2) 2i-1,3
D
Q2i-1, IQ2i-1, Q2i-l} 4 1 3
r(2)
D
0-1, w2i-l, Q2i-l} 2 3 4
2i-l',
2i-1,4
(15.34) and 6 faces with dimension 1:
r(l)
D
IQ2i-1, Q2i-l} 1 2
r(l)
D
Q2i-l} f Q2i-1, 4 1
2i-l',
2i-1,3
r(l)
D
2i-1,5
Q2i-l} f Q2i-1' 2 4
The truncations
are
C2( i)-1,2 I 2( i)-1,4 r2(i-)1,6 1
D
Q2i-1, Q2i-l} 1 3
D
QN-1, Q2i-l} 2 3
D
jQ2i-1' Q2i-l} 3 4
easily derived. The normal
cones are
a
U(3) 2i-
JP:
P2
=
P2i
=
P2i+l
d
U(2) 2i-
fP:
P2
=
P2i
=
P2i+1
P2 > P2i+2 I
c
U(2) 2i-1,2
=
1P:
P2
=
P2i
=
P2i+2
P2 > P2i+l
e
U(2) 2i-1,3
=
1P:
P2
=
P2i+l
b
U(2) 2i-1,4
=
JP:
P2i
h
U(2z)1
=
JP:
P2
P2i
=
fP:
P2
=
1p:
P2
=
fP:
P2i
,
1
k
U(21)1
j
UM
g
U(2z-1,4 )
2
2i-1,3
P2i+1
=
P2i+2
=
P2i+2
(15.35)
P2i+2
P2 >
)
)
P21j
P2i > P2
P2 > P2i+l
P2 > P2i+2 I
P2i+1
P2 > P2i
P2 > P2i +'2}
P2i+2
P2 >
P2 > P2i+1
,
P2i+1
P2i > P2
P2i >
I
P2i+2}
Polyhedra
15.3 Newton
UM 2i- 1,5
f
UM 2i-1,6
---:
---
---
:
JP:
P2i
IP:
P2i+1
15.3.3 Arc
Equations:
We consider
now
corresponding In that term X4
P2i > P2
P2i+2 -'
P2i+2
P2i+1 > P2
7
particular
to the first basic
P2i+1 >
i
P2i} (15-36)
(15.4),
negligible compared
to the
=
arc.
O(xi)
term
0 in
f,
O(M)
=
is
-
=
A 1, 0'..., 0)
=
(0, 0, 11 01
=
(01 01 01 11 01
This is
P2i+l}
of the first equation
case
The point Q2i-1 of the general case has 2 inant subset S, contains only 3 points:
Q11 Q31 Q144
P2i >
)
Initial Arc
equation, the Y1
=
the
--
101
....
=
0)
....
E2
The minimal dom-
I
E3
=
0)
disappeared.
=
(15-37)
E4.
equivalent to omitting the 0()
terms in the
equation, which reduces
to X4
-
G2X2
G3(1
+
-
K)X3
(15.38)
0
The Newton polyhedron is a triangle. We obtain the faces simply by eliminating from the lists (15.33) to (15-35) of the general case the faces which contain
QN-1. 2
]p(2) 13
D
IQ', 3 Q1} 1 Q1, 4
r(112)
D
IQ',,Q3'}
The
X4
113
X4
-
-
The normal
d C
b
G2X2
(15-39)
,
r(113)
,
corresponding
j(2) A(31) a
There remains:
D
1
IQ,,Q41}
truncations
+
G3(1
-
P161)
G2X2
16
,
12
X4 +
G3 (1
fR
P2
=
P3
'1)
U 12
JR
P2
=
P3
P2 >
P4}
,
1( 3) U11116)
='Ip:
P2
=
P4
P2 >
P3}
,
IP*
P3
=
P4
P3 >
P2}
.
=-
I
(15.40)
7
-
-G2X2
K)X3
+
G3(1
-
K)X3
(15.41)
cones are
U(2) 13 U
I
IQ3 Q41
are
K)X3 --
D
P4
-
(15.42)
Approach
15. The Newton
102-
Equations: Final Arc
15.3.4 Are
We consider
now
corresponding
the
particular
of the last equation f2,,-, = 0 in (15.4), case is symmetrical of the previous
case
to the last basic
arc.
This
one.
equation, the
In that term X2n
The
=
Q2i-1 4
point
S'2n-1
inant subset
Q2n-1 1 Q2n-1 2 Q2n-1 3 This is
term
O(xi)
O(IL)
=
is
negligible compared
to the
Yn-1-
=
general case has disappeared. only 3 points:
(0, 1, 0'..., 0)
=
E2,
7
0, 1, 0)
==
E2n
,
(0, =
of the
-
-
The minimal dom-
contains
-
,
(0'... 0, 1) ,
equivalent
to
(15.43)
E2n+l
=
omitting the 0 ()
terms in the
equation, which reduces
to
f2ln-1
-X2n +
=
The Newton from the lists
polyhedron
(15-33)
r(2) 2n-1,1
G2 SnX2
to
+
is
a
(15-35)
Q2n-li Mn-l, 1 2
]P(1) 2n-1,2
D
Mn-lMn-lj 1 3
]['(1) 2n-1,4
D
w2n-l, Q2n-l} 3 2
The
corresponding
.p(2)
KSn)X2n+l
triangle.
(15.44)
0
We obtain the faces
by eliminating
Q2i-1. 4
There remains:
the faces which contain
G28nX2
-X2n +
G28nX2
j'(1) 1,2
G2SnX2
+
hn-1,4
-X2n +
G3(1
n
n-
The normal
(15.45)
truncations
X2n +
J2n-l,l
P2
+
Q2n-1 Q2n-1, w2n-l, 3 1 2 D
2n-1,1
G3 (1
G3(1 +
are
+
G3(1
+
KSn)X2n+l
+
KSn)X2n+l
(15.46)
KSn)X2n+l
cones are
a
U(2) 2n-
c
UM 2n- 1,1
d
b
JP:
P2
=
P2n
P2n+l}
=
JP:
P2
=
P2n
P2 >
UM 2n- 1,2
=
IP:
P2
=
P2n+1
UM 2n- 1,4
=
IP:
P2n
=
P2 >
,
P2n+1
P2n+l}
,
P2n}
P2n >
P2}
(15.47)
15.3 Newton
Polyhedra
103
15.3.5 Additional Relations: General Case
We must also consider the additional relations
general case, i.e. the central 2. n 2 4 with i = 2 to n
equation f2,,-,+i
(15.7). We begin (15.7)), which
0 in
=
with the exists for
-
O(xi)
The term
=
0(p)
is
negligible
and X2i+4 = Yi+1 The minimal dominant subset
Q2n-l+i 1 Q2n-l+i 3 This is
=
=
Q2n-l+i 2 Q2n-l+i E2i+3 4 E2,
=
,
equivalent
to
with respect to the terms X2i
S',, 2 -1+j E2
contains the 4
points
+1
(15.48)
E2i+4
=
Yi-1
omitting the 0()
terms in the
equation, which reduces
to
f2in-l+i
=
-X2i + X2i+4 +
G3(1
+
Ksi)X2i+l
+
G3(1
-
Ksi)X2i+3
=
0
(15.49) The 4 points
not
are
coplanar.
We obtain 1 face with dimension
The Newton
3,
polyhedron
is
4 faces with dimension
a
tetrahedron.
2, and 6 faces
with dimension 1: r
r
(3) 2n-l+i,l
D
(2)
Q2n-l+i,, Q2n-l+i} M2n-l+i, Q2n-l+i 4 3 2 ,
Q2n-l+i Q2n-l+i} fQ2n-l+i, 3 2 1
2n-l+i,l
,
]p(2) 2n-l+i,2
D
Q2n-l+i Q2n-l+i} IQ2n-l+i, 4 2 1
]p(2) 2n-l+i,3
D
Q2n-l+i IQI2n-l+i, Q2n-l+i 4 3
][,(2)
D
Q2n-l+i Q2n-l+i} iQ2n-l+i, 4 3 2
r(l) 2n-l+i,l
D
IQ2n-l+i, Q2n-!+i } 2
2n-l+i,2
r(l) 2n-l+i,3
D
Q2n-l+i Mn-l+i, } 4 1
2n- 1 +i,4
2n-l+i,4
2n-l+z,5
D
,
,
,
1
D
(1) (1)
Q2n-l+i IQ2n-l+i, l 4 2
2n-
+i,6
2n-l+i
M
,
Q32n-l+t}
Q2n-l+i iQ2n-l+i, 3 2 D
Q2n-l+i IQ2n-l+i, 4 3 (15.50)
The truncations
are
A
U(3) 2n-
D
U(2) 2n- l+i,l
B
easily derived. The normal
cones are
JP:
P2i
=
P2i+1
=
P2i+3
P2i+4}
=
IP:
P2i
=
P2i+1
=
P2i+3
P2i >
P2i+4}
U(2) 2n- l+i,2
=
JP:
P2i
=
P2i+I
=
P2i+4
P2i >
P2i+31
C
U(2) 2n- l+i,3
=
JP:
P2i
=
P2i+3 =,P2i+4
P2i >
P2i+l}
E
U(2) 2n- 1+i,4
=
JP:
P2i+I
=
G
U(I) 2n- 1+-
=
JP:
P2i
=
P2i+1
,
P2i > P2i+3
,
P2i >
P2i+4}
H
UM 2n- l+i,2
=
JP:
P2i
=
P2i+3
,
P2i > P2i+1
,
P2i >
P2i+4}
Z' 1
P2i+3
=
P2i'4 +
,
P2i+1 >
P2i}
15. The Newton
104
Approach
F
UM 2n- l+z,3
JP:
P2i
K
UM 2n- 1+i,4
JP:
P2i+I
P2i+3
P2i+l > P2i
P2i+1 >
P2i+4}
I
U(I) 2n- l+i,5
IP:
P2i+1
P2i+4
P2i+l > P2i
P2i+1 >
P2i+31
j
UM 2n- l+i,6
IP:
P2i+3
P2i+4
P2i+3 > P2i
P2i+3 >
P2i+l} (15.51)
Here
---:
P2i+4
P2i > P2i+l
i
the letters A to K to label the
we use
7
P2i >
P2i+3}
cases.
15.3.6 Additional Relations: First Relation
0 particular case of the first additional relation f2n basic first two the arcs. to (15.7), corresponding The term O(xi) O(p) is negligible with respect to the term x6 Y2. The point Q2n-l+i of the general case has disappeared. The minimal contains only 3 points: dominant subset Sn 2
We consider
now
the
--,:
in
=
=
On 2
This is
Q2n=E 6 4
Q2n=E 5 3
E3
=
equivalent
to
omitting the 0()
(15-52)
terms in the
relation, which reduces
to
An
::--
X6 +
G3(1
K)X3
(15.50)
+
G3(1
+
K)X5
=
0
(15.53)
a triangle. We obtain the faces by eliminating general case the faces which contain Q2n-l+i
polyhedron
The Newton
from the list
-
is
of the
There remains:
][,(2),
D
Q2n' Q2n.} {Q2n, 4 2 3
r(l) 2n,2
D
IQ22n, Q42n}
2nl
The truncations
j(2) '2nj
-
X6 +
2nj
I'M 2n,3
D
D
w2n, 2- Q2n} 3
Q2n} jQ2n' 4 3
('15.54)
are
G3 (1
K)X3
-
+
G3 (1
+
K)X5
P,1) nj -G3(1-K)X3+G3(1+K)X5, -
-
2n,2
i(i) 2n,3
-
-
X6 +
G3(1
-
K)X3
X6 +
G3(1
+
K)X5
The normal
cones are
A
U(2) 2n,
D
UM 2n,
JP:
U2(n),2 U2(1n),3
B
C
(15.55)
P5
P6}
P3
P5
P3 >
JP:
P3
:P6
{P:
P5
P6
P: P3
P3
P6}
>P5}
P5 >
P3}
(15-56)
15.3 Newton
Polyhedra
105
15.3.7 Additional Relations: Last Relation 0 now the particular case of the last additional relation f3,,-2 (15.7), corresponding to the last two basic arcs. This case is symmetrical
We consider in
of the
,:--
previous
one.
O(p)
O(xi)
The term
with respect to the term X2n-2
negligible
is
Yn-2-
The
point
Q2n-l+i 4 S'3n-2
dominant subset
Q3n-2 I This is
E2n-2
=-:
Q3n-2 2
1
equivalent
general
of the
contains
to
case
The minimal
disappeared.
has
only 3 points: E2n-1
=
omitting the 0()
Q3n-2 3
)
terms in the
(15.57)
E2n+1
relation, which reduces
to
f3n-2
-X2n-2+G3(1+K,5n-l)X2n-l+G3(1-KSn-l)X2n+1
=
The Newton
(15.50)
from
r(2) 3n-2,1 r
(1) 3n-2,2
polyhedron
Q3n-2} jQ3n-2' 3 1
-X2n -2
-
G3(1
The normal
r3(n-2,3
.(15.58)
by eliminating
There remains:
FM-2,1
D
Q3n-21 jQ3n-2' 2 1
jQ3n-2' Q3n-2} 2 3
D
+
KSn-l)X2n-1
+G 3(1 +
KSn-l)X2n-1
G3(1
KSn-l)X2n+l
G3(1
X2n-2 +
-
.
the faces
0
(15-59)
are
-X2n-2 +
'1 n -2,1 3
3n-2,3
Q2n-l Fi 4
D
,F(2)
If')
triangle. We obtain
Q3n-2, Q3n-21 fQ3n-2, 3 1 2
J3n-2,1
J
a
D
The truncations
J3n-2,2
is
the faces which contain
::--
-
KSn-I)X2n-1
+
+
G3(1
-
+
G3(1
-
KSn-l)X2n+l
KSn-l)X2n+l
(15.60)
cones are
A
U(2) 3n-2,1
1P:
P2n-2
P2n-1
B
UM 3n-2,1
JP:
P2n-2
P2n-1
P2n-2 > P2n+1
(1)
==
P2n+1
C
U
3n-2,2
JP:
P2n-2
P2n+1
P2n-2 >
P2n-l}
D
UM 3n-2,3
JP:
P2n-1
P2n+1
P2n-1 >
P2n-2}
15.3.8 Additional Relations: Case
n
=
(15.61)
2
particular case, we have the single relation (15.8). and Q2n-l+i of the general case have disappeared. points Q2n-l+i 4 1 The minimal dominant subset S'4 contains only 2 points:
In that
The
Q4=E 3 2 This is to
Q4,3
equivalent
=
to
E5
(15.62)
.
omitting the 0()
terms in the
relation, which reduces
Approach
15. The Newton
106
f4l
G3 (1
=
-
jQ4' 2 Q4} 3
D
There is
a
K)X5
-
U4,1
segment. There is
a
face
single
(15.64)
K)X3
=
(15.63)
0
=
.
+
G3 (I
normal
corresponding
A
is
+
truncation, which
one
G3 (1 The
G3 (1
+
polyhedron
The Newton
]p(l) 4',
K)X3
fP:
P3
=
+
(15.63):
is identical with
K)x5
cone
(15.65)
.
is
(15.66)
P5
Intersections with the Cone of the Problem
15.4
The next step in the
analysis (Bruno 2.8)
for which the intersection of the normal
consists in
retaining only the faces cone of the problem K
with the
cone
is non-empty. This turns out to be the case for all faces found easily shown by exhibiting in each case a vector which belongs
(see (15.16))
above. This is to
the normal
(15.30),
cone
we can
and to K.
for
U(2) 2ij
p,
=
for
U(1) NJ
P1
=
P2i+3
=
-2
for
U(1) 2i,2
P1
=
P2i+1
=
-2
for
U2(1j 3
P1
=
Similar vectors
Up
to
equation, cones
which
equation. P must
can
we
-3
,
of the normal
case
cones
,
-1
P2i+2
=
P2i+3
=
P2i+l
=
P2i+2
=
-
,
P2i+2
=
P243
=
-
,
P2i+2
=
P2i+3
=
-
P2i+l
P2i+I
be found in the other
(15.67)
cases.
Boundary Subsets
analysed the conditions satisfy one equation in (15.4)
have
now we
(15.9)
-2
tan
15.5 Coherent
form
in the
For instance,
take
under which considered
have found that the vector P must
belong
solution of the
a
separately.
to
one
For each
of the normal
have identified, while X satisfies the corresponding truncated 0, the vector instance, in order to satisfy the equation f,
we
For
belong
X satisfies the
=
to
one
of the 4 normal
corresponding
truncated
cones
(d
Uj) 3
listed in
(15.42),
while
PI) 13
given
01 with
equation
by (15.41). Now
we
must find the solutions
(15.9)
which
satisfy simultaneously
all
fundamental equations (15.4). Bruno (1.9) sketches a procedure for doing this. Essentially, we consider all possible combinations formed by: the cone of the
problem (15.16);
one
normal
cone
for
fl;
one
normal
cone
for
f2;
...
;
one
Boundary Subsets
15.5 Coherent
normal
for
cone
For each
combination,
we
107
determine the intersection
of the mB + 1 cones. If that intersection is empty, the combination can be eliminated. If the intersection is non-empty, we have a valid combination. The
intersection
called cone of truncation (Bruno 2.6) and designated by H. corresponding combination of boundary subsets is called by Bruno (1.9) coherent a aggregate of boundary subsets. Unfortunately, the number of combinations, and thus the amount of work needed, grow exponentially with the number of equations MB In the case of the system (15.4), there are 4 possible normal cones for each of the equations 07 0; 11 possible normal cones for each of the equations f2 f2n-2 1 and 0) 0; 4 normal cones for the first equation f, f3 f2n-3 4 normal cones for the last equation f2n-l 0. Therefore thenumber of possible combinations is is
The
-
...
i
...
)
=
4n+1 For
n
=
becomes
lln-2
X
(15.68)
.
this equals 64, 2816, 123904, 2, 3, 4, impracticable. ..
We consider of normal
.,
now
.
.
.
.-
So the method
how to find the intersection H for
a
given combination
We remark that the definition of each normal
cones.
quickly
cone
is
a
set of
inequalities on the coordinates pj of P. This is also the case for the cone of the problem. Therefore, for any given combination, we need to solve a system of linear homogeneous equalities and
linear
homogeneous equalities
and strict
inequalities for the pj. an example, we consider the case 1P2, and we select the normal cone d for the initial arc equation (equ. (15.42b)), the normal cone a for the encounter equation (equ. (15.30a)), and the normal cone d for the final arc equastrict
As
tion
(equ. (15.47c)).
The vector P must
(15.16)
This
will be called "dad"
case
belong
and these 3 normal
below(see
to the intersection of the
cones:
cone
its 5 coordinates must
Sect.
of the
satisfy
the
15.5.4).
problem following
set of conditions
PI < 0
7
P2 < 0
P3 < 0
Y
P4 < 0
Y
-P2 + P3,7-- 0
-P2 + P4 < 0
-P1 + P3 + P4
0
-P2
+P5
There
are
---:
4
0
7
-P2
2
P5 < 0
7
-Pl+N+P5 0) +P4
(15.69)
< 0
equalities and 7
strict
inequalities. We need
to find all solu-
tions of this system.
15.5.1 The
An a
algorithm
Motzkin-Burger Algorithm exists for the solution of
system of non-strict inequalities
a
somewhat similar
problem. Consider
15. The Newton
108
aj1P1 + aj2P2 +
-
-
-
Approach + ainBPIIB ::
The set of its solutions forms
t
+
N(C) is called
a
B(C)
cone
C
(Bruno 1.1).
It
be
can
E pj Bi
(15.71)
j=1
.
where the Ni and the Bi and each tij
value;
polyhedral
9
E AjN' i=1
zero
(15.70)
j":_1)-*)M)
(Bruno 1.4)
written in the form
P
a
0)
fixed vectors; each Ai can take any positive take.any real value. The set
are
can
IN',-, Ntj
=
(15.72)
fundamental system of solutions,
while the set
IB',...,B'}
=
or
(15.73)
maximal linear subspace (Bruno 1.4). Motzkin-Burger algorithm allows the computation of the Ni and the Bj, thus giving the solution of the system. It works by adding the inequalities one by one, starting from the trivial inequality defines
a
The
QP1
+
-
-
+
-
OPn,,,
< 0
for which the solution is t
j
1,
=
.
Our
.
.
,
nB
case
(any
P is
a
differs from
(15.74) 0 (the set N (C) is empty), s solution). (15.70) in that it does not have =
=
nB
,
and Bi
non-strict
Ej,
inequal-
ities, but instead a ,mixture of equalities and strict inequalities. We need to adapt our equations so as to be able to use the algorithm. First, we replace
equality by two non-strict inequalities, in which the '=' sign is replaced and '2 !' respectively. This does not change the set of solutions. Second, we replace each strict inequality by a non-strict inequality, i.e. we replace In so doing, we enlarge the set of solutions, i.e. we obtain all the soluby tions of our problem, plus some parasitic solutions. Later, we will eliminate these parasitic solutions (Sect. 15.5.2). We modify slightly the procedure described by Bruno: we start, not from the inequality (15.74), but from the set of nB inequalities (15.16) which define the cone of the problem. (These strict inequalities are replaced by non-strict inequalities pj :! 0, as explained above.) Then, initially or at later stages of the algorithm, a solution (15.71) never includes B vectors, i.e. the set B(C) is empty. C is a polyhedral forward cone (Bruno 1.1). This is immediately seen from the fact that the cone of the problem does not include any straight line. The set of solutions can.then be described simply as' each
by
t
P
AjN';
Ai
The set of the Ni forms
1.1).
0,
a
i
1,
...,
skeleton of the
(15.75)
t.
polyhedral forward
cone
(Bruno
15.5 Coherent
particular,
In
in the
starting state,
have t
we
Boundaxy
=
nB
Subsets
and Ni
109
=
-Ei,
inequalities, coming from the normal cones, one by some point in the procedure we have a system of inequalities (15.70), and its solution (15.75), defined by a set (15.72).'We add one inequality Next
one as
we
add
the other
follows. Assume that at
l(P)
=
+ anBPnB :S 0
alp,
We call C' the
of
new cone
(15.76)
solutions, and
we
want to deduce
funda-
a new
mental system of solutions N(C). This is achieved by Theorem 4.1 in Bruno 1.4. This theorem is considerably simplified by the absence of the Bi and
reads here: Theorem 15.5.1. The system N(C') is composed of those vectors Ni of the set (15.72) for which l(Ni) :! 0, and in the case t > 1 also of the vectors
N(k, V)
=
N
k
JI(N k')j
for each pair of elements following conditions: 1.
l(Nk)1(N,")
+
Nk' JI(N k)l
N k, N
k'
(15.77)
of the system N(C) which satisfies the
< 0.
inequalities of the system (15.70) which reduce to Then no other element of the all these inequalities to equalities.
We consider the
2.
equalities for
the two elements N k and N k'
system N(C) reduces Another
l(P)
=
simplification
can
.
be made when
alp, +... + anBPnB
=
we
add
an
equality:
(15.78)
0
As
explained above, in principle we by introducing successively the inequalities l(P) :! 0 and l(P) 2! 0. In other words: assume that we already have a system of inequalities (15.70) and its solution (15.75). We add first the inequality I(P) :! 0. We obtain a new cone of solutions C', and the system of solutions N(U) given by Theorem 15.5.1. Next we add the inequality -l(P) < 0. We obtain a new cone of solutions C" and anew system of solutions N(C"), by applying again Theorem 15.5.1. It is easily shown that the two operations can be fused into a single one, described by do this
two
Theorem 15.5.2. the set
(15.72) for
N(k, V)
=
N
k
The system
which
l(N')
1.
l(Nk)l(Nk')
is
composed of those
in the
case
t > 1 also
N k, N
k'
vectors
of
Ni of
the vectors
(15.79)
of the system N(C) which satisfies
the
< 0.
We consider the
equalities for the
0, and
11(N k') I+ Nk' JI(N k)l
for each pair of elements following conditions2.
N(C") =
inequalities of the system (15-70) which reduce to Nk and Nk'. Then no other element of the
two elements
system N(C) reduces all these inequalities
to
equalities.
the
Approach
15. The Newton
110
This theorem is very similar to Theorem 15.5.1, the only difference replacement of 1(N ) :! O'by 1(N') = 0 in the second line.
being
15.5.2 Elimination of Parasitic Solutions
We go back now to our original system with strict inequalities. Its solutions are a subset of the set (15.75). Consider one of the strict inequalities, of the form
1j (P) If
we
=
substitute the
general
(15.80)
+ ainB PnB < 0
aj1P1 + aj2P2 +
solution
(15.75),
we
obtain
t
Ai 1j (N')
ij (P) Since Ni is
We call
of the form
(15.75),
there is
(15.82)
the set of the i such that
Ij
lj(N') (15-81)
particular solution
one
< 0
lj(N)
,
(15-81)
.
(15.83)
< 0.
becomes
Ai 1j (N')
1j (P)
(15-84)
ieij
inequality Ij (P) < 0 is satisfied if and only if at least positive. This condition can be written
It follows that the strict
Ai with i
one
Ai
E
Ij
is
(15-85)
> 0
iEIj
Each strict
form
inequality
(15.85),
As
a
in the
original system gives
which eliminates
particular
case, it may
some
rise to
of the solutions
a
condition of the
(15.75).
happen that 1j (Ni) vanishes for
all
i; the
set
Ij is empty. In that case 1j (P) = 0 for any solution (15.75), and the strict inequality cannot be satisfied. The combination of normal cones is -not valid and
can
be eliminated.
again the case dad, corresponding to the set Motzkin-Burger algorithm it is found that 2 the skeleton consists of the two vectors N' (- 1, 0, 0, 1, 0) and N (- 2, 1, 1, 1, 1) (see Table 15.2). The first strict inequality is 11 (P) -1, 11(N 2) -2, 1, 11, 2}, and the condition (15.85) pi < 0. So 11(N1) As
an
example,
of conditions
we
(15.69).
consider
From the
=
-
-
-
=
-
=
=
-
=
is
Al
+
/\2
> 0
(15.86)
15.5 Coherent
In the
same
way, the 6 other strict
inequalities
in
Boundary Subsets
ill
(15.69) give respectively
the conditions
A2
> 0
)
A2
> 0
A2
> 0
)
A,
>
A, +A2 A,
0,
> 0
(15-87)
> 0.
The whole set of 7 conditions reduces to
A,
A2
> 0
general
Thus the
P
solution of the system
EAjN'
=
(15-88)
> 0
Ai>O,
(15.69)
is
(15-89)
i=1,2.
Program
15.5.3
algorithm was implemented in a computer program. This program finds all valid combinations of normal cones for a bifurcation On, for an arbitrary value of n, on the basis of the normal cone definitions The
(15-30), (15-36), (15.42), (15.47). tors N is
For each valid
combination, the
set of
Table 15.1 shows the number of valid combinations found and the
puting
time in seconds
(on
HP 720
a
workstation)
for
n
=
2 to 6. It
steeply possible combinations, given by (15.68); however, it still exponentially with n.
number of to grow
Table 15.1. Number of valid combinations for number
n
a
com-
can
be
than the
that the number of valid combinations increases less
seen
vec-
printed.
appears
bifurcation On.
computing
of valid
time
combinations
(seconds)
2
12
0
3
39
2
4
138
24
5
505
1143
6
1920
69600
15.5.4 The Case IP2 From of
n
now on we
could in
principle
however, the In the
equations here:
restrict
attention to the
amount of work grows
case are
our
be also treated
02, there
are
nB
(15.38), (15.26)
=
for i
simplest
case
M.
by exponentially.
5 variables and MB =
Larger values described;
the methods about to be
1, (15.44)
for
n
=
=
3
equations.
These
2. We regroup them
15. The Newton
112
fl' f21 A
---:
I
X4
-
G2 X2
X4(X5
I
=
-
-X4 +
+
X3)
G2X2
The number of
ApProach
G3 (1
+
-
Gjxj
G3(1
+
K) X3
--
=
0
+
K)X5
0
=
7
0
(15.90)
-
possible combinations of normal
cones
is 4
x
4
x
4
=
64
(see (15.68). The program finds 12 valid combinations, which are listed in Table 15.2. Each combination is identified in column 1 by a set of 3 letters, which
are
the labels of the selected normal
equation respectively, n
=
as
defined in
2. Column 2 is the index i of
cones
in the
first, second and third 1, and (15.47) for
(15.42), (15.30) a vector N ; the components for i
=
of Ni
are
listed in column 3. The number t of the vectors Ni varies from 1 to 3.
Table 15.2. Valid combinations for the bifurcation 1P2.
Case
i
N'
dimIl
d
aaa
1
-2-1-1-1-1
1
4
acc
1
2
3
2
3
2
3
3
2
2
3
3
2
2
3
3
2
3
2
2
3
3
2
aba
dad dbd
cac
ccc
cda
cbc
bab bbb
0
9
0-1
2
-2-1-1-1-1
0
0
0
0
1
-1
2
-2-1-1-1-1
1
-1
0
2
-2
1 -1 -1 -1
1
-1
0
0- 0
0
2
-1
0
0-1
0
0-1
0
3
-2-1-1-1-1
1
-1
2
-2-1-1-1-1
0-1
0-1
1
0
2
-1
3
-2-1-1-1-1
1 2
cdc
0
1
0 0 0-1 0-1
0 0-1
0-1
0
0
-2-1-1-1-1 0 0-1
0
0
2
-1
3
-2-1-1-1-1
1
-1
0
2
-1
0-1
3
-2-1-1-1-1
1
0-1
0-1
0-1
0 0
0
0
0-1 0
0
2
-2-1-1-1-1
1
-1
2
3
0
0 0
0
0-1
0 0
0
-2-1-1-1-1
Column 4 gives the dimension of the x n matrix formed
is the rank of the t
cone
by
of truncation II. This dimension
the components of the Ni. In the
15.6 Truncated
of
113
Equations
2, the table shows that dim II is always equal to t. This for n > 2, cases exist with dim III < t. We remark that the vector (- 2, 1, 1, 1, 1) is always present.
present'case, not
Systems
n
=
is
generally true, however;
-
-
-
-
Systems of Equations
15.6 Truncated
To each normal cone is associated a truncated equation. Therefore, to each combination of normal cones is associated a truncated system of equations
(Bruno 2.6).
For each of the 12 valid
combinations,
we
this system and try to solve it. The dimension of the truncated system, which we codimension of the cone of truncation (Bruno 2.6):
must
now
designate by d,
This dimension appears in the last column of Table 15.2. It will important role below.
ystem
is made of 3
equations
j
expressions of the fi, read from (15.41), (15.29) for i are given by Table 15.3 for each valid combination. Table 15.3. Truncated systems of
X4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
bbb
X4
can
6.2).
G3-(l
=
-
-
K)X3
X4(X5
-
X3)
G2X2 + G3(1 + K)x5 -X4 + G2X2 -X4 + G2X2 + G3(1 + K)X5 G2X2 + G3(1 + K)x5 G2X2 + G3 (I + K)x5 -X4 + G2X2 -X4 + G2X2 -X4 + G2X2 + G3(1 + K)X5 -X4 + G2X2 -X4 + G2X2 -X4
+
-X4 + -X4 +
G3(1 G3(1
+
+
K)X5 K)x5
Degeneracy
15.6.1
It
+
=
f3P
-
-
-
an
1, 2, 3. The 0, i 1, (15.46) for n 2,
f2P
K)X3 X4(X5 X3) + Gjxj G2X2 + G3(1 acc G2X2 + G3(1 X4 X4(-X3) + Gixi K)X3 aba X4 G2X2 + G3(1 X4(X5 X3) K)X3 dad -G2X2 + G3(l X4(X5 X3) + Gjxj K)X3 dbd -G2X2 + G3(l X4(X5 X3') K)X3 cac G2X2 X4(X5 X3) + Gixi X4 ccc G2X2 X4 X4(-X3) + Gixi cda G2X2 X4 X4(X5) + Gixi cdc G2X2 X4 X4(X5) + Gixi cbc G2X2 X4 X4(X5 X3) bab X4(X5 X3) + Gixi X4 + G3(1 K)X3 aaa
play
equations.
f1p
Case
is the
(15-91)
d--nB-dimll.
The truncated
consider
happen.
that the truncated system is degenerate (Bruno 2.6, Remark case here in the 4 cases cac, ccc, cdc, fbc: the first
This is indeed the
equations are identical. remedy is indicated by Bruno (ibid.): one of the original equations 0 should be replaced by an appropriate combination, which provides in fi a sense a finer description of the system.'Here the nature of the degeneracy and third .
The
=
Approach
15. The Newton
114
we should replace f3 by the combination f, + f3. We have already considered this combination: it is the additional relation (15.8). In each of the 4 degenerate cases, we must replace the last normal cone c, defined in (15.47), by the normal cone A, defined by (15.66). We label the 4 cases thus redefined: caA, ccA, cdA, cbA. We apply the Motzkin-Burger algorithm to these new caseg. We find that only the two cases caA and cbA
indicates that
are
valid combinations.
In the truncated system of equations, the third equation should be replaced by (15.63). Thus the entries cac, ccc, cdc, cbc should be crossed out in Tables 15.2 and
15.3, and
new
entries should be inserted
as
indicated
by
Tables 15.4 and 15.5.
Table 15.4. Additional valid combinations for the bifurcation IP2.
N'
Case
i
caA
1
-1
2
-2-1-1-1-1
cbA
0-1
1
-1
0
2
-1
0-1
3
-2-1-1-1-1
0
dim III
d
2
3
3
2
0-1 0
0
0-1
Table 15.5. Additional truncated systems of
Case
_7
f, P
,
caA
X4
cbA
X4
-
-
_f2P
f3 P
X4(X5 X3) + Gjxj X4(X5 X3)
G2X2 G2X2
equations.
-
-
(1 (1
-
-
K)x3 K)x3
+
(1
+
+
(1
+
K)X5 K)x5
15.7 Power Transformations In each of the 10 cases, we must now study the truncated system of equations. It will be convenient to make in each case a change of variables, which Bruno
(1.11, 2.3, 2.7) Xi
I
use
=
calls
...
WnB
wi for the new
fusion with my there is Wi
:i4
power
3inB
Oil
W,
a
own
,
i
transformation, defined by
=
(15.92)
1,...,nB
variables, instead of yj as in Bruno, in order to avoid conquantities yi. Note that according to (15.9) and (15.10),
(15.93)
0.
Theorem 7.1 in Bruno 2.7 states that for
tions, with dimension d,.it is possible
to find
a
truncated system Of MB equachange of variables (15.92) in
a
15.7 Power Transformations
such on
way that the
a
d of the
system again has
new
variables wj. The simp 'lest method for
MB
115
equations, but depends only
new
finding this change
of variables in the present
appears to be Method 2 described in Bruno 1.11:
case
1. We select
(dim][I) linearly independent
this amounts to
selecting
vectors
N' (in the
all vectors Ni since in that
dim
case
2,. 11'always
case n
=
equals t). 2. We write these vectors
as
(dim III)
the last
(flij). Since all components of the Ni convenient to use -Ni rather than Ni.. 3. We fill the
remaining columns,
are
columns of the matrix
negative
or
zero, it will be
i.e. the first d columns
(see (15.91)
in
way that the matrix 0 is regular, i.e. det # 54 0. If possible, it is convenient to do this in such a way that the matrix is unirnodular, i.e.
such
a
1; then all exponents Oij are integers and I det 0 1 and the in also inverse change of variables w(x). (15.92)
all
=
Rearranging the system
integers,
new
in
truncated
(Bruno 2.8, (8.5))
as
9i (W1
wi variables if necessary, we can write the
are
7
...
7wd)=Oi
i=1)...
)MB
(15.94)
-
If d < MB , there are more equations than variables, and in general the no solutions. If d 2 MB, we must first compute the Jacobian
system has
0(911 a(WI
1
MB) Wd)
(15-95)
(Bruno 2.8, (8.7)). Any point (W1 less than TnB is ment
a
(Bruno 2.8).
Wd)
where this Jacobian has
solutions may also have to be rejected because they contradict We apply now this method to each of the 10 cases in turn.
15.7.1 Case
=
4. We write
case
N', given by
(E2, E3, E4, E5, -N1)
The
(15.93).
aaa, where the dimension of the truncated
Table 15.2, in the last column of
fill the first 4 columns with the vectors
There is det
rank
aaa
We consider first the is d
a
critical point. Such critical points require a special treatFortunately, no critical points will be found below. Some
the
we
E2, E3, E4, E5:
0
0
0
0
2
1
0
0
0
1
0
1
0
0
1
0
0
1
0
1
0
0
0
1
1
regular but not unimodular. change of variables are then
2: the matrix is
equations of
system and
(15-96)
Approach
15. The Newton
116
2
X1 =W5
X2
,
W1W5
X3
7
W2W5
=
X4
7
W3W5
=
7
X5
=
W4W5
-
(15.97) in the truncated
Substituting W3W5
G2W1W5
-
GO
+
equations given by Table 15.3,
K)W2W5
-
W3W5(W4W5-W2W5)+Gl W25 -W3W5 +
or,
G2W1W5
+
obtain
0,
=
0,
K)W4W5
(15.98)
0;
=
-respectively (we recall
that the wi
never
vanish),
G2W1+G3(1-K)W2=0,
91=W3 93
G3(1
2, w5 W5, W5
dividing by
92
+
=
we
`
W3(W4
=
-W3 +
W2)
+
Gi
G2W1
+
G3(1
-
We have 3 equations for 4
G3 (1
-G2
ag aw
-W3
G2
0
GO
-G2
-
0
The minor formed
G2
0
=
K)
0
as
(15 .99)
-
expected.
1 W4
The Jacobian
(15.95)
is
0
(15.100)
W3
W2
-
G3(1
-1
2, 4 has
+
K)
determinant
a
0
G3(1
(15-101)
2G2G3W3 0 0
W3
always ha's
Thus the matrix
K)W4
variables,
K)
-
-W3
+
the columns 1,
by
0
0,
=
+
K)
rank 3: there is
no
critical
point.
Solutions of the system (15.99) form a one-parameter family. It is convenient to take W3 as parameter; we have then a system of 3 linear equations for wl, W2, W4, which is W3 W1
+
=
G2
(1
-
easily solved:
K2)GiG3
2G2W3
(1
W2
+
K)Gj
(K
W4
2W3
-
1)Gj
2W3
(15.102) The value W3 = 0 is excluded according to (15.93). Thus wi, W2 and W4 are always defined. Also, W2 and W4 are always non-zero. However, w, vanishes for the two values
_1)GjG3 FLK2 -
W3
when
(15.103)
-
2
IKI
> 1. These two values Of W3 must then also be excluded.
Going back X1
=
X3
=
W
2 5
W5
,
(1
to the
X2
+
=
original W5
K)Gj
2W3
(W3
G2 X4
variables xi with +
=
(15.97),
we
obtain
(1-K 2)GjG3 2G2W3 W3W5
,
X5
=
W5
(K
-
1)Gj
2W3
(15.104)
15.7 Power Transformations
This is
(15.2),
tions with W5
system with 2 parameters
now a
Al /2
:--
AC
p
=
Aal
obtain
W3
=
Y11L_1 /2 (1
-
1/2
+
G2
It
=
we
1/2
(1
+
K)Gj '
2y,p-1/2
I
V3 and w5.
Going
117
back to my nota-
K 2)GjG3
2G2Y1 p-1/2 A a2
=
Y
1/2
(K
-
1)Gj
2y,/_1-1/2
(15.105)
*
a single vector Ni (-2, 1, 1, 1, 1) and we have from (15.75): -A,, with A, > 0. The elimination of parasitic solutions -2Aj, p2 (Sect. 15.5.2) gives A, > 0. Thus from (15.14) we have
There is p,
=
-
-
-
-
=
=
1 I/=
(15.106)
-
2
Therefore the present case aaa should correspond to the case v 1/2, the in detail in studied and 12.5 in Sect. established which of were equations and of variables the and 13 14. (12.111) changes Indeed, applying Chaps. =
(12.112), W
obtain
we
YJ
=
1
-
+
K2
Yl
X,
,
1 + K =
2Y,
'
X2
K
-
1
=
2Y,
(15.107)
'
equation of the characteristic (13.41) for the case 1P2, and the equations for X, and X2 deduced from (13.1a),with one proviso: in the 0 case JKJ > 1, the two intersections of the characteristic with the axis W This because is our the treatment. covered present by (see Fig. 13.1) are not fundamental assumption (15.9), (15.10) requires all variables to be non-zero. These points require a different change of variables; this will be covered in We
rec 'over
the
=
case
(Sect. 15.7.7).
bab
Additional computations would give the error terms (see Bruno example of these computations will be given in the case dad (Sect. 15.7.2 Case
2.8). An 15-7.4).
acc
We consider next the 6'cases where the dimension of the truncated, system is d = 3. We write N' and N 2, given by Tables 15.2 and 15.4, in the last
fill the first three columns with
two columns of
fl,
vectors. We
to obtain relations
try
and
we
correspond respectively We obtain in the for Which In the
0
=
we
expect
case
acc,
a
variables
a
possible
appropriate
unit.
for x, et X2, which
parameters p et AC. system of 3 equations for 3 variables,
take -N 2)
change of variables truncated equations become same
as
finite number of isolated solutions.
(E2, E3, E4, -N1,
This is the
simple
to the fundamental
new
we
as
(15.108) as
in the
case aaa
(see (15.97)).
The
15. The Newton
118
I
"
2 3
W3
G2 W1
-
-W3 +
=
0,
G2W1
=
0
This system has
contradicts
G3 (1
+
G,
-W3W2 + =
Approach
no
(15.93).
K) W2
-
"
0
(15.109)
-
solution: the first and last equations give W2 = 0, which addition, the second equation cannot be satisfied since
In
G, 0 0. 15.7.3 Case aba We take
0
(E3, E4, E5, -N',
=
Here det
variables
#
(15.110)
1: the matrix is unimodular. The
=
equations of the change of
are 2
X1
-N 2).
W4W5
X2=W5,
X3=WlW5,
X4=W2W5)
X5=W3W5.
(15.111) The truncated equations become
1
=
W2
-
G2
+
G3(1
K)wl
-
=
0,
2=W3-Wl=O)
3
=
G2
-W2 +
This system has
GO
+
+
K)W3
=
(15.112)
0
unique solution
a
W2=G2.
Wl=W3=0,
However, this solution
(15.113)
must be
rejected because
it contradicts
(15.93).
15.7.4 Case dad We take
(E3, E4, E5, -N1, The
-N 2)
(15.114)
equations of the change of variables X1
=
'27 W4W5
X4
=
W2W4W5
The truncated
j
2 93
=
-G2
+
W5
)
X5
=
)
W1W5
X3
(15-115)
W3W5
equations become +
W2(W3 G2
=
X2
are
GO -
wi)
GO
This system has
+ a
-
+
K)wl G,
K)W3
0, 0, 0
unique solution
(15.116)
119
15.7 Power T ansformations
V)j,
wi
i
(15.117)
1, 2,3,
=
with
G2
3(1
K)
-
G2
03
GO
+
02-
K)
GiGO
-
K 2)
2G2
(15.118) The Jacobian is
GO
Og
K)
-
09W
0
0 W3
-W2
GO
0
0
(15.119)
W2
W1
-
K)
+
Its determinant is
G 2(1 3
-
K2) (W3
WI)
(15.120)
-2G2G3 54 0
=
(15.117)
Thus the solution for i
-
is not
a
critical point. It is
acceptable
since
Oi 54
0
1, 2,3.
Going back
to the variables xi,
GiG3(1
K 2)
-
GO
-
k)
W5
G2 -4-0
2G2 now a
X3-
X2=W5,
=
This is
obtain
G2
2
Xl=W4W5,
X4
we
X '5
)
=
system with 2 parameters
-
*
Z 3(1 + K)
W4 et W5.
W5
Going back to
(15 121) .
my notations
with (15.2), we obtain the relations between these two parameters and the fundamental parameters 1L, AC:
W5=ACi
W4
LAC-2
=
(15-122)
and
Aa, Aa2
G2
G3(1
_
GO + K )
Here there
N'
K)
-
G2 =
=
are
-A,
-
2A2,
N
=
-A,
-
A2
The elimination of
A2
0,
(15.14)
we
2 G2
(15-123)
.
,-
N2= (-2, -1, -1, -1, -1) ,
(15-124)
(15.75):
=
From
AC
(-1, 0, 0, -1, 0)
p,
>
Y
,
two vectors
and therefore from
A,
GiGO K 2) -/1AC -
-AC
=
P2 P5
,
=
=
-A2
P3
=
-A2
,
-A2
(15A25)
parasitic solutions (Sect. 15.5.2) gives > 0
have
(15-126)
Approach
15. The Newton
120
A2 V
and
./\j
(15.127)
2/\2
+
using (15.126): 0
1:
GI(K2-1)
GI(K2-1)
1)
2
(15-155)
-
2G3 W,
V -2
K
-
W3=
1
2G3
K+1
(15-156) The determinant of the Jacobian is
GO
K)
-
W3
-W2
0
1
GO
-1
0 The solutions
are
4G3W2
W2
W1
-
+
not critical points.
(15-157)
0
K) They
are
acceptable
since wi
0,
Z
1,
2,3-
Going
X1
X4
=
back to the variables xi,
2
W5
,
X2=W4W57
V LIG3
2
( 2
-
1)
we
X3=
obtain
V Gl(K2-1) 2G3
K
-
1
W5
,
/2i7j LK2-11
W5
,
X5
V
2G3
K + 1
W5
(15-158)
124
15-. The Newton
This is
a
Approach
system with 2 parameters
(15.2),
W4 et w5.
Going
back to my notations
obtain the relations between these two parameters and the fundamental parameters p, AC: with
W5
P
-`
we
1/2
-1/2 AC
W4
,
(15-159)
.
We have also
K
2-1
2G3
Aa,
IL
K- 1
L -j V-G-l(.KF
1/2
,Y1
=
VelG3(K
2
1/2
2
2
2G3
Aa2
K+1
There
N1
1/2
(15-160)
-
two vectors:
are
=
A
(0, -1, 0, 0, 0)
N2
,
=
(-2, -1, -1, -1, -1).
(15-161)
Therefore pi,
-2A2
=
P2
i
=
-Al
-
A2
i
P3
=
-A2
-A2
P4
P5
=
-A2
(15-162) The elimination of
A,
Al and
> 0
(15-163)
-
(15.14):
Rom
V
A2
0,
>
parasitic solutions gives
+
A2
(15-164)
=
21\2 obtain
we
1 V
>
(15.165)
2
Thus the present
case
bab, corresponds to the
case v
>
1/2
considered in'
Sect. 12.6.
Finally, applying
the
changes
--1)/2 V-(K2
X1
K
::F'V(K2
Y1
-
,
1
-
-
1)/2,
X2
of variables
=
T-
V(K2
(12.111) -
and
(12.112),
we
have
1)/2
K + 1
(15.166)
with
'
=
sign(cos 0)
.
(15.167)
125
15.7 Power Transformations
15.7.8 Case dbd
the 3
finally
We consider is
where the dimension of the truncated system N3, given by Tables 15.2 and 15.4, in the
cases
N', N2 and
2. We write
last three columns of
fl, and
we
unit vectors. We obtain in the
variables, which
should not have solutions; this is indeed what
general
in
fill the first two columns with appropriate variables a system of 3 equations for 2
new
we
will find. In the
case
dbd,
-N 2, -N 3)
(E3, E4, -N1,
=
2
W3W4W5
.
,
(15.168)
of variables
change
The equations of the X1
take
we
are
X3=WlW5,
X2=W5i
X4=W2W4W5.)
X5=W3W5-
(15-169) The truncated
gi
=
-G2
+
equations become
GO
-
K)wl
=
0,
2=W3-Wl=07 93
=
G2
+
GO
+
K)W3
=
(15.170)
0
0, and with the Combining the first and last equations, we obtain W1 + W3 this contradicts 0. But (15.93). It also equation we have w, W3 contradicts the first and last equations. Thus this system has no solution. =
second
=
=
15.7.9 Case cbA We take
(E3, E4, -N1, The
-N
2
y3)
,
(15.171)
equations of the change of variables 2
W3W4W5
X1
X2
,
=
W5
X3
,
=
are
W1W4W5
X4
W2W5
,
X5
=
W4W5
(15.172) The truncated W2
2 g3
=
1
(1
-
-
G2
W1
-
equations become 0
,
0,
Kjwl
+
(1
The last two equations solution.
+
K)
=
(15.173)
0.
give two.different values for
wl.
The. system
has
no
126
Approach
15. The Newton
15.7.10 Case bbb We take
(E3, E5, -N1, The
of the
equations X1
2
W3W5
=
The truncated
91
This
G3(1
=
I +
2
=
W2
3
=
-1, +
WI
-
=
W4W5
X3
7
K)wl
-
Up
to
case
W1W5
=
X4
W5
X5
,
W2W5
=
GO
0,
=
0,
K)W2
+
0
=
(15-176)
-
(Sect. 15.7.8).
15.8 Total Bifurcation of
the
are
equations become
is similar to dbd
case
(15.174)
of variables
change
X2
,
-N 2, -N 3)
The system has
Type
no
solution.
1
in the present
to
a
as
chapter we have applied the Newton approach partial bifurcation of type 1. We now show how it can just well be applied to the case 1Tn, i.e. a total bifurcation of type 1. now
of
The orbit is then made of
basic
n
(see
arcs
6.2.1).
Sect.
n
must be
even
(Sect. 6.2.1.2). There X1
=
X6
=
are
P
nB
X2
7
=
=
2n + 2
I kC
Y2
variables, which
we
Aal
=
X3
i
X2n+1
=
Aan
=
X4
,
X2n+2
)
write in Bruno's notations: Y1
=
x5
-
7
Yn
Aa2
=
,
-
(15.177)
(12.32)
The fundamental equations
fl'
X4
=
II
f2i
f 2i-l
=
I
=
with the yn,
=
X2n+2
-
X2i+2 (X2i+3
=
f2n
-
X2i+2
-
X2n+2 (X3
G2X2
-
X2i+l)
X2i + -
1,
X2n+l)
Aan+1
G3 (1 +
G2 SiX2
cyclical boundary YO
+
=
+
-
G, x, +
become
K)X3 =
G3 (1
G, x,
=
=
0
;
0i +
n
-
1,
Ksi)X2i+l0 (15-178)
0
conditions
Aal
(15.179)
.
Their number is MB = 2n. We again have MB = nB 2. The additional relations (12.33) are still needed. Their number is -
n.
We
describe them with indices from 2n + 1 to 3n:
f2ln+i I
=
-X2i + -T2i+4 +
G3 (1
+
Ksi)X2i+l
+
G3 (1
-
Kgi)X2i+3 n
.
=
0
(15.180)
15.8 Total Bifurcation of
particular
In the
A
G3 (1
=
case n
K)X3
-
2, there
=
G3 (I
+
is
a
K)X5
+
single additional =
Type
127
1
relation
(15.181)
0
applicable without made in Newton the of the polyhedra use study again change. Sect. 15-3. There are 4 possible normal cones for each of the encounter equa0 in (15.178), and 11 possible normal cones for each of the arc tions f2i 0; thus the number of possible combinations of normal f2i-1 equations The method of solution described in Sect. 15.2 is
We
can
=
=
is
cones
(15.182)
44n
The program implementing the Motzkin-Burger algorithm can again be used to'find all valid combinations of normal cones. Table 15.6 shows the number of valid combinations and the computing time in seconds (on a HP 720
workstation)
for
n
=
2 to 5.
Table 15.6. Number of valid combinations for
bifurcation iTn.
computing
number
n
a
of valid
time
combinations
(seconds)
2
32
0
3
85
10
4
300
487
5
1095
26079
15.8.1 The Case 1T2
We consider the 'MB
=
4
equations.
fl'
-=
X4
f2
==
X4(X5
=
X6
=
X6(X3
I
f3 I
f4
case
-
-
X6 -
-
-
G2X2
X3)
X4 +
IT2
These
+
+
+
G3(1
Gixi
G2X2
x,5)
as an example. equations are
+
=
-
Gjxj
K)X3
:--
0
K)X5
=
0)
are nB
=
6 variables and
1
0
G3(1 =
There
+
(15.183)
0
The number of possible combinations of normal
cones
is 11
x
4 x 11
x
1936.
4
The program finds 32 valid combinations, which are listed in Table 15.7. Each combination is identified in column 1 by a set of 4 letters, which are the labels of the selected cones'(see (15.36) and (15.30)). Other columns have the same
meaning as in Table 15.2. 4 only: the 5 but dim III Here, in the case kbkb, there is t is there formed by the components of the Ni is singular. Indeed =
N'
-
N
2 -
N
4
+ N
5
=
0
=
matrix
(15-184)
15. The Newton
128
Approach
Table 15.7. Valid combinations for the bifurcation M
Case
i
aaaa
1
accd
1
abab daeb
dbeb
caca
cccd
cdac
cbcb
ebda ebdb
baba
kaka kakb
0
2
-2 -1 -1 -1 -1 -1
0 0
0
0
0-1
0
0
0 0
1
-1
2
-2 -1 -1 -1 -1 -1
1
-1
0
0
0
0
0
0
0-1
0
0
2
-1
3
-2 -1 -1 -1 -1 -1
1
-1
2
-2 -1 -1 -1 -1 -1
1
0
0-1 0
0-1
0
0
0-1
0
0-1
0-1
dim 11 d
Case
i
1
5
kbka
1
-2 -1 -1 -1 -1 -1
2
4
2
-1
2
4
N'
3
kbkb
1 2
2
3
4
3
jbha 2
4
jbhb 3
3
,
-
0
0
dim 1111 d 0
0-1
0-1
-1
0
0-1
-1
0
0
0
0
0
-1
0 0 -1
0
0
3
-2 -1 -1 -1 -1 -1
4
-1
0
0
0
0-1
5
-1
0
0-1
0-1
0-1
0-1
1
-1
2
-2 -1 -1 -1 -1 -1
3
-1
0
0
0
1
-1
0
0
0 0
0
2
-1
0-1
0-1
0
0
2
-1
0
3
-2 -1 -1 -1 -1 -1
-2 -1 -1 -1 -1 -1
4
-1
1
0
0-1
0
0
0
1
0
0-1
0
0
0
-1
0-1
0-1
0
3
-2 -1 -1 -1 -1 -1
1
-1
0
2
-1
0-1
0
0
0
0
0-1
0
3
-2 -1 -1 -1 -1 -1
1
-2 -1 -1 -1 -1 -1
2
-1
0
0
0
0-1
1
-1
0
0
0
0
2
-2 -1 -1 -1 -1 -1
3
-1
1
0
0
0
0
0-1
0-1
0
0
0
0
0-1
0
0
0
0
0
0
0
0-1
0
-2 -1 -1 -1 -1
1
1
-1
gaib
3
gbib
1
3
fafa 2
4
3
3
2
4
3
3
fcfd
fdbc
0
0
0
0
0
0-1
0
0
3
-2 -1 -1 -1 -1 -1
1
-1
0 0
0
0
0
0-1
0-0
0
0
0
0-1
0
0
0
3
-1
4
-2 -1 -1 -1 -1 -1
1
0-1
0
0
0
0
0-1
0
2
-1
3
-2 -1 -1 -1 -1 -1
0-1
1
0-1
0
0
0
0
2
0
0
0-1
0
3
-1
0-1
0-1
0
4
-2 -1 -1 -1 -1 -1
0
1
0-1
2
0
0
0
0
0
0-1
0
0
0
0-1
2
0
0
0
0
0
0-1
0
0
0
0 -1
0
0
0
0
0-1
0
0
0
0
3
-I, 0 -1
-2 -1 -1 -1 -1 -1
4
-2 -1 -1 -1 -1 -1
1
-1
1
-1
0
-1
0-1
3
-2 -1 -1 -1 -1 -1
1
-1
0
0
0
0
0
0-1
0
0
0-1
0
0-1
0
0
0-1
0
2
-1
0
3
-1
0-1
4
-2 -1 -1 -1 -1 -1
1
-2 -1 -1 -1 -1 -1
2
-1
0
0-1
0-1
1
-1
0
0-1
0
0
2 -1 -1 -1 -1 -1 -1
0
0-1
0-1
3
3
3
fbfb
2 4
2
ibga
2
4
ibgb
3
0
0
0
0
0
0
0
0
0-1
0
3
-1
-2-1 -1 -1 -1 -1
0-1
0-1
0
0
0
0
2
-2 -1 -1 -1 -1 -1
3
-1
0
0
0
0-1
1
-1
0
0
0
0
0
0-1
0
0 0
0
2
3
0
0-1
4 1
4
2
3
3
4
2
3
3
4
2
3
3
4
2
3
3
4
2
4
2
3
3
4
2
.
-2 -1 -1 -1 -1 -1
1
0
3
3
0-1
-1
3
fdfc
0
?
2
3
0
0-1
0
2
3
3
-2 -1 -1 -1 -1 -1
3
2
4
-2 -1 -1 -1 -1 -1
2
1
2
3
0-1
3
2
hbjb
0-1
-1
3
hajb
0
-2 -1 -1 -1 -1 -1
2
bbbb
0
1
2
bcfd
0
2
2
cdcc
N' -2 -1 -1 -1 -1 -1
3
-2-1-1-1
4
-1
0
0
0
1 -1 0-1
15.9 Conclusions
We remark that the vector We do not pursue here the much space.
(- 2, study
-
1,
-
1,
-
1,
of these 32
-
1)
is
129
always present.
cases as
it would take up too
15.9 Conclusions With the help of the Newton approach, we have thus recovered in a more rigorous way the results of the previous Chapters 12 to 14, at least in the simplest case IP2. In principle at least, bifurcations 1Pn with larger values of n, and also bifurcations
1Tn, could be treated in the
same
way.
However, the Newton approach has one fundamental shortcoming: it can be only applied for a specific value of n. Our objective is to obtain general results, valid for any n; this objec'tive was indeed achieved in Chap. 12.
practical problem with the Newton approach is that it prostudy of all possible cases, one by one. Therefore, and very as has been seen, the amount of work grows exponentially with n, quickly becomes prohibitive. This is why we had to limit the use of this approach to the case 1P2. An additional
ceeds
by
enumeration and
Proving General Results
16.
approach, in which we retain part of the formalism of the Newton approach; but instead of considering the values of n and the valid combinations one by one, we try to prove general results, valid for any We
develop
now a
third
n.
16.1 Variables and Equations We
use
again
X2
JL'
X1
the variable
(15.2)
the variables xj defined in
(15.177).
and
We recall that
(16.1)
AC;
=
Aaj, corresponding
to arc
i, is represented by
X2i+l; and the.
variable yi, corresponding to encounter i, is represented by X2i+2 We reproduce here for convenience the fundamental equations with which have first the we will work, so as to make this chapter self-contained. We -
encounter
equation (15.26)
X2i+2 (X2i+3
Next
we
X2i+2
-
-
X2i+l)
have the X2i +
in
(16.3)
vanishes
arc
G2 SiX2
For the initial basic
arc
GO
For, the first relation in
If
n
in
G3 (1
a
(see (15.53)). (see (15.58)).
=
0
(16i2)
.
in the
+
general
Ksi)X2i+l
=
case
a
+
(15.32) (16.3)
0
partial bifurcation, with
i
=
For the final arc, with i
have the additional relation in the
-X2i + X2i+4 +
vanishes
+
(see (15.38)). (see (15.44)). we
=
equation
vanishes
Finally
vanishes
G, x,
+
Ksi)X2i+l
+
GO
partial bifurcation,
-
general
1, the second =
case
Ksi)X2i+3
with i
=
For the last relation, with i
=
M. Hénon: LNPm 65, pp. 131 - 148, 2001 © Springer-Verlag Berlin Heidelberg 2001
terms vanish
0
(15.49) (16.4)
1, the first term in (16.4) n 1, the second term
=
-
-
2, the first and second
term
n, the first term
(see (15.63)).
16.
13
Proving General
Results
16.2 Method of Solution We
again consider solutions of the xi
=
bi-rP'(1
+
o(1))
i
,
=
(15.9):
form
1,...,nB
(16.5)
,
and, guided by the results obtained in Chaps. 12 to 14, we try to derive general properties of the coefficients pi and bi directly from the equations. No reference is made here to Newton polyhedra: the present approach is algebraic rather than geometric. We will generally treat the cases of partial and total bifurcation together, although on occasion separate proofs are needed. For the comparison with the results of Chap. 12, we recall the rela-
(15.14):
tion
V
P2
(16-6)
=
Pi
Substituting (15.9) three terms
into
(16.2),
we
find that the asymptotic values of the
are
b2i+2b2i+3 TP2i+2+P2i+3
-b2i+2b2i+l TP2i+2+P2i+l
,
Gib,-rP'
,
(16.7)
We call cp the maximum of the three exponents of -r (cf. Bruno 2.2). The dominant terms are those for which the exponent of r equals cp. At least two terms must be'dominant.
obtain
TcP, dominant, we
a:
one
Deleting
P2i+2 + P2i+1
P2i+2 + P2i+3 and
b:
P2i+2 + P2i+1
P2i+2 + P2i+1
P2i+2 + P2i+3
P1
b2i+2 b2i+1
-
b2i+2b2i+l
-
b2i+2 b2i+1
In each case, the first-half
b2i+2 b2i+3
corresponds
to
+
one
and the second half to the associated truncation
A similar
+
b2i+2 b2i+3
+
b2i+2b2i+3
+
Gib,
decomposition
into
cases
takes
Gib,
We consider first
an
=
+
Gib,
0
0
0
=
0
(16.8)
.
of the normal
cones
(15.30),
(15.29).
place for the other equations (see
15.3).
16.3 Two General
are
> P2i+2 + P2i+1
and
Sect.
dividing by
which terms
P1 > P2i+2 + P2i+3
and d:
-
-:--
on
P2i+2 + P2i+3 > P1
and c:
the non-dominant terms and
equation for the bi. Thus, depending of the following 4 cases must hold: an
Propositions
encounter i. We show first that
16.3 Two General
Proposition P2i+2
-< MaX(P2
Proof.
we
Assume that
place,
16.3.1. For any i E pi /2)
,
133
C,
(16.9)
.
consider first the
(16.9)
Propositions
case
is violated at
of partial bifurcation. Then 0 < i < n. places. Let i be the first such
one or more
i.e. the smallest i such that
P2i+2 >
max(p2, pi /2)
(16.10)
Consider the equation (16.3) for the arc preceding the encounter i. The terms 2 and 3 are negligible since by hypothesis P2i+2 > P2j and P2i+2 > P2Therefore the terms I and 4 must balance: P242
-=
In the
P2i+1
b2i+2
)
particular
obtain the
same
case
i
-G3 (1
=
+
KSj) b2i+1
1, the second
=
(16-11) (16.3) vanishes,
term in
and
we
result.
We consider next the equation (16.2) for the encounter i. The term 3 is since P2i+2 + P2i+1 = 2P2i+2 > pi. Therefore the terms 1 and 2
negligible
must balance:
P243
=
We consider X2i+4
-
b2i+3
P2i+1 now
X2i+2
-
(16.3)
G2SiX2
The terms 2 and 4
[G3(1 This sum
sum
+
+
G3(1
of the
G3(1
Ksi)X2i+3
-
same
order;
=
their
Ksi)b2i+,],rP2i+l
-
=
must be balanced
P2i+1
b2i+4
7
We consider next
X2i+4(X2i+5 The term 3 is P245
+
for i + 1: 0
(16-13)
-
sum
=
is
2G3b2i+l rP2i+l
does not vanish. The term 3 is therefore
(16.14)
P244
Ksi)b2i+l
are
(16-12)
b2i+1
=
=
-
(16.2)
X243)
negligible
P2i+3
+
by
=
negligible,
(16.14) and the
the term 1:
(16-15)
-2G3b2i+l
for i + 1:
Gjxj
=
0
(16.16)
.
since P2i+4 + P2i+3
b2i+5
P241
=
2P2i+2
b2i+3
=
> pi, and
we
obtain
(16.17)
b2i+l
Continuing in this way, we find that all successive exponents P246 7 P247) are equal to P2i+l) while the coefficients never vanish and are ....
b2i+2j+l
=
b2i+2j+2
=
b2i+ 11
j
-G3U G3U
=
+
1, 2,
.
1)b2i+l Ksi)b2i+l
+ 1 +
for
j f6rj
odd
=
even=
1, 3, 5, 0, 2,4,
(16.18)
However, sooner or later we reach the final arc. Then the first term in (16.3) vanishes, and the arc equation cannot be balanced. We have thus reached an impossibility.
Proving, General Results
16.
134
We consider next the
violated at an
of
one
of
case
places.
more
same
reasoning
above
as
we
arrive at
an
not all
are
MaX(P2)pl/2)
encounter i such that P2i+2 >
(16.9)
total bifurcation. Assume that
a
If the P242
equal,
we can
and P2i < P2i+2, and
is
select
by the
impossibility.
If all P242 equal a common value p* (with p* > max (P2 i p, /2)), assume first that for some i there is P2i+1 > pi p*. Then the term 3 in (16.2) is =: obtain p*, b2i+3 = b2i+1 We can P241 > pi negligible and we P243 -
-
-
continue and we obtain P2i+2j+l = P2i+l, b2i+2j+l = b2i+1 for all j. If we sum the additional relation (16.4) over every other value of i, starting from an
terms I and 2
arbitrary value, the
all of the
same
G3 (1
+
are
eliminated. The terms 3 and 4
are
order and must balance:
Ksi) (b2i+l
+G3 (1
-
+
b2i+5
Ksi') (b2i+3
+
+
-
-
-)
b2i+7
(16.19)
0
+
which reduces to
G3nb2i+l
=
(16.20)
0
assumption (15.10). p* for all
This contradicts the fundamental
Finally,
we
have the
case
where P2i+1 :! pi > P2, the term 3 is
-
is P2i+2 = P2i = P* Since p* we have P2i+l < p, /2 < p* and the term 4 is .
1 and 2 must balance. So all
b2i+2
are
equal
i. In
(16.3),
to
a common
value b*. We write
X2i+1
'Y2i+l
=
there
negligible. Since p* > p, / 2, negligible. Therefore the terms
(16.21)
liM 7'-++00'rPl-P*
There is 72i+1
=
b2i+1
if P2i+l
0
if P2i+l < P1
Dividing (16.2) by b* (Y2i+3
Summing
over
Ginbi which is Next
-
=
72j+j)
we
P241
-
+
P1
taking
Gib,
all encounters,
=
0
we
-
-
P*) P*
(16.22)
the limit
r -4
+oo,
we
have then
(16.23)
.
obtain
(16.24)
0, M
impossible. have
Proposition
16.3.2. For any i E
MaX(P2
This follows P2i+1 >
-rP' and
=
i
A,
(16.25)
pi / 2)
immediately from (16.9): if
MaX(P2,pi/2),
then the term 4 in
a
basic
(16.3)
arc
i existed for which
would be the
only
dominant
term. M
Propositions 16.3.1 and 16.3.2 suggest that the solutions of the system depend critically on which of the two -quantities P2 and pi /2 is largest.
will
So
we
distinguish
3
cases.
16.4 The Case P2
16.4 The Case P2 As shown
If P2
16.4. 1.
p, /2
135
pj/2
=
by (16-6), this corresponds
Proposition
=
pi /2
The
proof
is somewhat
lengthy
and
141
involved, and has been relegated
to
Sect. 16.8. We remark that this proposition is similar to Proposition 4.3.2, which proved in Sect. 12.2. There are differences, however. Proposition 4.3.2
was was
about
ordinary generating orbits,
and T-arcs defined
by Definition
4.3.2.
The present result is about periodic orbits in the vicinity of a generating orbit, and T-arcs* defined in Sect. 16-6.2.2. (It can be noticed also that the
present proof is
more
complex.)
1 6.6.2.4 T-Arcs* Again. Finally we refine the estimates (16.64) for a Tarc*, running from node* i to node* i + 2. We consider first the case where both ends of the T-arc* are junctions with a S-arc*. We have then from
(16.71)
(16.72)
and
P2i+2
--
Assume
negligible, P2i+3
P1
P2
-
that
P2i+6 > pi
P2i+3
---:
-
P1
(16-73)
P2
-
p2. Then the term 3 in
-:--
b2i+5
P2i+5
-
b2i+3
0
(16.74)
-
The terms 1 and 2 in the additional relation a
consequence of
G3 (1
assumption, and the
our
Ksi) b2i+3
-
+
But the two equations is
for i +'l is
(16.2)
and the terms 1 and 2 must balance:
G3 (1
+
(16.74b)
Ksi) b2i+5 and
(16.4)
for i + 1
are
negligible
as
terms 3 and 4 must balance: =
0
(16.75)
-
(16.75) imply b2i+3
b2i+5
0, which
impossible. Therefore P2i+3
In the
P1
way it
same
P1
P2i+5
P2
-
can
P2
-
(16.76)
-
be shown that
(16.77)
-
We write
7243
X2i+3
liM
=
T
72i+5
++00 'rP1 -P2
Dividing (16.2) for
G2sib2(^/2i+5
-
i+1
Gib, = o
+
7- -+
and
using (16.63),
we
obtain
(16.79) -+
+oo, and using
(16.71), (16.72),
obtain
-b2i+2
+
b2i+6
We consider with ends of the
+
now
G3 (1 the
-
Ksi)'Y2i+3
cases
bifurcating
where
are.
G3 (1
one or
+
Ksi)-y2i+5
0
.
(16.80)
both ends of the T-arc* coincide as above, we obtain an equation
(16.80), but with one or both of the first two terms absent. We 0 if the encounter i is the beginning g' and g" as in Sect. 12.4.2: g' 1 if it is a junction with a S-arc* made Of Mb basic bifurcating arc;,g'
define a
+
Proceeding
similar to
of
- oo,
-
Dividing (16.4) for i+1 byrP' P2, letting -r we
(16.78)
'r-*+00 TP17-P2
by -rP', letting
^/2i+3)
X2i+5
liM
=
=
=
142
0 if the encounter i + 2 is the end of a bifurcating arc; g" g" junction with a S-arc* made of m,, basic arcs. Using (16.71) and (16.72), we can then generalize (16.80) into
arcs;
is
Proving General Results
16.
=
=
1 if it
a
single
a
formula which 9
Gib, (Mb
covers
+
all
cases:
Ksi)
9
G2sib2
+(1 Solving
the two
,
-
Gi bi (m,
Ksi)
-
G2sib2
Ksi)'Y2i+3
+
(1 +,Ksi)-(2i+5
equations (16.79) and (16-81) for
=
0
.
(16-81)
'Y2i+3 and -Y2i+5,
we
obtain
'Y2i+3
=
^/2i+5
=
Gib, [(Ksi
+
1)
-
91(Mb
+
Ksi)
+
g"(m,,
-
Ksi)]
+
g"(m,,
-
Ksi)]
2G2sib2 Gibi [(Ksi
-
1)
-
91(Mb
+
Ksi)
2G2sib2
In most cases, 72i+3 and 'Y2i+5
are
different from
zero.
(16.82)
Then the coefficients
are
P2i+3
=
P245
=
P1
-
P2
,
b2i+3
'Y2i+3
,
b2i+5
=
7245
(16-83)
17 Mb = 1; 911 = 01 ^(2i+3 vanishes. particular case g' the coefficients P243 and b2i+3 cannot be determined without further computation. Similarly, in the particular case g' = Q, g" = 1, m,, However, In that
in the
case
7245 vanishes and the coefficients P245 and
b2i+5
are
not known.
We can go back to the physical variables, using (15.2), (15.9), (16.63), (16-78), (16-82). We recover the main terms of the expressions (12.109) for a I
T-arc in the
case
0
P*
The first
(16.90)
>PI -P2
inequality results from (16.87); the second inequality holds because
otherwise there would be P2i+4j-1 P1 P2 for all j, and P2, P2i+4j+l P1 the terms 1 and 2 would be negligible in (16.2) for the inside encounters of -
even
-
rank. We have also
P2i+4j+2
P1
-
P*
j
(16.91)
144
Proving General Results
16.
otherwise the terms 1 and 2 would be
again because
the inside encounters of
even
We prove next: if P2i+4j-1
negligible
in
(16.2)
for
rank. =-::
P*, then P2i+4j+l
=
p* also;
and
conversely.
Moreover,
b2i+4j+l The
proof
(1.6.2)
3 in
-
(16.92)
0
b2i+4j-l
(16.85)
is immediate: from
for encounter i +
2j
-
1 is
and
(16.90)
negligible;
we
find that the term
therefore the terms 1 and 2
must balance.
The T-arcs* for which P2i+4j-1 = P2i+4j+l = p* form one or more subsequences inside the T-sequence. We consider one such subsequence, extending
2j'
from encounter i +
to encounter i +
2j". 2i'.
We consider the initial encounter i + the,
in which
T-sequence, P2i+4j'+2 a preceding T-arc*, with P2i+4j'+l case
there is we
find that the term 2 is
b2i+4j'+2b2i+4j'+3 In view of
(16-90),
we
P1
P2i+4j' +2
0
=
0, it is the beginning of (16.88). If j, > 0,
P2 from
p*; applying (16.2) =
pi
at i + -
2j',
p* and
(16-93)
.
have therefore in all
P*
-
=
0, there is a preceding T-arc*, with P2i+4j'+l < P*; 2j', we find that the term 2 is negligible, and therefore p, /2, 32i+4j'+2 p2i+4j' +2, and
#2i+2 applying (16.2)
P2i+4j' +2
From
=
0.
at i +
=
02i+4j'+2b2i+4j'+3
+
Gib,
=
0
.
(16.109)
We define
i.e.
g'
=
g'
=
H(j')', 0 if
j'
single equation:
(16.110) =
0, g'
=
1 if
j'
> 0. Both cases can then be written as a
Results
Proving General
16.
146
#2i+4j'+2b2i+4j'+3
g'Gibi
+
Similarly,we consider the final
0
=
(16.111)
.
encounter
i+2j"
and
we
g"
define
=
H(1-j").
We obtain
-fl2i+4j'I+2b2i+4jII+1 We consider and
lettingr
+
+oo,
we
#2i+4j+2 (b2i+4j+3
==
0
(16.112)
.
Dividing (16.2)
inside encounter.
now an
-+
g"Gibi
b2i+4j+l )
-
for i +
-rP'
2j by
obtain +
Gib,
j
0
=
j,
1'...'j"
+
-
1
.
(16.113) Dividing (16.4)
for i + 2j
-
by
1
-#2i+4j-2 + #2i+4j+2 + GO +G3 (I + Ksi) b2i+4j+l
-rP1 /2 and
=
This
is identical with
equation
-02i+4j-2
+
02i+4j+2
+
-
letting
-r
+oo,
obtain
we
Ksi)b2i+4j-l
0
(16-114)
+
(16.103). Using (16.92)
2G3b2i+4j+l
=
we can
0
+
1,
rewrite this .
.
.
'ill
as
.
(16-115)
(16.104).
This is identical with
ill, j'+I,. multiply now this equation by b2i+4j+l, we sum it for j add also (16.113) for j ill 1, and we use (16.92), (16. 111), i' + 1, We
we
=
=
and
(16.112).
-
-
I
-
We obtain
jil
R
L
2G3
+
2i+4j+l
Gib,
+
g1l
+
Ull
-
if
-
1)]
=
(16-116)
0.
i=il+l
The
b2i+4j+l
non-zero, and there is at least
are
therefore the E is positive. Also there is have thus reached an impossibility. 16.8-1.3
p*
P2i+4j'+2
pl/2. P1
-
We have from
>
0, g"
>
one
term in the sum;
0, ill
-
j'
-
1
0. We
(16.94)
(16-117)
P*
to the first T-arc*. Assume that P2i+4j'+6 > P1 Then the first term is negligible compared to the second. Also we have
Consider
p*.
pj/2 > p*; therefore the terms 3 and is impossible; therefore
P2i+4j'+6
P1
-
P2i+4j+2
P1
This
(16-118)
P*
Applying again (16.4) -
negligible.
to the
following T-arcs*,
we
obtain
generally
(16.119)
P*
We write
#2i+4j+2
liM ++00
T2i+4j+2
'
rpl -P*
7
i
=
.
(16-120)
16.8
As in the previous case, we obtain (16.111) and -rP' and letting -r -+ +oo, we obtain
#2i+4j+'2(b2i+4j+3
-
b2i+4j+l)
+
Gib,
147
.
(16.112). Dividing (16.2) by
0
=
No TT Node*
Appendix:
i
=
j,
+
-'j"
1,
-
(16.121) (16-113). Dividing (16.4) by -rP'-P*
This is identical with
+oo,
we
-#2i+4j-2 multiply
We
j62i+4j+2
+
now
this
Gibi[g'+ g"
+
=
0
letting
7-
(j"
-
=
(16.122)
+
,
equation by b2i+4j+l,
(16.121) for j and (16.112). We obtain add also
we
and
obtain
j'+ 1,...,j"
j'- 1)]
=
we sum
-
1, and
it for
j'+ 1,. -d" (16.92), (16.111),
j
we use
=
-
(16.123)
0.
g' > 0, g" 2 ! 0, j" j' 1 t 0. Thus the only way to satisfy (16.123) 1. But this means that the subsequence 0, g" 0, j" j' by taking g' contains only one T-arc*, and that its ends coincide with the ends of the partial T-sequence. Therefore the partial T-sequence itself contains only one There is
-
is
=
=
-
-
T-arc*.
16.8.2 Total In the
T-Sequence
of
case
a
bifurcation,
total
we
consider
now
the
case
of
a
total T-
sequence, making up the whole bifurcation orbit. The proof follows the same lines as in the case of a partial T-sequence. We consider one period of the
bifurcation of
orbit, from
encounter i to encounter i + n, with i the
T-arc*. We define
p*
beginning
(16.89).
There is P2i+4j-1 = P2i+4j+l = P* either for all T-arcs* or for a non-empty subset of them. In the second case, they form one or more subsequences. We consider one such subsequence. The a
as
in
proof proceeds then as above, with the simplification that there is always a 1. preceding T-arc* and a following T-arc*, and g' 1, g" In the first case (i.e. P2i+4j-1 for all we consider p* T-arcs*), P2i+4j+l again the three separate cases for p*. =
=
16.8.2.1
p*
plying (16.4) term 3 in
pi/2.
>
=
=
Assume that there is P2i+4j+2 > p* for some j. ApT-arcs*, we find that P2i+4j+2 > p* for all j. The
to successive
(16.2)
is then
negligible for
encounters of even
rank,
and the terms
1 and 2 must balance:
b2i+4j+3
b2i+4j+l
-
Using also (16.92),
b2i+4j+l
=
we
b2i+1
i
=
0
for all
obtain
b2i+4j+3
=
b2i+1
Summing (16.4) over every other value 0, which is impossible. G3nb2i+l
and
--
(16.124)
j
for all j
'(16.125)
of i, we obtain the equation Therefore
(16.19),
148
Proving General Results
16.
We define then we
for all
P
P2i+4j+2
#2i+4j+2
as
(16.126)
j
in
(16.100)
and
we
continue
as
in Sect.
16-8.1.1;
obtain n/2
2G3
E b2i+4j+l 2
(16.127)
0
j=1
impossible.
which is
pl/2. Reasoning
p*
16.8.2.2
p, /2
P2i+4j+2
We define then we
for all
#2i+4j+2
in
in the
previous
case,
we can
show that
(16.128)
j
(16.108)
and
we
continue
as
in Sect.
16.8.1.2;
obtain n/2 2
2G3
b2i+4j+l
which is
impossible. p*
16.8.2.3
or
if
A T-arc
T'
arc
and
it is
a
T'
arc
and
a
designated by Tf if
Will be designated by T9 if a
it is
a
Ti
1,
or
arc
and.
-< 1. it is
Te
a
arc
and
a
>
if
it is'A
< 1.
symbols f and g for the arcs'Tf an'd T9, in the same way that symbols i and e for the arcs Ti and Te in Sect. 6.2.1.3. A geometrical interpretation can be given. The unit circle (the orbit of M2) divides the (X, Y) plane in two regions, and the bifurcation orbit lies entirely in one of these two regions (see Fig. 4.1, type 2). Then a Tf arc is one which moves out of the region occupied by the bifurcation orbit, while a T9 arc is one which moves into the region occupied by the bifurcation orbit. By analogy with the side of passage o,, defined by (8.1), we define also the relative side of passage as We define
we
defined
or'.
=
sign(a
-
1)o,
=
sign(a
-
1)sign(xo
-
1)
(17.1)
.
a' is positive if the side of passage lies in the same region orbit, negative if it lies in the other region. With these of
new
notations,
it is
possible
to
as
the bifurcation
simplify somewhat
the results
Chap. 8 for bifurcations of type 2 and to make them independent of A. 1. For a node, instead of (8.43) and Table 8.2, we have: if sign(E'AC) = +1,
only S-arcs
are
present, And
(17.2) If
sign(c'AC) 2. For
an
=
-1, the value of a' is given by Table 17.1.
antinode,
instead of
M. Hénon: LNPm 65, pp. 149 - 179, 2001 © Springer-Verlag Berlin Heidelberg 2001
(8.75)
and
(8-77),
we
have: for
a
S-arc:
150
17.
Quantitative Study
Table 17.1. Value of a' for
a
g
Type
2
type 2 node, for
second f
of
sign(,E'AC)
-1.
arc
1, 2,
f
first
arc
9
1, 2,
U
I
and for 01
I
a
=
+
+
+
(17-3)
'"
-1
=
+
first species orbit E:
sign(c'AC)
(17.4)
8.4.2, it was noted that in a bifurcation of type 2, all sides of 1 is changed and i and e are change their sign a if the sign of a exchanged. For that reason, only the case a 1 > 0 was listed in Tables 8.12 and 8.18. We can now transform these tables into universal tables, valid for all values of a, as follows: (i) we substitute the symbols f and g for i and e respectively; (ii) we make the convention that the signs in the heading are 3. In Sect.
passage
-
-
now
the relative sides of passage.
17.2 Fundamental We will follow
Equations
again the general analysis presented
in
Chap.
11. We consider
bifurcation orbit Q of type 2, and we number consecutively the collisions and the basic arcs implied in the bifurcation as explained in Sect. 11.3.1. We a
consider also
neighbouring periodic
small but not
orbits
f2A
of the restricted
problem for
p
zero.
The time ti, introduced in a general way in Sect. 11.3.1, was precisely case of a bifurcation of type 1 as the time of intersection of
defined in the
the'true orbit with the unit circle
type 2, another definition turns
(the
of conjunction, i.e. the time when M3
velocity of M3 parallel to the
in
movingaxes
orbit of
out to be
at
a
M2) (see
preferable:
crosses
the
collision is
x
we
Sect.
define ti
12.1.1). as
For
the time
axis at encounter i. The
non-zero
(see (11.22))
and is
axis; this guarantees that the neighbouring orbit f2A has a well-defined intersection with the x axis in the vicinity of M2. We define hi as the oriented distance from M2 to M3 at the conjunction; in other words the abscissa of the intersection point is 1 +
hi
sign(hi)
=
x
=
y
.
(17.5)
There is ai
(17.6)
17.2 Fundamental
where ai is the side of passage, defined by (8.1). We consider now the intermediate orbit i (see Sect. sects the
axis at
x
ri(t'i'). We define also the radial velocity r, tion 11.3.1 we obtain
t1i,
Similarly ti,
at
hi
1 +
z
O(p)
+
It inter-
u'i'
as
i(t'i').
=
Rom
as
Proposi-
(17.7)
.
the intermediate orbit i + 1 has
a
conjunction in the vicinity of velocity are r +,, u +,, and
The abscissa and the radial
t +,.
time
a
r'i
O(A)
ti +
=
S
11.3.2).
We define the abscissa of the intersection
time t
a
151
Equations
there is
t +J
=
ti +
O(A)
1 +
hi
+
O(p)
(17-8)
.
17.2.1 Encounter Relations
We consider
matching
the
now
relation
(11.79),
0(pd)
O[p(u
which
we
project
on
the
x
axis:
d(u +,
-
2p cos W
0)
+
V
+
(17.9)
+1
where V is the angle between ib and the x axis. The quantities v, d, p are taken for instance on the intermediate orbit i, at the time t'i'. But there is d
=
p'i'cos W
=
hi
cos
O(p)
p +
(17-10)
.
Therefore
hi(u'i+l For
ho
a
=
2P
u'i')
-
71
z
0(phi)
+
V
partial bifurcation,
O(M)
h,,
,
the
O(p)
=
+
0[p(u'i+1 21
-
u'i')] 71
,
i E C
.
(17.11)
equations (11.71) give
(17.12)
.
17.2.2,Arc Relations We must
now
obtain relations between
these quantities orbit i.
correspond
to the
ri', u'i, ri" u'i' expressing
same
keplerian
the fact that
orbit: the intermediate
For 0, the intermediate orbit becomes identical with the bifurcation orbit between times ti-1 and ti. In fixed axes with origin in M, and an appropriate orientation, and also an appropriate origin of time, the equations
of motion
X
=
are
r:a(e" cos E
Y=ee',E"av/1 t
=
a
in H6non
given
3/2
(E
-
e
-
e)
(1968, Equ. (26)):
,
e2sinE, 11
esinE)
,
(17.13)
152
from which r
a,
=
of
Quantitative Study
17.
elle sin E
e"e cos E)
-
2
deduce
we
a(l
Type
u
=
r
,,Fa(l
the elements of the bifurcation
e are
(17-14)
= -
Elle
cos
E)
ellipse. The Jacobi
constant is
given
by (3. 11): C
2F_'V'a(1
=
-
0 +
given by
Y
cost,
=
The collisions i
.
a
The motion of M2 is X
(17-15)
-
sin t
=
1 and i take
-
(17-16)
.
place for
t
77; therefore
,r, E
we
have
the relations cos T
c:a(E" cos 77
=
sin T
ee'e" a V1
=
3/2
T
=
a
1
=
a(l
(n
0 0,
For M
-
E" e sin 77)
-
still
we
,
e2 sin 77
e"e cosq)
-
e)
-
,
(17-17)
.
use a
system of fixed
axes
with
origin
in
Mi,
such that
the X axis passes through the pericenter and the apocenter of the intermediate orbit. We take the origin of time when M2 passes through the point (1, 0). The motion of M2 is then still given by (17.16). The motion on the intermediate orbit i is X Y
cai
=
(Ell cos E
='ElElfllai
t-t!
=a
V1
3/2
(E
given by
-
ei)
ei? sin E
-
-
Ell ei sin E)
(17-18)
ti*, defined as the time of passage at pericenter (or generally non-zero. are now replaced by encounters. A conjunction happens at time
Note the appearance of
apocenter),
now
Collisions
t'i
-7-, for E
rl
cos
t
=
r sin t -t
=
r UZ
=
=
F-ai
ai
3/2
we
(1
-
we
have then
(Ell cos Ei'
ee'e"ai
=a
from which
Ei';
=
(Ei'
V1 -
-
-
ei)
,
ei?2 sin Ei'
f_"ei sin Ei')
(17.19)
deduce
Ell ei
cos
Ei')
ell ei sin Ei'
- /ai (1
We have similar
-
ell ei
cos
(17.20)
Ei')
equations for the conjunction
at time
ti"
c--
+,r, for E
Ei":
17.2 Fundamental
r'
cos
t'
r ' sin t '
t'
(e" cos Ei"
rl
ef'E" ai
=
t!
-
eai
=
a
3/2
(Elli
-
-
Equations
153
ei)
-
e?2 sin Ei"
e"ei sin Ei")
(17.21)
and ri
0
ai (I
=
-
e"ei cos Ei
ell ei sin Ei"
=
,,Fai (1
-
Ell ei
cos
(17.22)
Ei")
expand now these equations bifurcation orbit.
We the
in terms of the small
17.2.2.1 Bifurcation Orbit. We recall first
some
displacements
basic
from
equations for the
0, AC = 0, i.e. the bifurcation orbit of type 2 (see Sects. 4.4 p and 6.2.1.3). The bifurcation ellipse is characterized by two mutually prime case
=
integers I and J, not both equal to 1, and a direction of motion E' (Sects. 4.4, 6.2.1.3). Its semi-major axis and eccentricity are
() I
a=
2/3 e-
la
-
11
(17.23)
a
In fixed axes, for
a
collision in X
1
1,
Y
=
0, the components of the velocity
are
VX In
0,
=
rotating V.
axes, the
0,
=
vy
6
,
-
components
=
VY
-
e2)
=
6
La
(17.24)
a
are
Ca
1
(17.25)
1.
a
The modulus of the
V=
e',V a(l
Vy.
/2a_-1 V -a
relative'velocity -
is therefore
(17.26)
1
The Jacobi constant is
C
V
11',a::-:1:+1 a
The bifurcation orbit identical basic
arcs.
we can
in such
a
M3
choose the
way
or
the
Each basic
bifurcating arc corresponds
arc
consist of to
a
a sequence of n time interval 271 and
on its supporting ellipse. Therefore, for a given basic origin of time and the origin of the eccentric anomaly that the two end collisions take place at times
to J revolutions of
are,
(17.27)
a
Quantitative Study of Type
17.
154
t'ito
-7rI
=
Ej'o
=
-
(17.28)
7rI
tio
anomaly
to values of the eccentric
correspond
and
2
7r J
Ej'0
7
7rJ
=
(17.29)
.
t z vanish, and r z = r ?I/ = 1. Finally, On the other hand, all variables hi, u *1, 0, z e and e" in H6non (1968) it-is easily shown that
from the definitions of e
(-l)Isign(a
=
17.2.2.2
ti
6"
,
(-l)jsign(a
=
Expansions. We consider
ai =a+ =
1)
-
Aaj
+7rI +
We call AC
ei
,
e
=
Ati
Ej'
+
Aej
-irJ +
,
t/i
.
(17.30)
.
the intermediate orbit i. We write
now
=
1)
-
=
At ,
-71 +
Ej"
AEj"
=
7
+7rJ +
AEj"
.
(17-31)
the,displacement of the Jacobi constant for the true orbit ACi the displacement for the in-
with respect to the bifurcation orbit, and termediate orbit i. We have (see (12.15))
Aci
The A
(17.32)
O(P)
AC +
=
quantities
are
small. We recall that the quantities
We, simplify
z
T
and
with respect to the small
by expanding
the equations
now
quantities, and eliminating
variables. We obtain first from
some
(17.15)
(17.32) Vy
AC
-
a
We
hi, u'j, u'j', ti*
also small.
are
1L2
also express ej
can
Aej
VY -
21a
-
11
21a
Aa.as a
AC +
-
11
Aej
VY
+ 0
(Aa?, Ae?, it)
function of C and aj;
21a
-
11 (
VY
i
Aaj
-
a2
a
we
(17.33)
.
obtain then +
O(AC2, Aa?, p) (17-34)
This relation will be used later to eliminate
Subtracting (17.19c) a
3/2
[Eji
-
from
(17-21c),
Ej' -,E"ej (sin Ei"
This becomes upon substitution of 27rI +
At'j'
At'i AEj"
-
[2-x J +
=
-
we
(a
+
AEj'
Aej.
eliminate
-
sin Ei')]
t7:
(17.35)'
.
(17.31):
Aaj )3/2 -
f-" (e
+
J
Aej) (- 1) (sin AEj"
-
sin
AEi')]
.
(17-36) The last sin
parenthesis'can be
AEj"
-
sin
AEj'
=
written
(AEj"
Using (17.30b) and (17.23),
we
-
AEj') [1
obtain
+ 0 (AEj'
12, AEj 2)]
(17.37)
17.2 Fundamental
At'i 37rJVa-Adi +Vfa-(AEi" AEj') [I
At'j'
155
Equations
=
-
+
-
O(Aai,- Aej, AEi,12, AEir2)]
+
O(Aai2) (17.38)
Using (17.20a), sinti which
=
can
At [j or,
ee'e"Vl
be
+
write
we can
e22i 1
(17.19b)
as
sin Ej
'
ellei
-
cos
(17.39)
Ei'
expanded, using (17.30), (17.24), (17.31),
O(At 2)]
V/a-VyAEi[1
=
+
into
O(Aei,,AE,2)] i
(17.40)
using the equation itself to evaluate O(At 2)'
At'i
=
vI'a__VyAEj[1
+
1'7.41)
O(Aej, AE;2)]
Similarly, (17.21b) gives
At ,' =',,FaVy AEj" [1 We will need
AEj"
-
AEj',
+
O(Aej, AEi'12)]
accurate
a more
expression for the differences At'j'
(17.38).
which appear in
(17.42)
We subtract
(17.39) from
-
At'i
the similar
equation for sinti: sin 4'
ee'e "
sin ti
sin Ej
e2'
1
1
-
ell ej
cos
sin
Ej"
1
-
Ej
ellej
cos
Ej' (17.43)
The left-hand side is treated
pands
above. The expression between brackets
as
ex-
into
AE11 ( qO + q2 AE112 + + q4 AEj,14 i i
_AEr( i qO
+ + q2 AE12 + q4 AE14 i i
where the coefficients qj
are
functions of
(17.44) e"ej.
This
can
also be written
(AEII_AEt)[qO +O(AEI2'AEt/2)].
(17.45)
There is
(-W
qO
and
(17.43) A t'j'
-
can
=
expanded
equation
can now
37r JAaj
be
Va-Vy (A Ej"'
At
Note that this We
(17.46)
1)7
-
A Ej') 1 + 0 (A e j,
cannot be
eliminate
vz, (AEj"
into
-
At'j' A
-
Ej') [1
AE,,2, AEi,/2)]
deduced directly
At'i
from
(17.41)
(17.47 and
between the last equation and
+ 0 (Aaj,
Aej, AE; 2, Agil 2)]
+
(17.42). (17.38):
O(Aa?)
.
(17.48)
156
Quantitative Study
17.
of
Type
2
O(Aa?) can be eliminated by moving it to the left side, which 37rJAai[l + O(Aaj)], and dividing both sides by 1 + O(Aaj). consider now (17.20b). Comparing with (17.39), we obtain
The last term
2
becomes
We
ce'ej
U
_
2 -:7 e?)
N/aj (1
sin t
(17.49)
)
i
from which 1
a
U
At [l
=
aVY
In the
same a
0
=
a
0
u'
VY
(17.50)
At [l
+
O(Aaj, Aej, Ag/2)1
(17.5-1)
again we need a more accurate expression for the difference. This by subtracting (17.49) from the similar equation for 0 and t ,-
Here obtained
which
1
O(Aaj, Aej, At , 2)]
(17.22b) gives
way,
-
+
ec'ej
U
Va-i (1
expands
(sin t '
sin t
-
)
(17.52)
ej
into a
u
-
2
-
is
1
-
At )[l
+
O(Aaj, Aej, At/ 2, At 12)j
(17-53)
Y
We eliminate
Ati"
Ati'
-
/a-v K
37rJAai
a-1
with
(0
(17.47)
u ) [1 2
'
AEj"
and then
-
Aej, u , 2, U 12 A
+ 0 (Aaj,
z
with
AEj'
(17.48): (17.54)
-
replaced the quantities O(At 2)' O(At 12)' 0 (A Ej' 2), O(AEi'12) O(U 12)' using (17.41), (17.42), (17.50), and (17.51). by O(U 2 ) eliminate we Aej with the help of (17.34): Finally Here
have
we
S
71
and
37rJAai
=
a-1
We consider 1
hi-1
(0
now
Aaj
-
a
14
+O(AEi'
a
-
'
u ) [1 z
z
(17.20a). Using (17.8b) sign(a
-
1) Aej
a
+
12
,
Aaj, U 2' U 12'
+ 0 (AC,
AaiAei, AajAEj
-
2
and
expanding,
we
obtain
1AEi12 12
,
(17-55)
2
AeiAEi
,
p)
(17.56)
.
(The expansion to second order in AEj' will be needed later.) We AEj' with (17.41) and (17.50), Aej with (17.34), and we obtain 2(a In the
-
1)hi-l
same
2(a
-
way,
1)hi
=
=
aVyAC
+
vy
Aaj
+
au
2 +
a
(17.22a) aVyAC
and +
vy a
eliminate
O(AC2' Aa?, U 4'
(17-57)
O(AC2, Aa?, U 14' 11)
(17.58)
(17.7b) give Aaj
+ au, 12 +
z
71
17.2 Fundamental
Equations
157
equations (17.11), (17.55), (17.57), (17.58) form a system of 4 relations quantities Aai, u'i, u'i', hi; the 8 other quantities have been
The
for the 4 indexed eliminated. Here from
again,'we can obtain
a more
accurate relation
by subtracting (17.20a)
(17.22a).
hi
-
or, after
'hi
hi-1
=
-aic".ei (cos Ei"
Ei')
cos
-
+
O(p)
(17-59)
,
expansion
-
a
hi-1
1
-
2
I(AE
12
AEf2)[1
_
O(Aai, Aei, AEi2' AEi"2)]
+
+O(A) AEi', AEi",
and after elimination of a.
hi -.hi-i
12
(U
and
(17.60)
Aei
U 2)[l + O(AC, Aai, U/2' U/12)1
_
+
0(/_,) (17.61)
Finally hi
we
substitute
hi-1
-
u'
u
-
Aai(u
2vy
(17.55), obtaining
from
+
0)[1
+
O(AC, Aai U 2' U 12 A ,
+
O(P) (17.62)
This relation
be substituted to
can
of the two relations
one
(17.57)
and
(17.68). We collect and rearrange (17.11), the set of fundamental equations for
hi (u Z+I U '
-
2(a
U
-
0 D -
2A[l + 0(hi, u +,
=
--
-
1)
a2V, =
0))
-
z
z
v
37rI(a
1)hi-,
(17.55), (17.57), (17.62), 'bifurcations of type 2:
Aai[l
aVyAC
+
+
vy
Aai
+
aU 2 z
11 1
hi
hi
-
avy.
Aai(u
z
+
,
0)[1 z
+
+
ho
boundary conditions (17.12)
=
O(p)
,
Note that for
(17-57).
h,,,
=
(17,65)
O(p) we
(17.63)
,
i E A
t
iEA,
(17.65)
z
a
+
0(/,) (17.66)
,
partial'bifurcation
(17.67)
.
could
,
z
,
O(AC, Aai, U 2' U 12)]
for
obtaining
O(AC2, Aa?, U 4' P)
i E A
and the
i E C
O(AC, Aai, U 2' U 12' t,)]
a
31rI
,
thus
just
as
well have used
(17.58)
instead of
of
Quantitative Study
158
17.
17.2.3
Separation
Type
of the Case
n
2
=
1
=1, have very different properties from partial bifurcation (case 2P1) and for a total 1, general bifurcation (case 2TI). Therefore these two cases will be studied separately in Chap. 23, and for now we assume that n > 1. The
with
cases
the
both for
>
17.3 The Case We
apply
v
n
a
0
=
in Sect. 11.4. We consider first
general method described
the
now
basic arc,
single
a
case n
1L = 0, AC 0 0, corresponding to v = 0. As for type 1 (Sect. 12.3), this case is simple and it is not necessary to use the machinery described
the
case
in Sect. 11.4. The solutions
from the bifurcation orbit The first
species family
bifurcation with families. The
n
known:
are
along
one
is excluded here because it
1. Therefore
=
they correspond
to
a
we
have
only
corresponds
bifurcating arc or the bifurcation orbit is made of (If the symbolic sign of the branch,I c'sign(AC),
only S-arcs
We determine
u'j, u'j', hi. hi (u
-
This equation
hi
=
0)
0
total
a
species
a
sequence
is
positive,
6.2.1-3.) variables
Aaj,
(17-68)
0
be satisfied in two different ways:
or
The first solution
Sect.
magnitude in AC of the fundamental equation (17.63) reduces to
=
can
see
the orders of
now
The first
+,
present;
are
to
to consider second
of S and T-arcs. then
displacement
of the branches which emanate from it.
0 i
Ui+1
corresponds
(17-69)
*
to the
case
where the collision becomes
a
node
away from the bifurcation: the impact distance remains zero. The second solution corresponds to an antinode: the velocity does not change. as we move
For the two ends of
hi As
bifurcating
arc,
we
have also
(17.70)
=
0.
a
consequence,
equations,
a
which
can
.
we
obtain for each
be solved
arc
T
or
S
an
independent
set of
separately.
17.3.1 T-Arc
We consider first the
case
of
a
T arc,
running from collision
i
-
1 to collision
i. There is
hi-1
=
hi
=
(17.71)
0.
(17.66) gives 0
=
Aai(u'i
+
u'j')
.
(17.72)
17.3 The Case
corresponding
The solution
Aaj
0
=
U .1'
The radial 0
=
159
0
T-arc is
period is constant along a T-arc family. corresponds to a S-arc of length 1; this case u +u' the next section. (17-64) gives then
to the fact that the
will be treated in =
=
(17.73)
The other solution
U
a
.
corresponds
This
to
v
0
=
(17.74)
.
is the
velocity
at both ends of
same
a
T-arc.
Finally, (17-65) gives
O(AC2, U 4)'
(17.75)
Vf VYAC[1 + O(AC)l
(17.76)
aVyAC
+
au '
+
which reduces to
U
=
The +
U '
=
sign corresponds
to
Te arcj the
a
sign
-
to
a
Ti
arc.
17.3.2 S-Arc
We consider i +
m.
U
the
now
of
case
a
S-arc made of
For the intermediate antinodes
All basic
U
=
Z+j+l
Aai+j
same
=
over
the
-
1
i to
(1.7.69b) (17.77)
.
same
we
supporting ellipse, therefore they
write
basic arcs,
m
we
obtain
(17.79)
.
symmetry of
-U
=
we
running from
(17.78)
O(Aa)
Because of the
Ui'+M
semi-major axis;
U
-
1'...'M
basic arcs,
Aa,
Summing (17.64)
U'i'+M
j
i+j
in the S-arc have the
arcs
all have the
it
m
have from
a
S-arc,
we
have also
(17.80)
i+1
Therefore
U ,1+1, u'i'+m and
From
Z+j
U11
i+j
=
=
Aa
'-
= .
the
O(Aa 2)
Substituting
in
(17.65),
OVy
(17.64)
from
O(Aa)
(17.66), using
hi+j
(17.81)
O(Aa)
generally,
more
U
=
Ac[l
j
,
=
boundary condition hi j we
+
=
(17.82)
1'...'M.
0'...' M
0,
we
have then
(17.83)
obtain
O(AQ
(17.84)
Quantitative Study
17.
160
Incidentally,
(17.64)
and
U'i'+M
Type
2
verify that. all Aai+j have the summing again, we have we
3-7rmI(a
U
-
of
1)Vy
-
V2
AC[i
+
value.
same
O(AC)]
Substituting
in
(17-85)
.
Y
Using the symmetry (17.80), we can compute u +j and 0 (17.64) again and summing over the first j basic arcs, we -equations
U +j
=
(rn
-
2j +2)
37rl(a
-
1)Vy
2V2
AC[i
+
O(AC)]
separately. Using general
,,
obtain the
,
Y
Ulfi+ =(m-2j) j
37rI(a
-
1)Vy
2V2
AC[l
+
O(AQ
j
,
=
1,
m
(17.86)
-
Y
Finally, using (17.66) and summing
hi+j
=
91r 2J2 a(a
_j ('rn
-
1)V2 Y
2V4
over
the first
AC2[1
+
j basic
O(AC)]
arcs,
j
we
=
obtain
0'...'
M
.
Y
(17.87)
17.4 The Case 0 < We consider
try
now
the
case
1/3
0, AC 0 0, corresponding to v > 0. We will 0 to the previous section.for v
to extend each solution obtained in the
present
case.
for
orbits
We
M
use
This
corresponds
to the
:A 0, which are close general method
the
=
asymptotic branches
to the branches for M
=
Step
1. We estimate orders of
case v
=
(see Fig. 1.1).
described in Sect. 11.4. Reference
below to the successive steps. We continue to call T-arc a sequence of intermediate arcs which reduce to a T-arc vious
0
of the families of
magnitude by
or
S-arc,
is
for
made v
>
0,
S-arc for 1v -4 0. extrapolating from the preor
0.
(17.87) suggests that hi is of the order of AC2 inside a S-arc. (17.84), (17.86) suggest that Aaj, u , u,'j' are of order AC in a S-arc. (17.76) suggests k in a T-arc. that u , ui' are of order 0: hi at a node (see On the other hand, some quantities vanish for v of order Their T-arc magnitude for v > 0 (see (17.73)). (17.69)), Aaj in a and 8 later estimated be will only 9). (see below, Steps =
Step hi for
a
a
AC2Y!
suggests the following changes of variables:
Aaj
ACz!
(17.88)
A cxi,*
(17.89)
=
S-arc:
U'i for
=
2. This
=
Acxi*
T-arc:
U'i'
=
17.4 The Case 0
0, a solution being of the order of M 1/6
solution, the distance Steps 8 and 9. Werefine now the also go back to the physical variables.
totic
on
generally
(17.156)
T-arc, (17.66) gives then
Aaj
(17.65)
=
O(p 2/3)
and
(17.64) give
V---VYAC[i + O(P 1/3)]
U 'U ' For
a
(17.157)
(17.158)
R-region, (17.65) gives a2 VY
Aai
VY
AC[l
+
O(Pl/3)].
(17.159)
a R-region. Guillaume (1971, pp. 134equal within O(p2/3). Since the system (17.154) is not exactly solvable, it is not possible to give explicit expressions for the other variables. If a particular solution of that system is known, then the physical variables can be computed with (17.137)
The
Aaj
asymptotically equal
are
noted that the Aaj
135) already
to
in
are
(17.141):
o(pl/6)], ill+jftl/3 [I + 0(111/6)],
u Z+j
=
-Ljx %+jP 1/3[1
U11
=
-L, X
i+j
,2
hi+j
L-j yj+jj_t
2/3
+
[1
+
O(pl/6)]
V2 L, UlVy (a
17.6 The Case We continue to
E
j
E
j
E
1/3
Y
AC
j
-
1)
1/3
speak of
-
1/3
transition
(17-163)
For
are
still small for
>
results for
v
=
1/3 (see (17.156)) suggest
1/3. following changes
of variables:
T-arc:
a
Ui
=
(-e/Ac)1/2x,/*
which is identical with For
v
2. We make therefore the
I
a
Uz
U/I i
(-6 IAC)112XII* i
Aaj
ACZj* (17.164) ,
(17.137).
R-region: =
Aaj -
help
O(AC)
Finally, for the R-nodes, the that the hi
-
with the
still have
Aaj
-
=
z
*1
19.1-3.
it =
1/2(_EI'AC)-1/2X * ?I
ACzj*
U
P
1/2(_4EIAC)-1/2X /* (17.165)
For all encounters:
hi
pl /2(_ 6- IAC)1/2y! z
.
(17.166)
17.6 The Case
3. It will be convenient to make
Step
xil*
Yx! . f,E;-V
=
-L2X
Z*
-L2X
x *
a
Zi*
Yxi
xi
1, Assume -yj > 0. Then -yj -yo > 0. From 0. Using (19.47) again, we obtain -y3 72 > -
(19.47)
we
-
-
=
there exists
no
branch with
w -+
0-.
19.1
207
Properties
19.1.4 R-Jacobian
As shown in Sect. 17.5, Step 7, the Jacobian for the whole bifurcation orbit or bifurcating arc is equal, within a non-zero multiplicative constant, to the product of the R-Jacobians of the individual R-arcs. We must therefore
compute the R-Jacobian for the R-arc defined by
proceed
We x
1
1
X12
11
,x,Y1) 1
X11, 2 Y2
i
A
=
f2
=
Y1 +
f3
=
Y1(X12
xi
xi +
W
W(Xi
+
-
X11)
-
f3i-2 =X ' -X 71
f3i-1
=
f3i
Yi
=
Yi
(X Z+j
-
I
=
An
f3ft-1
=
YA
f3h
YA
S
X11)
-
-
+W
X Z)
Xn
0 0
1
-
=
X11, A YR
can
a
(19.48)
-
be written
W
YA-1 +
1
1
i
=
X Zl)
+
=
0
0
0,
W(4
+
4n
0
(19.49)
0
=
We have eliminated yo, substituting yo have kept yf, and the last equation yfj = 0. For
follows:
0,
W(X s -_
+
i
as
0,
=
Yi-1 +
-
f3ft-2
=
Xn
YA-1,
i... 7
equations (19.1)
The fundamental
(19.1).
in Sect. 13.1.4. We order the variables
as
given value of
w, this is
a
=
0 in the first
equation, but
we
system of M equations for M variables.
The R-Jacobian is
19(fl,
O(xl, X,,Y11 19.1.5
f3fi) 4n 4n YA)
(19.50)
I
...
I
I
I
Stability
We consider the system of equations (19.49) for a given value of w, , with 1 equations for M the last equation deleted. We have then a system of M -
variables. Starting from a given value xi and applying the equations one by one, we obtain successively x1', yi, ..., yfj. In the same way, starting from a variation
dxl,
critical R-arc
dyfj
dxj
we can as a
compute successively dxl', dyl,
dyf,.
We define
a
(19.51)
0
Proceeding
...,
R-arc for which there is
as
in Sect.
13.1.4,
we can
show that
19. Partial Transition 2.1
208
19.1.1. A R-Jacobian vanishes
Proposition
if and only if the corresponding
R-arc is critical. For the whole bifurcation orbit
or
bifurcating
are,
we
have then
Proposition 19.1.2. In a'partial transition 2.1, the Jacobian and only if the bifurcation orbit or bifurcating arc contains at least
vanishes one
if
critical
R-arc.
(19.49)
We have from
x j' dxj
dxi dxl
dyi
dyi-1
dxj dx *
dx,
+j
dx
=
.
dxi dxj
1
dyi
-
dx'
2
dx'
from which
dy, =
dY2
-
-4w
=
-6w
dY3 dx'
(19-52)
dx'
compute the successive derivatives
(19-53)
2w
=
dx'
Yi
we can
dxj
These
2w
4w2
(19.54)
-
y2 8w2
8w 2
Y21
2 y2
8W3
(19.55)
-
-
y 2y2 1 2
equations will be used below.
19.2 Small Values of A The characteristics of the families will be
19.2.1 ii The
=
are
immediately solved
W
For
(w, yj) plane.
into
-W
xi w
in the
I
equations xi
represented
(19.56)
=
2
0, the condition AC
=
O(IL 1/3)
away from transition 2.1 and into the Chap. 18. -
ceases
region 1/3
The characteristic is the line yj = 0. (19.53) shows that there is no critical R-arc for
to be verified:
3
The solutions found for h
suggest that for yj
=
larger
bi(_W2
where the bi and c solutions exist for R be solved
2 and h
values of h there
3
(equations (19.58), (19-65), (19.66))
might
'FW4
+
are
constants. It'can be
=
numerically.
exist also solutions of the form
CW)
4. It
seems
(19.69) shown, however, that
that the system
(19.1)
for h > 3,
no
such
can
only
19, Partial Transition 2.1
212
19.3 Positional Method
"positional method" similar to that of Sect. 13.3. iseasily shown that two characteristics in the (w, yl) plane again, which correspond to different values of n never intersect. This can be verified 1 divide the 2 and 3. The characteristics up to A on Fig. 19.2 for h of the and have into we Propositions 13.3.1 equivalent regions, (w, yi) plane We introduce
now a
Here
it
=
-
and 13.3.2:
Proposition Here
Two branches
19.3.1.
(ii) the
region and
same
again,
we
be
can
will determine in which
joined only if (i) they
branches lie in the
symmetrical
two
same
lie in the
region.
region a branch lies by studying the
relative position of the asymptotic branches. We need to know all asymptotic branches in the (w, yi) plane. For w -+ oo, we have shown in Sect. 19. 1.1 are no other asymptotic branches than those which correspond decomposition of the R-arc into S-arcs. For w -+ 0, similarly, we have shown in Sect. 19.1.3 that there are no other asymptotic branches than those which correspond to a R-arc. A last possibility would consist in a branch
that there to
a
oo,
y, -4
w
-+ wo,
i.e.
branch with
a
-
....
a
But in that case, (19.3) would give iyl, and we would never reach yfj yi
plane.
19.3.1 Branch Order for
This limit
corresponds
Sect. 19.1.1 that there
correspond
to a
to
Y3
(w, yi) =
3yI,
=
passage to the
a
other
decomposition of
case v
0, the order of the values of yi. for the two branches is given the order of the two values of m,,,+, in the sequence
If If
U,,+, Thus, if w
by
0;
3,2,1,
....
.
(19.75)
where the ,case of the R-arc ending in i,, is represented symbolically by mc,+1
0. If
=
w,
(19-75);
if
(19.76)
we use
two branches: w
is reversed:
0
1, 2,3,...;
Finally,
0, the order
E)(w-1)
(19.73a) =
we
we
(19.79)
have then
E)(w 2-3a)
(19-80)
.
a (previous section), the difference between the values of yj for the two branches is E)(w 2) Therefore (19.80) covers this case also, with the
For
0
=
.
proviso that dy, is not a small quantity anymore. As in Sect. 13-3.2.5, we find thus that the branches are organized hierarchically. The characteristics of all branches having a given first arc in common are at a distance O(w-') of each other; they form a first-order packet. Inside such a packet, the characteristics of all branches having in common their first two arcs are at a distance O(w-') of each other, and form a second-order packet; and so on. 19.3.1.5 Results for ii :S 5. The rules established in the preceding sections allow
to determine
completely how the characteristics of all branches are Figs. 19.3 and 19.4 show this ordering for w -* +oo and 5. In these figures, yj increases from w -+ -oo, respectively, up to A bottom to top, as in Figs. 19.1 and 19.2, and packets of first order, second order, etc. have been separated into different columns, with their filiation indicated by line segments. us
ordered in yi.
=
19.3.2 Branch Order for
This limit
corresponds
Sect. 19.1.3 that for
correspond
to
0
passage to the case v > 1/3. It was shown in 0, there are no other branches than those which
a
-+
w
w -+
to R-arcs. We
can
therefore
variable yj used in that Section has the and 17.6, steps 2 and 3). In the
case w
h > 1 there is
In the
branch
no
case
< 0
(e'AC
w
> 0
corresponds
to
the results of Section 18.1. The
sign
there exists no
as
only
branch
yj here
one
w -+
0-
(see
R-arc, for
are
use
same
1).
given by the
(19.81) The
where the R-arc ends in i + 1 has been
represented symbolically by 0, i.e. if the two branches differ already in yi. 4. Note that this Fig. 19.5 shows the ordering of the branches, up to h is essentially a lexicographic order, with the order of the three symbols given by (19.81). case
0. The rule
applies
even
if i
=
=
19.3 Positional Method
215
1112 1121 -113 1211 -
122
-13.1 -14
--2111 2 11 2 12 -2 1 221 -23 -2 311
-32 -3 -4 -5
a
Fig.
=
a
=
19.3. Order of the branches for
4
3
2
w
+oo, A
< 5
(y,
increases
5
upwaxds.)
19. Partial Transition 2.1
216
+
+14
---+
131
+13---
+ 121 +
1211
+
1112
+122 +
2 +
+ + 1
+ 1121 + 113 + 11
+2 +23 +221
+211 + 2111
+212
+3 +32 +311 +4 + 41-
+5
a
Fig.
=
a
=
2
19.4. Order of the branches for
3
w -+
< 5 -oo, R
4
(y,
increases
5
upwards.)
217
19.3 Positional Method
R+R+R
0
have
19.4.3. =
h
The situation is the
on
The branch and goes
on
the other
(Sect. 19.3,3.2).
+
to
1 in Table 8.12 passes
higher
values
of
through
the tran-
v.
hand, the characteristics
are
joined between
We have thus:
Proposition 19.4.4. All branches with a + sign and are joined between themselves at transition 2.1.
n
2! 2 in Table 8.12
branches, therefore, v increases from 0 to a maximal value 1/3, again to 0. It never goes on to higher values. Table 19.1 lists the junctions up to n 6, in the same format as Table 8.12, which was obtained using only qualitative -methods. It can be veriFor these
and then decreases
=
fied that there is agreement for all branches which appear in both tables n
In
(21.22),
term 4 is
compared to term 3. Therefore the latter must be balanced by 2, and at least one of the following inequalities is true: pi-1
>
3qj
and/or (21.23)
3qj
pi
,
negligible
terms 1
(a similar reasoning holds in the particular that the encounter i cannot be the end of the bifurcating arc. We distinguish several cases. (i) pi > pi-1. Term 2 in (21.22) is negligible, therefore terms 1 and 3 must Assume that the second inequality is true
case).
other
It follows in
balance: pi
In
=
3qj
(21.21),
ai
,
term 3 is
=
qj
Consider
(21.22)
and 3
of the
are
pj+j
=
Continuing pi+j The p some
=
3qj
,
,
=
bi
to term
2; therefore
(21.25)
.
with i increased same
in the
3qj
(21.24)
negligible compared
bj+j
'
qj+1
01
=
by
1. Term 4 is still
negligible;
terms 2
order. We obtain
aj+j same
=
202
way,
ai+j
=
we
(j
+
(21.26) have
1)0&
(21.27)
are equal and the a constantly increase. This is impossible because point we must reach the end of the bifurcating arc.
at
pi < pi-1. Term 1 in
(iii) ai
pi
-
=
obtain
we
again
an
impossibility.
(21-22)
We have from
pi-1
ai-1
(21.22) is negligible. Proceeding as in the previous
opposite direction,
but in the
case
Ot
=
243
Properties
21.1
(21.28)
Assume first that ai has the
sign
same
as
bi. Using (21.21) and (21.22),
we
obtain
qj+j
bi+j
qj
=
=
bi
pi+j
,
3qj
=
ai+j
,
=
ai +
jbi3
.
(21.29)
again an increasing sequence: we have reached an impossibility. If ai sign opposite to bi, then ai-1 has the same sign as -bi, and the same reasoning in the opposite direction again gives an impossibility. (iv) pi pi-1 > 3qj. We have This is
has
a
=
ai
=
Continuing, qj+j Here
-(21.30)
ai-1
=
qj
again,
obtain
we
bi+j
,
we
will
=
bi
reach Y,,
never
=
Therefore qj :! 1/2 for all i. 2. Assume now that there exists In
pi-1
(i) pi
Pi > pi-i =
1 + qj
(21.21), qj+1
=
qj
which is
=
ai
,
inequality
=
an
(21.31)
impossibility.
-1/2
< qj
1 + qj. This case is treated as above.
pi-I
Therefore
no
=
qj
can
-1/2 and +1/2. 1/2 with b? 54 1. In (21.22), the qj 1)W3/2. It follows asymptotic value: bi(b?
lie between
3. Assume that there exists of the terms 3 and 4 has
pi-I
ai-1
obtain
1 + qj
(ii) pi (iii) pi
at least
have
qi in the interval
negligible compared
bj+j
,
we
we
=
We have
term 3 is
Continuing, pi+j
.
ai+j
pi 2! 1 + qj
,
Assume that the second
In
a
term 3 is
> 1 + qj
,
0, and thus
negligible compared following inequalities is true:
(21.22),
the
Pi-1
Pi+j
,
one
> -
of the
=
following inequalities
3 2
as
a
A
> =
3 2
-
sum
that
is true:
(21.36)
21. Paxtial Transition 2.2
244
Proceeding
as
in the
previous
cases,
we
reach
again
an
impossibility.
We have thus shown that for any given i, either qi :! -1/2, or qi = 1/2 and bi = 1. We recognize in the second case the T-arcs, and we recover
Equ. (21.14).
i
4. We consider
sequence of basic
a
1/2.
for which qi
arcs
We
assume
that this sequence is maximal, i.e. it is not possible to extend it on one side or the other. For the interior encounters of this sequence, (21.21) shows that
1/2.
pi
by
i +
1,
Pi+1
A
=
(21.21) gives Pi+j We
1/2
Assume that pi > terms 3 and 4
then:
of them. In
qi+'2
1/2. Continuing
ai+j
ai
=
0; this
(21.22)
to term 2.
with i
replaced.
It follows that
(21.37)
ai
reach Y.,,
never
one
ai+1
A
=
for
negligible compared
are
and
,
is
in the
same
1/2
qi+j
Way,
we
obtain
(21.38)
.
impossible.
it
follows that pi = 1/2 for all interior encounters of the sequence. We recognize a R-arc, and we recover the asymptotic equations (21.15).
1/2 with bi junction of a T-arc and a R-arc, there is qi T-arc, and qi+l < -1/2 for the T-arc. (21.21) shows.then that
5. At the
for the pi
We
=
=
-1/2,
recover
ai
=
Equ. (21.18). junction of two
1
(21.39)
:F1
6. At the
=
T-arcs with
opposite signs,
we recover
similarly
Equ. (21.17). 21.1.2 Variational
Equations
We derive here variational
for W
--+
equations similar
+oo to those
Which
were
obtained for
type 1 in Sects. 13.1.2 and 19.1.2.
UN the successive R-arcs or T-arcs. positions of the nodes, with io ia7 07 iN that the from extends so arc U,, i,,-, to i,,,. The -number of basic arcs in n, 1 if U,, is a T arc.) U,,, is m,, i,, (m,, Rom the two initial values YO and X1, using (21.1a), (21.1b) and (21.16) in turn, we can compute successively the values of Z1, Y1, X2, Z2, We variational the we now assume compute equations: corresponding arbitrary infinitesimal variations dYO and dX1 and we compute the corresponding variations of Z1, Y, We have We
name
U1, U2,.
We call iO i il i
=
.
.'
...
U,
...
7
.
.
.
,
iN the
=
=
-
....
7
dZi
=
dXi+l
....
dYi
2XidXi, dXi
-
dYi y,2 i
=
dYi_1
+
ZidXi
+
XidZi
,
(21.40)
For a given arc U, we. compute now the final variations dXi.,, dYi,,, as functions of the initial variations dYi._,, dXi,,,-,+,. We consider first the
21.1
where U,, is
case
(21.40b)
or,
ia-1
-
=
1, and using (21.40a) and
have'
we
dXj.
i,,,
T-arc. Then m,,
a
245
Properties
dYi.-,
dYi.
dXi.-,+,
=
+
(Zi.
+
2Xj2 )dXj.
(21.41)
using the asymptotic expressions (21.12) and (21.13),
dYi.
dYi,,,-,
=
We consider
2WdXi.-,+, [1
+
the
now
+
O(W-2)].
where U, is
case
(21.42)
R-arc.
a
(21-40a)
and
(21.40b)
can
be combined into
dYj
Using
=
the
(21.40c) dYj
dYi-I
=
asymptotic expressions (21.15), dYi-1 =
=
rewrite this
-
WdXj[1 dyi
dXj
-
WY?3.
[1
+
O(W-2)],
+
O(W-2)],
-W-1ajdYj.,-, [] + bjdXi,,,-,+, cjdYi.-, []
dYj where
we
-
=
aj+l cj+l
i
bi
0, =
=
aj +
1
C,
,
Cj
bj+l
_2
Y3
and
dj+l
cj + aj+l
dXj.
dYi.
bj, we
i" and aj,
bj
=
dl
1,
(21.45) bj,
Pj,
dj
+
dj
=
1,
dj
yj2 +
bj+l
(21.46)
.
-W-1a,,,.dYj.-j [] + b,,,.dXi.-,+,[] [ ] Wdm,,, dXi,, +1 [ ] cm,,, dYi.
(21.47)
-
-
bi
i,
+
i,,-, +
dXi.-,+,
cj, dj are positive. have the final variations
=
.
-,
1
These equations cover also the (21.42), if we take in that case
=0,
i,,-,
and
recursively by
=
=
=
=
[1 + O(W-1)],
have used the abbreviation
Note that all aj, In particular
a,
i
WdjdXi.-,+,[]
numerical coefficients defined a,
equation
(21.44)
-
dXj
,
we can
i Starting from the initial variations dYi.-, j applying these relations, we find that generally
where
are
(21.43)
(Zl'+ 2Xi2)dXi.
as
dXi+l
and
+
1
C,
,
case
=0,
of
a
T-arc, described by (21.41a) and di
=
-2.
(21.48)
We compute now the initial variations of the next arc U,,+,. We already dYi,,,. Since i,, is a node, we have from (21.17) and (21.18): Yic,
know
E)(W-1/2). dXi.+,
From
(21.40c)
dYi,,,
YZ
we
obtain
c`dYj.-, [] + YZ
YZ
dXic,-,+,[]
(21.49)
246
21. Partial Transition 2.2
(21.47b)
The equations
(21.49) give
and
the initial variations of
functions of the initial variations of U,,. We we obtain
dYi,,,
=
dM2 dM3 WCi-1 CMI y2
dYo
WI
iterate these
dMI dM2
dm.
...
-
y2
Y!
dYo[]
+ W'
YZ dmi d112 Y2
U,,+,
as
equations and
dm'
...
Y2 il
cm, drn2 dM3
Wa -1
dXj. +1
drn,,,
Y2
can
...
dX,[] dm
dX,[]
Y!
(21.50) The orders of
19yic
magnitude
are
OYi.
()(W2a-2),
=
ayo
19Xi.+l.
'9xi.+1
()(W2a-1)
=
C9Y0
E)(W2a-1)
ax,
(w2a
ax,
(21.51)
The variations cm,
are strongly amplified, after each node. always positive. On the other hand, dmx is negative if U,
is
positive if
it is
is
a
T-arc,
R-arc.
a
21.1.3 Jacobian
We
proceed
as
the variables
in Sects. 13.1.4 and 19.1.4. To compute the follows:
X1A7Y1)x2iz2,Y2i The fundamental =
X12
f2
=
Y1
f3
=
Y1(X2
_
-
Z1_.W
ZIX1
=
Xi2
f3i-1
=
Yi
f3i
Yi(Xi+l
_
-
f3n-2
=
Xn2
f3n-1
=
Yn
f3n
Yn
=
=
X1)
-
f3i-2
=
...
7
Yn-1, Xni Zn7 Yn
equations (21.1)
fl
=
order
(21-52)
0,
0, -
1
=
0,
Zi_W=O,
Yi-1
-
=
can
-
_
-
xi)
zixi -
1
=
=
0
0
=
0
1
Zn _W=O,
Yn-1
ZnXn
=
0,
0
Yo, substituting Yo 0. kept Yn and the last equation Yn
For
we
be written
We have eliminated have
Jacobian,
as
=
0 in the first equation, but
we
=
a
given value of W, this
The Jacobian is
is
a
system of 3n equations for 3n variables.
21.1
49(h)
...
f3n)
i
247
(21.54)
X., Zn, Y.)
19(XI, Z1, Y1, 21.1.4
Properties
Stability
We consider the system of equations (21.53), with the last equation deleted. 1 equations for 3n variables. Starting from a We have then a system of 3n -
given value X, and applying the equations one by one, we obtain successively Y,,. In the same way, starting from a variation dXj, we can Z1, Y1, dYn. We define a critical bifurcating arc compute successively dZj, dYj, as a bifurcating arc for which there is ...'
...,
d
Y,,- 0.
(21.55)
dXj
Proceeding
in Sect.
as
13.1.4,
show that
we can
the Jacobian vanishes
Proposition 21.1.1. In a partial transition 2.2, and only if the bifurcating'arc is critical.
(21.53)
We have from
dYj
dYj-j
dXj
_jX-1
dXi+l
+
from which
dYj
dY2 dY3
dYj
Yj2
dXj
_
_.(3 X2
(21.56)
(21-57)
W,
-
W)
+
(3X22
-
W)
+
(3X22
-
1
dXj dXj
compute the successive derivatives
we can
(3X2
dXj
W'
dXj
(3X2
dXj
-
1
3X2
dXj
(3X?
dXj -
dXj
W) (3X22
-
-
-
(3X21
W)
-
W)
+
W)
-
W) (3X32
-
W)
-
W) (3X22
-
y2 Y
W)( 3X22 y2
(3X32
(3X22
-
y2 1
W) (3X32
-
W)
(21-.58)
W)
W) (3X32 2 Y
-
W)
1
1 +
,
(3X2
_
.
y2 1
-(3X2
if
-
2
Y'2 W)
2
(21.59)
21.1.5 Branch Notation
In
Chap. 18, RRR
the R-arcs have been
...
R
represented by
(21.60)
21. Partial Transition 2.2
248
-11
where there + or
-
are
sign
RRR
...
where there
Now
ft letters R. Each of them stands for
is the
sign
R
are
of
one
yi.
Similarly,
a
R-orbit
one
basic arc, and each
was
represented by
(21-61)
,
A letters R.
generalize this notation to an arbitrary bifurcating arc or bifurcation orbit of type 2, made of R-regions and T-arcs. It will be represented by a sequence of alternating letters and signs. Each letter represents one basic arc and is either R, inside a R-region, f for a Tf -arc, or g for a T9-arc. Each sign is the sign of the corresponding yj as before. An example of a bifurcating arc is thus: f -g+R+R-R-g. Not all sequences are permitted. As shown by Table 17.1, a f symbol is always preceded by a + sign (or by the beginning of the bifurcating arc) and followed by a sign (or by the end of the bifurcating arc), and the converse is true for a g symbol. If theses conditions are satisfied, then the sequence represents one and only one branch. we
-
21.2 Small Values of
n
The characteristics of the families will be
represented
in the
(W, Xj) plane.
Because of the symmetry V (Sect. 21.1), it will be sufficient to study the case X, > 0, i.e. the upper half of the plane. (For n > 1, we deduce from
(21.1)b
for i
21.2.1
n
This
=
=
1 and from
never
vanishes.)
1
belongs
case
Property 1, Sect. 21.1, that X,
to the bifurcation 2PI and will be treated
separately
in
Sect. 23.2. But it is of interest to consider it also here, in the context of the transition 2.2.
The
equations reduce
x21 -Z,-W=O, Therefore
X1
we
Z1X1
=
0
(21-62)
-
have two solutions:
0,
Z,
=
0,
X,
=
=
to
(21-63)
-W,
and
Z,
=
Vw-
(21.64)
.
Using the asymptotic values (Sect. 21.1.1), we identify easily the 4 branches (Fig. 21.1). The left and right branches correspond to a R-arc made of a single basic arc; the arc respectively. For W move
-4
upper and lower branches
correspond
a
Tf and
a
Tg
0, the condition AC O(IL 1/2) is not satisfied any more: we continuation will be studied in Sect. 23.2. =
out of transition 2.2. This
(21.57)
to
shows that there is
no
critical
arc
for W
-,6
0.
21.2 Small Values of
249
n
2
X1
R
f
R
0
9 -2 2
1
0
-1
-2
W
Fig.
21.1. Chaxacteristics for 2.2P1 transitions.
21.2.2
The
n
=
2
equations
are
X2_Z,_W,=O, 1 2
X24
Y1
-Y1
-Z2-W=Oi
This system
can
be
explicitly
O'= X1Z1 + X2Z2= Thus
a
first solution
Z1X1
-
-
solved.
(X1 +,X2) (X21
corresponds
to solutions with X, > 0,
Z2X2
we
X,
to
=
-X2
=
A/W
YJ
=
+
X1)
0,
(21.65)
0
Combining -
-
the
equations,
2
(21.66)
W).
X1X2
+
X2
X2
0.
Restricting
=
_
our
attention
rj!L-T ; W 2
1
Z1 = Z2
W T
,-2 W -2
--
)
2
V/W--2-2
(21-67)
2
The
XI
asymptotic branches for W
-4
12W
Z2
=
1
-X2
and for the lower
X1
=
X2
-
obtain
we
find -2
X1
YI(X2
0)
=
Z1
+oo are, for the
-W,
Y,
upper'signs, W
(21.68)
signs
"1W
Z1
=
Z2
2W'
Y,
2 N/W-
(21-69)
asymptotic expressions of Sect. 21.1.1, the numerical values of yj obtained in Sect. 18.1.4 and the branch notation defined in Sect., 21.1.5, we find that these two branches correspond to R-R and T-T respectively.
Using
the
250
21. Partial Transition 2.2
The
stability
d Y2
can
-8W (W2
=
dXj
This vanishes for W
for W
3/2
=
the second
be determined from
-
=
(8 W2
2)
,F2,
with the upper
12)VW2
-
which
(21.58): -
corresponds
2
to
(21.70)
.
an
sign, which corresponds
family (presently
described).
to be
A second solution is obtained
W,'and
extremum in
to the intersection with
by setting the
second factor in
(21.66) equal
to 0:
X2_X,X 2 1 Using (21.65)a
+
X2'2_W=O.
(21.65)c
to
(21.71)
to eliminate
X1, Y1, X2,
we
obtain
an
equation
for Zj:
Z41
+
WZ3
+
Z12
WZ1
+
+ 1
It will be convenient here to The other variables
W
=
V1 -
use
expressed
are
Z12 +1 Z, (Z21 + 1) +
-V 2
=
X,
(21.72)
0.
Z, as
as
independent variable instead of W. ZI, in the case X, > 0, by
functions of 1
-Z1 (Z2+ j) 1
-ZI ZI
Z1
-
Y,
The variations
X2
-
1
as
+1
Z,
-
=
from
runs
Z, -oo
to 0
are
1
Z2 shown
zi
by
Table 21.1.
Table 21.1. Variations of the variables when Zi increases from
zi
-00
+00
X, Y,
I
0
1
+00
2
'-
0
-1
I
,-
0
1,
,-
0
I
'-*&.
-00
V-2
X2
-00
Z2
0
V-2-
-oo
to 0.
,- +00
A/21
0
(21.73)
-
There
Z,
-4
0
X,
are two branches W --+ +oo, corresponding to Z, -+ -oo and respectively. The asymptotic values are, for the first branch:
_
X2_
W-3/2
zi
_W1/2'
Z2
-
and for the second branch:
-
-W,
-W-1
y11
_
_W-1/2
1
(21.74)
21.2 Small Values of
X
_
X2_
W1/2
_W-1
Z
_W-3/2,
Z2
-
251
n
_W-1/2
y
(21.75)
-W
Using the asymptotic expressions of Sect. 21.1.1, correspond to R-T and T-R respectively. The stability is determined from (21.58):
we
find that these two
branches
z13 (
dXj This has
a
+3 Z2 1
1)2(Z41
2(Z,2
dY2
7
,
2
+
1,
or
W
3/2, corresponding
=
shown
are
an
Here it is sufficient to show the
Fig. 21.2).
on
to both
family.
intersection with the other
an
The solutions
(21.76)
1)
double root for Z2 1
extremum and
+-1)
W > 0, X, > 0. We have a trident (Sect. 13.3.3.2), formed by the two branches of symmetric orbits R-R and T-T and the two branches of
quarter-plane
asymmetric
orbits R-T and- T-R.
X1 2
f-g f-R
R-R
R-g 0 3
2
1
0
W
Fig.
21.2. Characteristics for 2.2P2 transitions. Also shown: 2.2P1 transitions
(dotted lines).
21.2.3
The
n
=
equations X21
_
Z,
X2_Z 2 2 '
3
_
_
2
X3'- Z3
-
are
,
Y1
,
Y2
W
=
0
W
=
0
W
=
0)
-
-
-Y2
z1X1 YI -
-
=
Y1 (X2
0
Z2X2
Z3X3
=
=
0
0,
-
X1)
Y2(X3
-
-
1
=
X2)
0
-
1
=
0,
(21.77)
-
It seems that the general solution can only be obtained numerically. The equations can be solved, however, in the particular case of symmetric orbits. We have then, from (21.8): X1 Y2, X2 0, and the Z3, Yj -X3, Z, equations reduce to =
=
=
252
21. Partial Transition 2.2
X2_Z1 _W=0, 1 -Z2
-
W
from which
X1
Y1
The
X, y1
4
Using
Z1
X/W-24
-X3 y2
_
_X3
=
=
y2
W-1/2
ZI
I
_WI/2
_
_
X2
-
1
=
0,
Z2
=
0
=
W T
Z3
V W-2
Z3
(21.79)
-W
signs,
W,
-
Z2
1
4
2
oo are, for the upper
-+
=
=
=
-W
(21.80)
1
signs
W1/2,
_W-1/2
_
-Y1X1
0,
0,
>
PW2-::4:
and for the lower
y1
X,
asymptotic branches for W
=
X,
in the case
Y2
=
=
(21.78)
obtain,
we
zIX1
-
0)
-X3
=
=
=
Y1
Z1 X2
1
=
Z3
=
-
-W-1,
0,
Z2
=
-W
(21.81)
-
the
asymptotic expressions of Sect. 21.1.1, we find that these correspond to R-R-R and T-R-T respectively. The stability is determined from (21.59):
two
branches
dY3
2
dX1
W3(W2
-
This vanishes for W
4). 2 (W4
=-
-2 W2
-
24
6) ViW
2, which corresponds
to
an
(21-82)
extremum in
W, and
for
W
to
+2 V-(1(1-73) 3(1
+
03)
(21.83)
2
with the upper sign; numerical computations show that this intersection with another family.
corresponds
an
21.3 Positional Method We introduce
now
a
"positional method"
similar to those of'Sects. 13.3
and 19-3.
Here
again, it is easily shown that two characteristics in the (W, Xi) plane correspond to different values of n never intersect. This can be verified 1 and 2. The characteristics up to n on Fig. 21.2 for n 1 divide the (W, Xi) plane into regions, andve have which
=
Proposition same
21.3.1.
region and
(ii)
-
Two branches
the two
can be joined only if (i) they lie in the symmetrical branches lie in the same region.
21.3 Positional Method
21.3.1 Branch Order for W
253
+oo
-+
we will determine in which region a branch lies by studying position of the asymptotic branches, which in the present case correspond to W -+ +oo. We need to know all asymptotic branches in the (W, Xj) plane. For W -+ +oo, we have shown in Sect. 21.1.1 that there are no other asymptotic branches than those which correspond to a sequence of
Here
again,
the relative
T-arcs and R-arcs. It
there
(W,
branches with W
are no
0. A last
-+
possibility
-+
oo, W
Wo,
-+
i.e.
-
would
a
branch with
would consist in
a
vertical asymptote in the Xj) plane. But in that case, the equations (21. 1) would give successively yi _ iX3, y1 _ X3 2 X3, and we X2, y2 X2, X1, Z2 ...' 1 1 1 1 1, X2
branch X,
Z,
shown in Sect. 21.1 that W > 2- 1/2 , and therefore
was
a
-
never
reach Y,,
=
0: this is
given large value. of W, using the condition Yo given value of X1, we can compute the successive
2 1.3. 1. 1 Variations. For
from any
0 and
impossible.
a
starting Z1, Y1, X2, Z2, Y2, ...from (2 1. 1). We consider the value of X, which corresponds to some particular branch of order n (so that the computation ends in Y,, 0). We apply now a small variation dXj, -and, we compute the Me can use the computations made corresponding variations of Zl,'Yl, in Sect. 21.1.2, setting dY0 0. We obtain in particular values
=
....
dYj,
=
dXj
_W,,,d,,,jd
112
Y
We will need the order of
dYj, =
dXj where
C
'd,,,a.
(21-84)
y.2
magnitude and the sign
E)(W2a-1)
sign.
(dYi,,,
(21-85)
dXj
is te number of T-arcs in the
bifurcating
arc.
Relative Positions of Two Branches with Initial Common now two branches for which the arcs U, to U," are the same, but the continuation is different: either the arcs Uc,+, are differeAt in the two branches, or the arc Uc,+, does not exist in one of the two branches 21.3.1.2
Arcs. We consider
(the bifurcating
are
ends in
ic,).
ic, given by (21.12) asymptotic expressions of Xj, Zi, Yj up to i to (21.18) are the same for the two branches; therefore these quantities differ by O(W-112 ) between the two branches. For W large enough, this can be made as small as desired. Therefore these differences will be called dXj, etc., and the above results on small variations can be applied. We consider first the case where U,, is a R-arc. Then if U,,,+, exists, it must be a T-axc. From (21.17) we know that Yi. is positive (resp. negative) if U,,+, is a Tf arc (resp. T9 arc). If U,,+, does not exist, i.e. if the bifurcating 0. Thus the order of the values of Yi. for the arc ends in i,,,, we have Yi. two branches is given by the order of the corresponding symbols (for U,+,) The
=
=
in the sequence
254
21. Paxtial Transition 2.2
g;
0;
f
where the
(21.86)
case
of the
bifurcating
We consider next the be
Tf
a
arc or a
ending
R-arc. If it is
a
Tf
arc,
in
a
we
i,, is represented by a 0. arc. If U,,,+, exists, it must
T9
(21.17):
have from
'W-1/2 + O(W-5/2
Yi If it is
arc
where U,, is
case
(21.87)
2 a
R-arc,
we
W-1/2
yi
have from
_
W-312X,
(21.19) +0 (W-5/2).
(21.88)
Here x, is the value which appears in Table 18.1 and which corresponds to the first basic arc of the R-arc. The order of the values of x, for various R-arcs can be read from that table. If we consider for instance all R-arcs up to
n
=
3, the
x, increase
along
the
following
R+R+R, R+R, R+R-R, R, R-R+R, R-R, This is the
sequence:-
R-R-R
(21-89)
-
same lexicographic as in Fig. 19.5, read from top to bottom. Since x, is multiplied by a negative factor in (21.88), and considering also (21.87), we find that the order of the values of Yi. for the two branches is given by the order of the corresponding symbols in the sequence
0;
f;
order
R-R-R, R-R, R-R+R, R, R+R-R, R+R,
This sequence should be
(21.90)
R+R+R.
appropriately enlarged
if R-arcs with
n
> 3 are
present.
Finally,
if U,, is
a
T9 are,
a
similar argument leads to the sequence
R-R-R, R-R, R-R+R, R, R+R-R, R+R, R+R+R;
g;
0
We have thus determined the order of the values of
(21.91)
.
Yj.,
in all
cases.
Using
(21.85b), we obtain then the order of the values of X1, i.e. the relative position of the two branches in the (W, Xj) plane: if C, the number of T-arcs in JU,,..., U,,}, is odd, the sequence (21.86), (21.86), or (21.86)can be used as it
is; if C
is even, the sequence must be inverted.
21.3.1.3 Relative Positions of Two Branches with Different First Arcs. The method of the previous section does not work when the two branches differ already in their first arc (this can be taken to correspond a 0). In that case we compare directly the values of X1, which are given by (21.13) if U, is a T-arc, (21.15c) if it is a R-arc. We find that, taking again as an example all R-arcs up to n 3, the order of the values of X, for the two branches is given by the order of the symbols corresponding to U, in the
to
=
=
sequence
g;
R+R+R, R+R, R+R-R, R, R-R+R, R-R, R-R-R;
f
.
(21.92)
21.3 Positional Method
21.3.1.4 Packets. From
dYi,,, if
dYi.
(21.88)
we
have, for
>
a
0,
a
(21.93) R-arc in the two
branches; and
E)(W-1./2)
=
in all other
(21.94)
cases.
(21.85a)
From
and
E)(W-3/2)
=
exists and is
U,,,+,
-(21.87)
255
have then
we
dXj
=
E)(W-1/2-2ce)
(21.95)
dXj
=
E)(Wl/2-2ce)
(21.96)
or
respectively. For
a
0
=
(previous section),
(W-1/2)
the two branches is in all other
cases.
Therefore
the difference between the values of Yj for
if U, is
(21.95)
a
and
R-arc in the two
(21.96)
cover
this
branches, E)(Wl/2) case also, with the
dy, is not necessarily a small quantity anymore. again we find that the branches are organized hierarchically. The characteristics of all branches having a given first arc in common are at a distance O(W-3/2) of each other; they form a first-order packet. Inside such a packet, the characteristics of all branches having in common their first two arcs are at a distance o(W-1/2) of each other, and form a second-order packet; and so on. proviso
that
Thus here
n :! 3. The rules established in the preceding sections completely how the characteristics of all branches are 3. X, increases from ordered in X1. Fig. 21.3 shows this ordering up to n bottom to top, as in Figs. 21.1 and 21.2, and packets of first order, second order, etc. have been separated into different columns, with their filiation indicated by line segments. The left column reproduces (21.92). The figure has a symmetry: it is invariant if we exchange top'and bottom,the + and signs, and the symbols f and g. This is a consequence of the V symmetry (Sect. 21.1, Property 5).
21.3.1.5 Results for
allow
us
to determine
=
-
21.3.1.6 Other Method. Instead of
building the branches by successive adding a T- or R-arc, we can build them by adding each time one basic arc. Fig. 21.3 is then transformed into Fig. 21.4. Successive columns correspond now to successive values of n. A regular strucpackets, each
time
ture emerges.
(The
vertical order of the branches is of
course
the
same as
in
Fig. 21.3.) This structure
can
be explained
by reworking
the
building
lished in Sects. 21-3.1.2 and 21.3-1.3. We consider first the branches have initial
different.
Using
the
find that the order
common
basic
arcs
up to arc
i,
rules estab-
case
where two
but the continuation is
asymptotic expressions (21.17), (21.18), and (21.15b), we of the values of Yi for the two branches is then given by
256
21. Partial Transition 2.2
f
f-g f-g+f f-g+R f -R+R f -R+f f-R f -R-g
f -R-R
R-R-R
R-R-g R-R R-R+f R-R+R
R-g+R R-g+f R+f
R+f -g R+f -R
R+R-R.
R+R-g R+R
I+R+f
R
R+R+R
g+R+R
-::Z
g+R
9+R+f
g+R-R g+f-R g+f -g 9
=
Fig.
1
a
=
21.3. Order of the branches for W
increases
upwards.)
--+
2
+oo,
3
n
:! 3, found using packets.
(Xi
21.3 Positional Method
257
f
f -g+f
f-g+R f -R+R f -R+f
f-R f -R-g f -R-R R-R-R R- R-g
R-R R-R+f R-R+R
R-g+R R- 9+f R
-
g
R
R+f R+f -g R+f -R R+R-R
R+R-g R+R
R+R+f R+R+R +R+R
g+R+f gtR
g+R-g g+R-R g+f
-
R
g+f-g g+f 9
n
Fig.
n
=
2
21.4. Order of the branches for W -+ +oo,
increases
upwards.)
n
3
n:! 3, found using basic
arcs.
(Xi
258
21. Partial Transition 2.2
the order of the
symbols representing the sign of Yi
and the next basic
arc
in
the sequence
-R, -g, 0, +f, where the
case
If the basic
only
(21.97)
+R
bifurcating
of the
arc
ending
in
i, is represented by
a
0.
T9 arc, the following sign cannot be a -, and therefore the last 3 elements of the sequence are present. Similarly, if the basic aTc
i is
a
i is a Tf arc, only the first 3 elements of the sequence are present. We deduce the relative position of the two branches in the (W, X1) plane in Sect.21.3.1.2: if (, the number of T-arcs among the basic arcs 1 to i, is
arc
as
odd, the
sequence
(21.97)can
be used
as
it
is; if C
is even, the sequence must
be inverted. If the two branches have different first basic arcs, the order of the values given by the order of the symbols representing the first basic arc in
of X, is
the sequence g, R, f
This
x
construction method shows that the number of
new
branches for 8
(21-98)
.
3
a
given
n
asymptotic
2 is
n-2
(21.99)
21.3.2 Results
Symmetries. We use now the positional method systematically junctions. We will make use of the symmetry E; it inverts the sequence of symbols and exchanges the symbols f and g. We will also need the symmetry E': it exchanges the signs + and and the symbols f and g. Finally the symmetry EV inverts the sequence and changes the signs. An orbit cannot be symmetric under V (except in the singular case n 1) since the signs are changed. But an orbit can be symmetric under EV, provided that n is odd. An example is: R-g+f -g+R. (For n even, this is not possible since the central sign changes.) Exactly as was the case with E (Sect. 7.3), this property is invariant along a characteristic. Therefore a branch of symmetric orbits under EV must be joined to another branch of symmetric orbits under EV, and a branch of asymmetric orbits must be joined to another branch of asymmetric orbits. This property will be used 21.3.2.1
to determine the
-
=
below. n = 1. In the special case n 1, there are three branches for +oo, whose relative position is given by the first column.in Fig. 21.4, and in addition there is one branch for W, oo (Fig. 2 1. 1). They form a
21.3.2.2
W
-
trident.
21.3 Positional Method
2. In the
21.3.2.3
n
there
only
are
respect
=
to the axis
schematically Fig. 21.2). For
n
X,
=
0
case n
2,
we
There
1,
we
have W > 0
Since the
(symmetry V),
figure
is
(Sect. 21.1),
a
4
new
branches,
and
symmetric with
it will be sufficient to represent
quarter-plane
W >
0, X,
represent the already known junctions
are
These branches form
>
-4 +oo.
the characteristics in the
=
(Fig. 21.5).
generic
branches for W
259
all of which lie in the
trident, and their junctions
can
as
> 0
(as
in
solid lines
same
be established
region.
(dashed
lines).
f.
f-g
17
f-R R-R
R-g R
Fig.
21.5. Junctions for
n
1
R
(full lines)
and
n
=
2
(dashed lines); X1
> 0.
are 3 groups of 4 branches. Each group lies in one The central region (Fig. 21.6). group forms a trident. The upper group is made of asymmetric orbits only. Here we make use of the second part of
2 1.3.2.4
n
=
3. -There
Proposition 21.3.1. The symmetry E sends the 4 branches intothe lower half-plane X, < 0; therefore it is more convenient to use the symmetry EV. Under that symmetry, two of the branches (f -g+f and f -R+f) are invariant and remain in the upper group, while the two other branches (f -g+R and f -R+R) are
changed
into
R-g+f and
R-R+f and
Therefore, the first two branches must be
to the lower group. and the last two also. A
move
joined,
similar argument establishes the junctions in the lower group. We remark that two joined branches always have the same
generally I
true and is
Four branches
a
are
consequence of Broucke's
signs. This
is
principle.
made of orbits symmetric under EV: f -g+f, f -R+f, Sect.. 21.3.2.1, these branches are joined be-
R-R+R, R-g+R. As predicted in tween thems1eves.
4. In Fig. 21.7, the already known junctions are repres'ented 2 and n 1 lie outside of them and do 3; the junctions for n not contribute any further division of the plane into regions. This appears to be generally true: it is sufficient to consider the already known junctions for 1 when we try to find the junctions for n. n The new branches form 5 groups of 4 branches, which we label G, to G5, and 2 groups of 8 branches, H, and H2. We study these groups in turn. If we 2 1.3.2'. 5
only for
-
n
n
=
=
=
=
260
21. Partial Transition 2.2
f
f-g f -g+f f -g+R
I 4-
f -R+R f -R+f f-R f -R-g f -R-R R-R-R
R-R-g R-R R-R+f R-R+R
R-g+R R-g+f R-g
RI
Fig.
21.6. Junctions for
R
< 3
n
(full lines)
and
n
=
3
(dashed lines); X1
> 0.
apply the symmetry EV to G1, we find that two branches go over to G2 , while the two other branches go over to G4; the junctions are thus established. The groups G2 7 G4 7 G5 are solved in a similar way. G3 is a trident. We turn now to the groups of 8 branches.
Applying
that 4 branches remain in
first 4 branches form is
joined
a
the symmetry E to the group H, we find , 4 other branches go over to H2. The
H1, while the
trident and therefore the
junctions
to f -R+R-g,' and f -g+R-g to f -R+f -g.
been drawn in
Fig. 21.7, we branches,
are
solved: f -g+f -g
After these junctions have find that the last 4 branches are now separated
into 2 groups of 2
so that their junctions are determined. (Their characteristics cannot intersect those of the previous trident, because the symmetrical of a common orbit would have to lie both in H, and in H2.) The
last group H2 is solved in
a
21.3.2.6
!! 5,
n
=
5. For
n
similar way.
establishing
hand becomes tedious and error-prone, and to generate this list automatically. For
the ordered list of branches a
computer program
was
by
set up
5, there are 108 branches with X, > 0. Using the same methods find again that all junctions can be established. We omit the details and give only the results on Fig. 21.8. (That figure is in three parts, represented here side by side for convenience, but which should be rearranged one above the other, in descending order, to represent the (W, X1) quarteras
n
before,
plane.)
=
we
21.3 Positional Method
f-g+f f-g+f-g f-g+f-R f -g+R-R f-g+R-g f -g+R f-g+R+f f-g+R+R
I I I
I
I
I I
I
I
'--
I
f -R+R+R -f-R+R+f f -R+R
H
C
f-R+R-g f -R+R-R f-R+f-R
H
f-R+f-g f -R+f
f-R-g f-R-g+f f -R-g+R
I "--
f -R-R+R -f-R-R+f f-R-R
.,-
-f-R-R-g
1
f -R-R-R R-R-R-R
I
C
G
2
3
R-R-R-g R-R-R R-R-R+f R-R-R+R
I
R-R-g+R R-R-g+f R-R-g
I
G
4
R-R+f ------
-
R-R+f-g R-R+f-R
r+----- -R-R+R-R 1
11 I
r---
-
I
I
1 1
------
Fig.
21-7. Junctions for n3
2
-R-R+R-g
I /,-
H
-
(full lines)
R-R+R R-R+R+f R-R+R+R
R-g+R+R R-g+R+f R-g+R R-g+R-g R-g+R-R R-g+f-R R-g+f-g R-g+f
and
n
=
4
C
H
2
(dashed lines); X,
> 0.
261
262
21. Partial T ansition 2.2
f -R-g+f
f-g+f-g f-g+f-g+f 'f-g+f-g+R f -g+f -R+R f -g+f -R+f f -g+f -R f-g+f-R-g f -g+f -R-R f -g+R-R-R f-g+R-R-g f-g+R-R f-g+R-R+f f -g+R-R+R f-g+R-g+R f-g+R-g+f f-g+R-g f -g+R+f f-g+R+f-g f-g+R+f-R f-g+R+R-R f-g+R+R-g f-g+R+R f-g+R+R+f f -g+R+R+R
I I
"
I
-
-
f-R-g+f-g f -R-g+f -R f -R-g+R-R f -R-g+R-g
f-R-g+R f-R-g+R+f f-R-g+R+R f -R-R+R+R
f -R-R+R+f f -R-R+R f -R-R+R-g f -R-R+R-R f -R-R+f -R f -R-R+f -g f -R-R+f
f-R-R-g f -R-R-g+f f-R-R-g+R f-R-R-R+R f-R-R-R+f f- R-R-R f -R-R-R-g f -R-R-R-R
f -R+R+R+R f -R+R+R+f f -R+R+R
R-R-R-R-R
f -R+R+R-g f -R+R+R-R f -R+R+f -R
R-R-R-R+f R-R-R-R+R
f -R+R+f -g f -R+R+f f -R+R-g f -R+R-g+f
R-R-R-R-g R-R-R-R
R-R-R-g+R R-R-R-g+f R-R-R-g -
-
-
-
-
-
-
--
-
-
-
-
-
-f-R+R-g+R
/
I
f -R+R-R+R f -R+R-R+f f -R+R-R f -R+R-R-g f -R+R-R-R f -R+f -R-R f -R+f -R-g f -R+f -R f -R+f -R+f f -R+f -R+R
f-R+f-g+R f-R+f-g+f
R-R-R+R-g R-R-R+R R-R-R+R+f R-R-R+R+R
-
-
f -R+f -g
Fig.
21.8. Junctions for
n
=
4
(full lines)
R-R-R+f R-R-R+f -g R-R-R+f -R R-R-R+R-R
and n5
-
-
-
-
-
R-R-g+R+R R-R-g+R+f R-R-g+R R-R-g+R-g R-R-g+R-R R-R-g+f -R R-R-g+f -g R-R-g+f
(dashed lines); X1
> 0.
21.3 Positional Method
R -
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
7--,---
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
R + f -g
R-R+f-g+f R-R+f-g+R R-R+f -R+R R-R+f -R+f
R-R+f -R R-R+f -R-g R-R+f -R-R R-R+R-R-R
R-R+R-R-g -
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
7
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
R-R+R-R R-R+R-R+f R-R+R-R+R
R-R+R-g+R R-R+R-g+f R-R+R-g R-R+R+f R-R+R+f -g R-R+R+f -R R-R+R+R-R
R-R+R+R-g R-R+R+R R-R+R+R+f R-R+R+R+R I
f---
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
I I
---
'-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
---
-
-
-
-
-
-
-
-
-
R-g+R+R+R R-g+R+R+f R-g+R+R R-g+R+R-g R-g+R+R-R R-g+R+f -R R-g+R+f-g R-g+R+f R-g+R-g R-g+R-g+f R-g+R-g+R R-g+R-R+R R-g+R-R+f R-g+R-R R-g+R-R-g R-g+R-R-R /R-g+f-R-R R-g+f -R-g R-g+f-R R-g+f -R+f R-g+f -R+R R-g+f -g+R R-g+f-g+f R -g+ f -g
Fig.
-21.8.
(continuation)
263
21. Partial Transition 2.2
264
A
kind of trident is observed here in two
new
exactly
the
same as
for the standard trident
(see
The situation is
cases.
13.3.3.2),
Sect.
with the
symmetry E replaced by EE'. It involves thus two branches of orbits symmetric under EV and two branches of asymmetric orbits; the latter two branches are changed into each other by EV. One such trident consists of the branches R-R+f -R+R, R-g+f -g+R, R-R+f -g+R, R-g+f -R+R; the other of R-R+R-R+R, R-g+R-g+R, R-R+R-g+R, R-g+R-R+R. The characteristics of the
symmetric and asymmetric orbits by a dot. 21.3.2.7
n
=
6. There
are
intersect in
a common
orbit, represented
324 branches with X, > 0. We do not show the
results here as they would take up too much space. All junctions can be established by the same methods for two groups of 4 branches, which are represented established junctions are shown as full lines.
on
Fig.
as
above, except already
21.9. The
R-R+f-R+R-R R-R+f-R+f-R
R-R+R-g+R-R R-R+R-g+f -R R-g+R-g+f-R R-g+R-g+R-R R-g+f -R+f -R R-g+f-R+R-R
Fig.
21.9. Some hard branches for
n
=
6.
a new kind of argument can be used here. We remember that a sketch of the (W, X1) plane, and we notice that the essentially figure the groups is encircled by the region containing the of one region, containing other group. On the other hand, every characteristic has a minimal value of W. That minimal value is the same for two characteristics symmetrical of each other. Suppose now that in the outer region, the branch R-R+T-R+R-R is joined to the branch R-T+T-R+T-R. Applying E, we find that in the inner region, R-R+R-T+R-R is joined to R-T+R-T+T-R. But it is clear from the figure that these two characteristics cannot have the same minimal value of W, since the first characteristic encircles the region which itself contains the second characteristic. We have thus a contradiction. Similarly, we find that
However,
the
is
21.4 Results for Bifurcations of
R-R+T-RtR-R cannot be to
R-R+T-R+T-R, and all
265
2
must therefore be
joined
T. There are 972 branches with X, > 0. A detailed study that, using the above methods, all junctions can be established except
21.3.2.8
shows
joined to R-T+T-R+R-R. It junctions are established.
Type
n
=
for 40 branches.
A numerical computation seems necessary for these cases. Thus, the positional method is not able to solve all cases. However, its success in n
=
the present
(transition 2.2)
case
An alternative would be to find all values of a
is
spectacular:
all
junctions
up to
6 have been established.
junction
n.
Some
regularities
is not involved in
a
a
prescribing the junctions for computed cases. If symbol single changes between the
set of rules
can
be observed in the
trident,
a
two. branches. Moreover, that symbol cannot be one of the two end symbols. If the junction is involved in a trident, two symbols change; they occupy
symmetrical positions and
n
+ 1
-
in the sequence, i.e.
they correspond
to basic
arcs
i
i.
21.4 Results for Bifurcations of
Type
2
give the junctions between branches at transition 2.2. We come back to our original objective and deduce the corresponding junctions between branches in a partial bifurcation of type 2. Only branches with a sign are involved (Sect. 21.1, Property 3). For these branches, the path followed through the partial bifurcation of type 2 is as follows. When the branch approaches the bifurcation v increases from 0. When v 1/3 is reached, the branch goes through a transition 2.1, with 0 w > (Sect. 19.1, Property 5). This transition was studied in Sect. 19.3.3.1 and was found to have a simple and systematic structure: successive S-arcs coalesce to form a single R-arc, while T-arcs are not affected; and the family continues toward higher values of v. When v 1/2 is reached, we have a transition 2.2,- and the branch is joined to another branch as described by the above results. v starts then to decrease and the above evolution proceeds now in reverse, going through another transition 2.1. The whole junction is shown schematically in Table 21.2.
Figs.
21.5 to 21.8
use now
these results to
-
,
=
=
Table 21.2. Partial bifurcation of type 2:
junction between
two branches with
sign. V
0
Transition
S,
Arcs
Example
f 31
1/3
1/2
1/3
2.1
2.2
2.1
T
R, f -R-R-R+R
R,
T f -R-R-g+R
'-*. 0
S,
T
f2gl
T
-
a
-
21. Partial Transition 2.2
266
Figs. 21.5 to 21.8 corresponds thus to one junction partial bifurcation of type 2. The symbolic names of these branches are easily found: each R-arc corresponds to a sequence of S-arcs; each + sign inside the R-arc corresponds to a node and thus separates Each junction shown in
between two branches in
two consecutive
a
S-arcs(Sect. 18.1.2).
example, consider the branch f 31 (see last line of Table 21.2). When it reaches transition 2.1, the two S-arcs represented by the symbols 3 and I fuse into a R-arc of length 4, and the orbit becomes f -R-R-R+R. At transition 2.2, that branch is joined to f -R-R-g+R, as shown by Fig. 21.8. This branch contains now two R-arcs of lengths 2 and 1. When transition 2.1 is again visited, these two R-arcs change into S-arcs, and we end on the branch As
-
an
-
f 2gl.
The junctions of the branches obtained in this way up to n = 5 are listed in Table 21.3. This table is in the same format as Table 8.12, which was obtained
using only qualitative methods.
It
can
be verified that there is agreement for n ! 4). (For this
and 2 :! all branches which appear in both tables (sign be should 8.12 in Table comparison, the symbols i and e -
g
respectively;
see
Sect.
for
n
2 to 5. It
can
f and
17.1.)
Tables 19.1 and 21.3 =
replaced by
be
give
all
seen
junctions for partial bifurcations of type 2, quantitative approach is much more
that the
qualitative one: all the cases left undecided in Table 8.12 (the majority) are now solved; in addition all junctions are established for 5. Nothing prevents in principle the solution of higher values of n, using n 6 are listed in Table 19.1 for numerical computation. (The junctions for n branches with a + sign. They were also computed for for branches with a sign (see Sect. 21.3.2.7), but not given here.) However, the amount of work
powerful
than the
=
=
-
grows
exponentially.
21.4 Results for Bifurcations of
Table 21.3. Partial
branches,
2P2+S -
-
11
determined
2P3-+A -
gf
-
2P2+A
-
-
-
if -
-
gl
-
2P2-S -
-
2 fg
2P2-A
-
-
fgf fil
fgl
Igf 2f
Igl
2P4++-A -
-
fig
2P3--A -
-
-
-
lit
g1f.
2g f2
2P4+++S
gil
2P4 ... A -
-
gig gfg
g2 gf I lfg Ilg
-
If2
-
13
-
-
-
-
-
-
glig glfg
g12 glf 1
-
-
-
-
g2g gf Ig g3 gf2
2P4-++A
lfgl
--lglf
121
-
-
-
-
-
-
g2f gfgf
2P4+-+A
-
-
-
If gf
Ilgf
Igll
-
-
-
f
lif
fgIf f Ill
fgIl
-
-
12f
-22
-
-
-
-
-
glgl g2l gf 11 gfgl
2P4--+A -
-
-
12
2gf 3f
2gl
-31 f2f
-
-
lgf I f Ilg
fgfg
f2l
flgl 2P4 -
-
-
---
S
4
f2g
2P4 -
Ifl
fg2 fgfl
flgf
if If
glgf gf If
flfg fglg f 12
-
2P4-+-S
-
-
lglg 21g
f If I
ifil
Ilgi
Ig2 2f I
-
21f
-211
Igfg 2fg
-
-
ilif
gill
-
-
-
gIlf
If Ig 12g
-
lill
lif
2P3+-A'
-
2P4-+-A
112
-
-
-
-
2P3++A -
2P4+--A
267
2
between the
llfl
2P4+-+S
3
junctions
-
2P3--S -
2 to 5:
-
-
2P3++S
llfg IlIg
-
Ig
-fl
f if
=
-21
-
-
bifurcation, type 2, n by the quantitative study.
Type
---
3g f3
A
-
268
21. Paxtial Transition 2.2
Table 21.3.
2P5 .... S -
-
11111
g1lif
2P5 .... A
(continuation) 2P5++--A -
-
-
Ilf Ig
112g 11f2
-113 111if -
g1III 2P5+++-A
-
-
Mfg 1111g
-
-
-
-
-
-
g13 gIf2
-
-
2P5+-++A
11ifI -
-
1112 -
-
-
-
-
gIIIg glIfg
g112 g11f 1
2P5++-+A
-
-
-
-
-
-
-
7
11fgf 111gf
-
-
-
-
I1fgI 111gl
-
lifil
-
1121
-
lif if
g1g1f g21f g1g1I g211 gfIIf gfgIf gf 111 gfgll If gif
-
-
-
112f
-
-
-
-
-
-
-
If lif
Ifg1I
-
if ill
121f
ligif
-
1211 -
-
-
-
-
-
-
gllgf g1fgf gligi gifgl
g121 g1f 11
11g1l
-
-
gl2f gifif.
g22 g2fl gflfg gfgfg gf If I gf 12
gf 11g gfglg
gfg2 gfgfI lfgfg 11gfg Ifgf I 11g2 If g2
11gf1 If gig
11g1g If 11g
121g
-
If 12
-
12fl
-
If If I
-
122
-
-
-
-
1f1fg 12f g
g3f gflgf If IgI
131
2P5+--+A
-
g2gf gf2f .
-
-
-
-
-
-
-
-
-
-
-
-
g'g1g g21g
-
-
-
-
-
gIgfI g1g2
-
-
-
-
-
g1gf g g2fg
2P5+--+S
-
-
-
-
g12g gIf1g
2P5+r-+-A
-
g 2gI
g3l gf21 gf Igl If lgf
12gf If2l
12gl
-
If 2f
-
Of
2P5+---A -
-
g3g gf2g -
-
-
g4 gf3 -
-
-
1f2g 13g -
-
1f3
-
14
21.4 Results for Bifurcations of
Table 21.3.
2P5-+++A -
f 111f
fgIlf -
f 1111
fgI1I
1gl1f 211f
1g11I 2111
2P5-++-S
(continuation) 2P5-+-+A -
-
-
-
f Mg
fglfg Ig1f1
-
-
-
-
-
f Ifif
-
-
f 12f
-
-
f 121
-
-
-
-
-
-
-
-
f1fgI
f 11gl
fglgf fgfgf f g1gl
fg2l
fg2f fgf if fgf11 fgfgl
1gfgf 2fgf
-
-
-
-
-
-
-
-
-
-
-
2fI1
Igf 11 2fgl
-
1g12
1gfIf
-
2f if -
-
-
-
-
-
1g2f 22f
Ig2l 21gl lglg!l
Igf 1g Ig2g
lgf2 1g3 22g 2f Ig M
2glf 31f
2g1l
-311, -
-
fg3 fgf2
2P5--++A
-
-
-
Igl1g 211g
lgfgl
fg2g fgf Ig
-23
-
21fl
-
f13
-
1gIf g 21fg
-
fIf2
flifi
fg12 fg1f I
f 12g
-
-
f 112
f if 1g
-
-
f11fg fgllg
f 11gf
2P5--+-A
f if ii
-212 2P5-++-A
f if gf
2P5-+--A
-
-
-
-
Type
-
f21f f Ig1f
f211 f 1g1l
-
2gfg 3fg
2gf I 3f 1
2g2
-32 -
-
-
-
-
-
-
-
-
-
2gig 31g f2fg f1gfg f 2f I
f Igf 1 f22
f1g2 f2lg f 1g1g
2P5---+A -
-
-
3gf 4f
3gl
-41 -
-
-
-
f3f
f2gf f3l
f2gl
2P5 -
-
----
S
5
f3g
-221 2P5 -
-
Ig1gf 21gf
----
4g f4
A
2
269
22. Total Transition 2.2
The
equations X2i
Zi
Yi
Yi-1
-
IYi(Xi+l
are
_W=0, ziXi
-
-
Xi)
-
=
1
0,
=
0
(22.1)
1
where i is to be taken modulo
n
and takes all values from 1 to
n.
equations form a system of 3n equations for the 3n + 1 variables W) Yo to Yn-1, X1 to Xn) Zi to Zn. As in the case of partial bifurcations (Chap. 21), we expect one-parameter families of solutions, which correspond These
to
ordinary one-parameter families of orbits.
22.1
Properties
properties are essentially the -same as for partial transition 2.2 (Sect. 21.1). The proof of Property 2 is slightly different: we use the fact that a continually increasing or decreasing sequence of Yj cannot be periodic. A lower bound can -again be established for W, but it is not the same as The
(21.7).
(22.1c). Xi+,. (21.4) is
Two successive values X must be different in view of
there exists a Xi such that Xi-1 < Xi and Xi > and we have then from (22.1c):
Yi_1
>
1W-1/2, 2
Yi
1
0. Equating to 0 the first factor in (22.14),we obtain a
first
family YO Z1
=
of solutions
-Y1
Z2
=
X1
8
V/W-2
_W :F
-
4
(22.15)
2
These solutions exist for W > 2. The
for the upper
YO
V/W_
YJ
X1
2
and for the lower
YO
asymptotic
branches for W
+oo are,
signs.
-Yi
=
W2 -4
:F
-X2
-X2
V/-W-
Zi
Z2
-
-W
(22.16)
signs 1
-
2 v/W_
X1
-X2
VW)
Z1
Z2
(22.17) Using
the
asymptotic
expressions of Sect.
21.1.1, the numerical values of
yo
and yj obtained in Sect. 18.2.3 and the branch notation defined in Sect. 21.1.5, we find that these two branches correspond to +R-R+ and +f -g+ respectively.
22. Total Transition 2.2
274
The
stability
IOX2
C9X2
OX0
ax,
IOX3
aX3
'9XO
ax,
is
computed from (22.10), which gives 0
01
1 U2
-1
1 U1
-1
Y2 0
0
U2
y2 0 U1
+
y2 0
(22.18)
UIU2
y4 0
with
2YO2
ul
stability
The
z
=
-1 +
-
3X21
+
W,
1
-
=
8
=
3/.\,/2-
the second
3X22
+ W
(22.19)
.
UIU2
(22.20)
2 Y04
[(W2
gives
4)(2W2
-
This vanishes for W W
-
index is
In the present case, this z
2YO2
U2
=
3)
-
(2W3
::F
7W) VW2
-
(22.21)
-
2, which corresponds to an extremum in W, and for sign, which corresponds to the intersection with
with the upper
family (presently
described).
to be
by setting the second factor in (22.14) equal Eliminating variables with the help of (22.12), we obtain an equation
A second solution is obtained to 0.
for Zj:
Z41
+
WZ31+2 Z21 +2WZ,+4=0.
It will be convenient here to
The other variables
There 0
zi
are
as
Z,
--
_y11
as
independent Z1,
as
functions of
1
X2
=
Z1
V;2:
-oo
to 0
_
-W,
The
asymptotic
W-1/2 X2
X,
_W1/2
and for the second branch:
case
Yj
>
0, by
+2
_Z1 Z1 1
from
variable instead of W. in the
Z1
Y,
2 +2
'
Z2
2
(22.23)
Z,
shown
by Table 22.1. corresponding to Z, -+
are
two branches W -+ +oo,
respectively.
=
runs
Z,
YO
V- Z1 (Zl2+2)
The variations
y0 f
expressed
2
X,
Z,
use
Z14 + 2Z2 +4 Zj(Zj2 + 2)
W
-+
are
(22.22)
-oo
and
values are, for the first branch:
2W -3/2,
Z2
-
-2W-1
(22.24)
22.2 Small Values of
Table 22.1. Variations of the variables when Zi increases from
Z,
-
oo
_-I
-
+00-"..
W
V 2_
-'
3
--",+Oo
72
0
2-3/4
X1
0
2- 1/4
X2
-oo
-2- 1/4
Z2
0
YJ
YO
YO
=
Zi
-
-11
to 0.
0
0
";r +00 0
1
-00
72
W-1/2
-2W-1,
-oo
275
n
wI/2
x,
-2W -3/2
X2
-W
Z2
(22.25)
asymptotic expressions of Sect. 21.1.1, we find-that these branches correspond to +R-g+ and +f -R+ respectively. The stability index is computed from (22.20), which gives here
Using
z
the
(Z12
-
2 )2(Z4 1 +
6Z12
+
4)
(22.26)
-
This has
Z4I a
double root for Z2
extremum and
an
=
2,
or
W
two
=
31V2_, corresponding to
intersection with the other
both
an
family.
are shown on Fig. 22.1. It is sufficient to show the quarter0, Yo > 0. We have a Mdent (Sect. 13.3.3.2), formed by the two branches of symmetric orbits +R-R+ and +f -g+ and the two branches of asymmetric orbits +R-g+ and +f -R+. The latter two branches have the same projection on the (W, YO) plane.
The solutions
plane W
>
+R-R+
i
YO +R-f +
0.5
+g-R+ +g-f +
0
0
Fig.
1
2
3
22.1. Characteristics for 2.2T2 transitions.
W
4
22. Total I ansition 2.2
276
22.2.3
n
3
=
It
3 the general solution can only be obtained numeriseems that for n cally (see below Sect. 22.2.4). The equations can be solved, however, in the particular case of symmetric orbits. One of the crossings of the symmetry axis is at the junction of two basic arcs. We take it as origin (h Y2, X2 -X3, 0, X, 0). There is then Y, and the equations are easily solved; for Yo > 0 we obtain =
=
V VW K__
YO
Z1
Z3
X2
0)
The
z
-
2
-6
XI
Y1
2 =
-W
W:2
-X3
=
_2 6 -W::F V/W-
Z2
=
=
=
=
6
qW-2 6
Y2
1
(22.27)
-
computation of the stability gives 1
27 =
2
(
Y04 ) ( T88y8
1 24
This vanishes for Yo of W, and for
1
1 +
0
=
24-1/1,
8Y
4
,F6,
W
(22.28)
-3
which
corresponds
to the minimum
1/4
YO
(22.29)
144
which should correspond to bits.
an
The characteristic is shown
intersection with
on
Fig.
a
family of asymmetric
22.2. The two branches
fied from the asymptotic expressions. Two other families shifting the origin, i.e. rotating the indices.
22.2.4 Numerical
For
n
>
2,
a
can
are
or-
identi-
be obtained
by
Computation
numerical computation
seems
necpssary. A program similar to
those described in Sects. 14.2.2 and 20.2.5.2 of 2n
was used. It solves,the system equations for the Yi and Xi formed by (17.205a) and (17.207):
Yi(x+l
-
X)
-
1
=
0,
Y
Y
_Xi(Xi
2 _
W)
=
0
(22.30)
The computation of a given branch is begun at a large value of W and proceeds toward smaller values. The method of computation differs depending on whether the branch corresponds to a R-orbit or not.
22.2 Small Values of
n
277
+R-R-R+
YO 0.5
+f -R-g+ 0 3
2
1
0
5
4 W
Fig.
22.2. Characteristics for 2.2T3
22.2.4.1 R-Orbit. In the
case
transitions, symmetric orbits, h
of
a
=
R-orbit, the asymptotic form
0.
is defined
(21.15c), with numerical values for the yj and xi which by (21.15b) characterize the branch; it is thus well isolated from other branches. On the other hand, the stability index tends toward a finite value (see Sect. 18.2.2). and
immediately use a shooting method; it is not necessary to relaxation method (which in any case would be inapplicable begin since there are no nodes). Starting from the approximate value& of Yo and X, given by (21.15b) and (21.15c): Therefore
we can
with
Y, 0
a
wl/2VO,
W-112X,
X,
(22-31)
with yo and x, determined in Sect. 18.2, we compute successively Y1, X2, Y, X,,+,. We compute also the variations, as functions of dY0 and dXj. This allows
obtained
index
an
(Y,,
iterative correction of the initial
=
Yo, X,,+,
=
Xj).
It allows also
values, until a periodic orbit is a computation of the stability
z.
Containing T-Arcs. In that case, we start with a relax(Sects. 14.2.2, 20.2.5.2), based on the asymptotic deinto T- and. R-arcs. The asymptotic expressions of orbit the composition that show E)(Wll') at an Yj E)(W-1/2 ) at a node, Yj 21.1.1) (Sect. 22.2.4.2 'Orbit
ation method
as
usual
=
=
antinode
(inside
1. We take
as
a
R-arc).
initial
The steps
are as
approximation Yj
=
follows: 0 in nodes.
2. For each arc, we compute the values of the internal 3. For each node, we recompute Yj, using (22.30a). 4. We go back to step
Some details
are
compute Xi from
Yj and Xi (see below).
2, until the solution has converged.
necessary for
(22.30b),
third-degree equation has
point
of a T-arc, we must given. For large W, this 0, +W1/2, _W1/2 respectively.
2. In the
where Yj and Yi-1
3 real roots, close to
are
case
22. Total Transition 2.2
278
As indicated by
Tf
arc,
_W112
In the
case
(21.14), in the
of
a
we
case
+W1/2
select the root close to
of
a
T9
in the
case
of
a
arc.
R-arc, the end values Yi and Yi+j
are
known,
and
compute the intermediate values Yi+,, ..., Yi+j-l and Xi+,, ..., start from an approximate value of Xi+,, given by (21.15c): X,+1 -
we
must
Xi+j.
We
W-112X1.
We compute then successively Yi+,, Xi+2 7 .., Yi+j with (22.30). We compute also the variations, as functions of dXi+,. This allows a correction of the .
initial value of the
given
Xi+,,
and
we
continue until the final value
Yi+j
agrees with
value.
The branch is followed towards
decreasing
W.
Usually
at some
point
the iteration does not converge anymore, because the decomposition into Tand R-arcs ceases to be a good approximation. We shift then to a shooting
method, which works well for moderate values of W where the branches well separated.
22.3 Results for Bifurcations of All branches
Type
are
2
were computed and joined up to n 5. Showing characteristics (W, YO) plane would not be useful because the number of branches is too great and the figure is crowded. Instead, detailed print-outs of the branches were. made, from which the junctions were found by inspection. As in the case of partial bifurcation (Sect. 21.4), we can use the knowledge of these junctions at transition 2.2 to deduce the corresponding junctions =
in the
a total bifurcation of type 2. These junctions follow again path illustrated by Table 21.2, going successively through a transition 2. 1, a transition 2.2, and another transition 2. 1. In each transition 2. 1, a sequence of S-arcs is replaced by a R-arc (or a R-orbit if the whole orbit is involved), or conversely. The rules are the same as in Sect. 21.4. We do not list here the junctions computed for the transition 2.2 since they represent only an intermediate step. Instead we list in Table 22.2 the final results of interest, i.e. ..the, junctions between branches for the total bi-' furcation. This table is in the same format as Table 8.18, which was obtained ,using only qualitative methods. It can be verified that there is agreement
between branches in
the
for all branches which appear in both tables (sign and the 8.18 and i in Table should be e symbols comparison, -
n
4 and
present chapter.
separately study The junctions for these two cases have been established in Chap. 8 (Tables 8.12, 8.18), so that we could dispense with their study if we were only interested in the junctions. But we will pursue the quantitative approach for these two cases also, for homogeneity, and in order to obtain in all cases a quantitative approximation of the families of periodic orbits in the neighbourhood them
we
of the bifurcation.
23.1 Total Bifurcation of
Type 2, (17.63)
We have the,4 fundamental equations Aal, ul, ul', AC. For p -
1
=
0, there
assume
symmetric. U1
=
(17.66)
is
that for y >
Then there
arriving
(M)
=
to
(17.66)
for the variables
at the bifurcation:
+
E,
-
E,
ho, +
1,
orbits.
symmetric generating 0, the corresponding periodic orbits are also
All these branches
(Table 6.5).
We will
4 branches
are
I
n
are
made of
is
(23.1)
_UI
identically satisfied,
and the system of fundamental
equations
re-
duces to
houl
V
IL[l - 0(ho, ul)]
3,7rI(a 1) Aal [1 2a2V,
,
-
U1
-
2(a
-
1)hO
=
aVyAC
23.1.1 The Case
+
+
O(AC, Aal, ul
!LAal a
v
=
+
au
12
+
2 ,
AA
,
O(AC2, Aa 2, U/ 4, tl)
(23.2)
0
0. As in case p = 0, AC 54 0, corresponding to v Sects. 12.3 and 17.3, this case is simple and it is not necessary to use the machinery described in Sect. 11.4. The solutions are known: they correspond,
We consider first the
M. Hénon: LNPm 65, pp. 283 - 296, 2001 © Springer-Verlag Berlin Heidelberg 2001
=
23. Bifurcations 2TI and 2P1
284
to -
a
E,
displacement from the bifurcation orbit along 1. The first equation (23.2a) reduces to 1,
+
one
of the branches
+
E,
-
houl
0
=
(23-3)
.
Using (23.2b)
We have two solutions.
(23.2c),
and
we
find that the first
so-
lution is U
0
=
11
Aa,
,
ho
0
=
=
2(a
-
1)
AC +
O(AC2),
(23.4)
and the second solution is
ho =0
Aa,
,
a2 VY -
AC +
VY
37rI(a
U
1)Vy
-
2V2
AC +
O(AC2),
O(AC2)
(23.5)
Y
These two solutions
E) and that (23.5)
(branches remark for
m
to orbits of the first
correspond respectively
second species is a particular
(branches case
of the
species
1). In the latter case, we may equations (17.84) and (17.86),
1.
=
23.1.2 The Case 0
to extend each solution obtained in the
We
use once more
U
ACX1*
1
general
ho
7
= -
method described in Sect. 11.4. v
ACy*
37rI(a
-
1)Vy
=
a X
Y
,
Substituting
4.
powers of =
2x'+ Y
-
=
z
-
member,
O(AC) =
case
1 V
1y
'
change Z*
of variables
a2 VY -
-.
V
(23.7)
Y
equations (23.2), collecting the dividing by the appropriate
and
obtain
+
0(p)
,
O(Ac' pAC-1)
1/2
Fig. 23.1 shows that w vanishes in two points. As for type 1, Sect. 12.6, the limit w -+ 0 corresponds to moving from the case v 1/2 to the case v > 1/2, which should in principle be studied separately. However, the solutions found =
in this way turn out to be
a
subset of the solutions of the system
(23.25),
corresponding to -AC'= 0. It is not necessary to consider these solutions separately; they are not fundamentally different from the other solutions. 23.1.5
The
Recapitulation
quantitative study of the bifurcation
2TI is'now
complete.
For
a
given
small /.z, consider for instance a family of periodic orbits coming along the + E branch. It corresponds at first toa small value of v as the distance AC to the -bifurcation is still
comparatively large; the orbits are approximately by the equations (23.14) and (23-15) of Sect. 23.1.2. When v reaches the value 1/2, the orbits are approximately described by the equations (23.25) of Sect..23.1.3. The branch + E joins there the branch 1 (Fig. 23.1). As we continue along the family, v decreases from 1/2 to small values; the orbits are approximately described by the equations (23.16) and (23.17) of Sect. 23.1.2, described
-
away from the bifurcation. 23.2 shows in log-log plots the variations in order of
as we move
Fig. magnitude of and functions of AC. quantities -ho, Aal, ul ul' as This figure for the 2T1 bifurcation differs much from the figure for the general case n > 1 (Fig. 17.2). Here there is a single transition at p 1/2, instead of the two transitions at p 1/3 and p 1/2. This explains why a particular treatment was necessary. The junctions are shown in Table 23.1 in the usual format. They agree with those established in Chap. 8 with the help of Broucke's principle (Table 8.18). The Tables 20.1, 22.2, and 23.1, taken together, give all junctions for total the
=
=
=
1 to 5. bifurcations of type 2, for n In a sense the bifurcation 2T1 is completely described =
by the equations equations of Sects. 23.1.1 and 23.1.2 as limit cases. The bifurcation 2TI was studied by Guillaume (1971, pp. 125-126). It can be verified that his equation (IV-33) is identical with the above equations.
(23.25),
which include the
23.1 Total Bifurcation of
AC
Type 2,
AC
ho
E
=
I
(M)
289
.AC
1/2
1/2
n
1/2
Aa
S
U
S
UPI 1
A
1/2
A
E'\,
Fig.
axe
logarithmic.
while E represents values at
Table 23.1. Total tative
In the left an
1/2
E
ho, Aal, ul and u"1 as functions of AC. panel, SS represents values of ho at a node,
antinode.
bifurcation, type 2,
study.
2T1++O E
k4
23-2. 2TI bifurcation: variations of
Both scales
+
1/2
2T1--O -
E
+
1
n
1:
junctions determined by
the
quanti-
290
23. Bifurcations 2TI and 2P1
23.2 Partial Bifurcation of
Type 2,
(M)
n
We have the 5 fundamental equations (17.64) to (17-67) for the variables ho, hi, Aal, ul, ul, AC. However, from (17.67) we have ho = O(p), hi = 0(p). In
(17.65)
and
(17.66), ho
and h,
side. We have thus 3 equations for U
U1 0
=
37rl(a
, _
1
a
aVyAC
2
1)
-
vy
+
au
a
0
37rl =
2avy
Aal (u,
23.2.1 The Case
v
=
We consider first the solutions
correspond
ul') [1
+
12 1
+
ul',
2
112
Aa, (ul Therefore
Aal
+
we
ul)
Aal, ul 2, U 112)]
+
a
(23.26)
0. The 0, AC 54 0, corresponding to v p displacement from the bifurcation orbit along one =
=
are +
1,
-
1,
0
U
/,--VYACI' + O(Anl
=U
=
-
e
(23.28)
i and
we
-
e).
The
-
sign
have then
E)(Aal)
and from
i,
(23.27)
.
-
=
-
reduces to
solution corresponds to the T-arcs (branches corresponds to a Ti arc, the + sign to a T' arc. The'second solution is ul -ul. From (23.26a)
Aal
O(tl)
0
This
ul
,
have two solutions. The first solution is
0,
=
=
tz)
'u 1
of the branches which emanate from it. These branches
(Table 6.9). (23.26c)
right-hand
AC:
U11 4, 11) O(AC2, Aa 2, 1
+ 0 (AC,
case
to
ul,
Aaj[1+O(AC,Aaj,u'1
Y-Aaj
+
then be absorbed in the
can
Aal,
(23.29)
(23.26b) VY
=
-
.-
VY
AC[I
+
O(Ac)]
(23.30)
(23.26a) gives.-then 37rl(a
U/
-
I)Vy
2V2
AC[i
+
(23'.31)
O(AC)]
Y
(23.28), (23-30), (23.31) are particular (17.73), (17-76), (17-84), (17.86) (for m 1).
The results
We consider
try
the
cases
0, AC
0, corresponding to v > 0. We will 0. This previous section for v the asymptotic branches of the families of orbits for JL :A 0, now
case
p >
to extend each solution obtained in the
corresponds which
are
to
of the equations
close to the branches for p
=
0
(see Fig. 1.1).
=
23.2 Partial Bifurcation of
Type 2,
n
=
(2P1)
1
291
use the general method described in Sect. 11.4. The intervals in v and appropriate changes of variables turn out to be different for the T-arcs and for the S-arcs. Also we know already that the two symmetric branches + i and 1 and, 1 (S-arcs) are joined, and the two asymmetric branches e (T-arcs) are joined (Tables 7.2, 8.12). Therefore we consider the two families separately.
We
the
23.2.2 T-Arcs: The Case 0
