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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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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.
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Berlin
Heidelberg
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Springer
Quantitative study of bifurcations. 2001 (Lecture notes in physics: N.s. M, Monographs ; 65)
2.
-
ISBN 3-540-41733-8
ISSN 0940-7677
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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
...................................
12.4.2
Second
12.5 The Case
v
=
12.6 The Case
v
>
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 <
12.4.3
............
...................................
12.2 Exclusion of Successive Identical T-Arcs =
.................
14
*.............................
35
......................................
36
..
-*
I*
,
,
VIII
Contents
13. Partial Bifurcation of
13.1
Properties
1
.............................
13.1.1
Asymptotic
Branches for W
oo
13-1.2
Variational
Equations for W
oo
13.1.3
Jacobian
13.1.4
Relation with
13.1.5
Asymptotic
13.2 Small Values of 13.2.1
n
=
13.2.2
n
=
13.2.3
n
=
2
n
39 40
...............
41
.......................................
43
Stability
..........................
JWJ
Behaviour for
-+ oo
...............
.......................................
44 47
47
..............
47
3
.........................................
48
4
.........................................
50
...........................
......................................
51
.......................................
51
13.3.1
Principle
13.3.2
Branch Order
13.3.3
Results
..................................
54
........................................
60
14. Total Bifurcation of
Properties
39
...............
13.3 Positional Method
14.1
Type
.............................................
Type
I
...............................
79
.............................................
79
14.1.1
Jacobian
14.1.2
Relation with
14.1.3
Asymptotic
14.2 Small Values of 2
n
.......................................
Stability
..........................
JWJ
Behaviour for
79 80
...............
81
.......................................
82
e................
82
..................
83
-+ oo
14.2.1
n
14.2.2
Numerical
14.2.3
n
=
4
.........................................
84
14.2.4
n
=
6
.........................................
86
=
14.3 Conclusions.for 15. The Newton
.......................
Computation:
Type
Approach
15.1 Partial Bifurcation of
15.1.1
..................................
87
...................................
93
Type
1: Variables and
93
............................
95
.....................................
95
......................................
98
Additional Relations
Polyhedra Encounter
Equations
......
15.2 Method of Solution 15.3 Newton
1
Method
15-3.4
Equations Arc Equations: General Case Arc Equations: Initial Arc Arc Equations: Final Arc
15.3.5
Additional Relations: General Case
...............
103
15.3.6
Additional Relations: First Relation
...............
104
15.3.7
Additional Relations: Last Relation
...............
105
15.3.8
Additional Relations: Case
.................
105
15.3.1 15.3.2
15.3.3
............................
99
.......................
101
........................
102
n
=
2
15.4 Intersections with the Cone of the Problem
................
106
..............................
106
...................
107
15.5.1
Boundary Subsets The Motzkin-Burger Algorithm
15.5.2
Elimination of Parasitic Solutions
15.5 Coherent
98
.....................
.................
110
Contents
15.5.3
Program
15.5.4
The Case 1P2
.......................................
..................................
Systems of Equations Degeneracy
15.6 Truncated 15.6.1
111 111
..........................
113
....................................
113
15.7 Power Transformations
..................................
114
Case
aaa
.......................................
115
15.7.2
Case
acc
.......................................
117
15.7.3
Case aba
......................................
118
15.7.4
Case dad
......................................
118
15.7.5
Case caA
......................................
121
15.7.6.
Case cda
15.7.7
15.7.1
.......................................
122
Case bab
......................................
123
15.7.8
Case dbd
......................................
125
15.7.9
Case cbA
......................................
125
15.7.10
Case bbb
......................................
15.8 Total Bifurcation of
Proving
Type
The Case 1T2
15.8.1
15.9 Conclusions 16.
IX
I
126
..................................
127
................................
General Results
16.1 Variables and 16.3 Two General
16.6.1
No Arcs*
16.6.2
Arcs*
129
131
.................................
131
.....................................
132
Propositions
16.4 The Case P2 = p, /2 16.5 The Case P2 < p, /2 16.6 The Case P2 > 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 <
v
17.5 The Case
v
1/3:
17,6 The Case
1/3
=
<
<
v
1/3 .................................... Transition 2.1 <
1/2
....................
I
....
.................................
160
166 171
X
Contents
17.7 The Case
v
=
17.8 The Case
v
>
18. The Case
18.1 R-Arc
1/3
179
1/2
<
v
..................................
181
..................................................
Properties
18.1.2
Number of Solutions
.....................................
182
183
...............................
184
...............................................
184
Small Values of A
.....................................
185
...........................
186
...............................
188
.................................
191
...................................
192
.....................................
199
18.2.2
Properties Stability and Jacobian
18-2.3
Small Values of
18.2.4
Sign Sequences
Study
181
181
...........................
Stability
18.1.4
,
of the Mapping
19. Partial Mransition 2.1
n
Properties 19.1.1 Asymptotic Branches for w Variational Equations for w 19.1.2 19A.3 'Asymptotic Branches for w
.............................................
199
................
201
...............
203
..................
204
....................................
207
.......................................
207
19.1.4
R-Jacobian
19.1.5
Stability
oo
oo 0
.......................................
208
1
19.2 Small Values of A
19.2.1
A
=
.........................................
208
19.2.2
A
=
2
..........................................
209
.19.2.3
A
=
3
.........................................
210
A > 3
.........................................
211
19.2.4
19.3 Positional Method
......................................
212
212
19.3.1
Branch Order for
w
oo
19.3.2
Branch Order for
w
0 .......................... 214
19-3.3
Results
......................
........................................
19.4 Results for Bifurcations of
Type
2
........................
217 221
................................
221
............
223
.......................................
225
19.4.1
The Case
w
> 0
19.4.2
The Case
w
< 0
20. Total Transition 2.1
20.1
............................
and Jacobian
18.1.3
18.2.1
19.1
176
........................
18.1.1
18.2 R-Orbit
18.3
<
.........................
1/2: Transition 2.2 1/2 Does Not Exist
....................
.............................................
225
............................
226
...........................
226
Properties 20.1.1 Asymptotic Branches 20.1.2 Stability and Jacobian
.......................................
228
1
20.2 Small Values of
n
20.2.1
n
=
.........................................
228
20.2.2
n
=
2
.........................................
229
20.2.3
n
=
3
.........................................
230
20.2.4
n
=
4
.........................................
232
xi
Contents
20.2.5
n
> 4
.........................................
20.3 Results for Bifurcations of
........................
237
237
w
> 0
.................................
20.3.2
The Case
w
< 0
................................
237
.....................................
239
Properties 21.1.1 Asymptotic Branches for W -4 +oo 21.1.2 Variational Equations for W -4 +oo
.............................................
...............
239 241
244
246
.......................................
247
.....................................
21.1.4
Stability
21.1.5
Branch Notation n
................
.
Jacobian
21.2 Small Values of
I.
................................
247
.......................................
248
21.2.1
n
=
1
.........................................
248
21.2.2
n
=
2
.........................................
249
21.2.3
n
=
3
.........................................
251
21.3 Positional Method
......................................
21.3.1
Branch Order for W
21.3.2
Results
252
......................
253
........................................
258
21.4 Results for Bifurcations of 22. Total Transition 2.2
+oo
-
........................
265
.......................................
271
Type
2
.............................................
271
...........................
272
.......................................
272
Properties 22.1.1 Stability and Jacobian
22.2 Small Values of
n
273,
22.2.1
n
=
1
.........................................
22.2.2
n
=
2
.........................................
273
22.2.3
n
=
3
.........................................
276
22.2.4
Numerical
Computation Type 2
.........................
276
.........................
278
...................................
283
22.3 Results for Bifurcations of 23. Bifurcations 2T1 and 2P1
23.1 Total Bifurcation of The Case
23.1.2
The Case 0 <
23.1.3
The Case
v
=
23.1.4
The Case
v
>
v
=
0 v
(M)
..................
283
.................................
283
Type 2,
23.1.1
<
n
=
1/2
1
............................
1/2 1/2 ................................ ...............................
Recapitulation Partial Bifurcation of Type 2, 23.1.5
23.2
236
The Case
21.1.3
22.1
2
20.3.1
21. Partial Mransition 2.2
21.1
Type
..................................
(M)
288 290 290
1
23.2.2
T-Arcs: The Case 0 <
23.2.3
T-Arcs: The Case
23.2.4 23.2.5
S-Arcs: The Case 0 < v < 1 S-Arcs: The Case v 2! 1
23.2.6
Recapitulation
0
288
.................................
=
The Case
=
286
.................
n
23.2.1
v
284
v
>
v
<
2/3
2/3
....................
291
.......................
292
......................
292
.........................
294
..................................
294
XII
Contents
23.3 Conclusions for
23.4
Type
2 ................................... 294
Approach
23.3.1
The Newton
23.3.2
Proving General Results
Type
3
294
.........................
295
.................................................
295
...........................................
297
............................................
299
....................................................
301
Index of Definitions Index of Notations References
..........................
Equations
11. Definitions and General
11.1 Introduction
The present volume II begins with Chapter 11, as it is the direct continuation of Chapters 1 to 10 of volume I (H6non 1997). This avoids any ambiguity about cross-references. All references to Chapters 1 to 10 (for instance, the reference to Sect. 2.1 below) are to be found in volume I, while all references to Chapters 11 to 23 are to be found in the present volume II. In the present chapter, we describe the general method for the quantitative study of bifurcations, and we derive equations common to the three 'types of, bifurcations. Later chapters will be devoted to detailed treatments -
of individual types I and 2. Some bifurcations have already been studied
(1971), 98),
in the
and
and
orbits. Here
partial
or
of
case
Perko
by
we
quantitatively by Guillaume
orbits and total bifurcation
symmetric
(1977a, 1981a, 1981b), again
consider
more
generally symmetric
in the
or
(ibid.,
pages 9
of
symmetric asymmetric orbits, and case
total bifurcations.
11.2 The 0 Notation
0()
The notation
briefly described in Sect. 2.1, and some simple uses of place in Chapters 4 and 5. In the present volume, we will
was
that notation took
much
intensive
of that notation. Therefore
make
a
some
detail its definition and properties.
more
use
recall here in
we
11.2.1 Definitions
1989, Sect. 9.2, with adaptations to our problem. We x"; we are interested involving small quantities xj,
We follow Graham et al.
consider expressions in the limit x, -+ 0,
..
...,
Definition 11. 2. 1. The expression 0 [g (xi, such that , xn)
functions f (xi,
1h for
some
C191 set
.
-
.
.
.
of positive
1xi 15
constants
M. Hénon: LNPm 65, pp. 1 - 16, 2001 © Springer-Verlag Berlin Heidelberg 2001
C,
-`
1xn1
61, el,
.
.
.
-
,
6n
En
as
follows:
xn)] represents
,
-
for
-,
We define the 0 notation
x.,, -* 0.
-
the set
of
all
11. Definitions and General
f
Equations
and g themselves do not have to tend to zero in the limit, or even 2 example, the notation O(11xj) is perfectly valid: it
to remain finite. For
represents the g
set of all functions which grow
no
for x, -+ 0. product of powers:
In the present work, the function g will generally be a Xqj Xq,,, The exponents qj do not have to be positive;
=
...
to be
n
11x 2I
faster than
.
nor
do
they
have
integers.
consider expressions involving also other variables yi, which remain finite in the limit ej -+ 0. The above definition applies .... y,,,, then for every choice of the yi. The constants C, ei, ..., 6n are functions of
generally,
More
Yi,
..
-
7
Ym.
The 0 notation cases,
we
however,
be used
can
alone,
it will appear inside
for instance in: yo = O(A). In many expression. The rule is then
as
an
Definition 11.2.2. An expression containing one or more instances of the 0 notation represents the set of the expressions obtained by letting each 0() range
over
its domain
For instance, the
(given by Definition 11.2.1).
expression
(11.2)
0 [91 (X)l + 0 [92 (X)l 0 [93 (X)l
represents the set of all functions of the form f, (x) + f2 (x) f3 (x), where fl,
f2, f3 belong respectively to the sets defined by 0[gi(x)], 0[92(X)I, 0[93(X)IThis will be called a O-expression. For completeness, we include also the case of an expression containing no 0 notation; in that case the set reduces to a single element. The 0 notation cannot be used recursively: in O(g), the expression g cannot itself contain instances of the 0() notation, because Definition 11.2.1 would then be meaningless. Apart from that restriction, the 0 notation can be freely used in an expression. We consider next equations involving the 0 notation, i.e. equations of the form f, f2, where fi and f2 are O-expressions. We call this a O-relation. =
The fundamental rule is then Definition 11.2.3. In
an
equality Containing
the '=' sign should be taken to
Thus, the O-relation f, is
a
=
f2
mean
means
11.2.2
Computation
Many rules
can
Of
the notation
0,
that the set of functions defined
subset of the set of functions defined
is commented upon in Graham et al.
instances
'C'.
by f2.
Note the
by f, dissymetry (which
1989).
Rules
be derived for the
manipulation of O-expressions and 0-
Graham et al. 1989, Sect. 9.3). We quote only some of the most important rules which will be needed later. To simplify the equations, we adopt the following conventions: f represents a O-expression; g represents a relations
(see
3
11.2 The 0 Notation
Xn; h
function of x, i.e. of the small quantities xj, of y, i.e. of the variables yl, ..., yn. 1. First
=
a
function
have, trivially:
we
g
represents
O(g)
(11.3)
-
2. Substitution: If the two O-relations appears inside
O-expression fl'
f2,
f2
f,
then it
and
can
f2' hold, and replaced by f2.
fl'
be
=
if the
This follows from the transitive property of the subset relation. Note, however, that only the substitution of a left-hand side fl' into a righthand side f2 is allowed. 3.
O(g)
O(g)
+
4. Elimination of
hO(g) 5.
Elimination of identical terms:
Simplifications:
Using (11.3),
Ih9 6. Sum:
a
very
frequently
used relation:
(11-6)
.
-
0(91 The
(11-5)
.
obtain
we
O(g)
=
multiplicative factor:
a
O(g)
=
(11.4)
O(g)
=
92)
+
reverse
0(91)
=
+
(11-7)
0(92)
relation is not true.
7. Product:,
0(91)0(92) Here the
,
=
relation is also true:
reverse
0 (9192)
8. Powers: if
O(Xql I
0 (91) 0 (92)
=
q'1 ...
>
qj,
q'
.
n
Xq
a
(11.9)
-
2 qn, then
-
O(X1qj
n
9. Elimination of
(11-8)
0(9192)
...
composite
term:
o(gl)+O(gl'g12-')+0(92) As
an
example,
we
10. Truncation: Consider
f
can
=
ho
be +
a
expanded hlxl
+
O(X2) 2
O(X2) 1
+
for
0 < A:!
O(X2). 2
function as a 2
h2X 1
Using the previous rules, < 0: q
0(91)+0(92)
1
.(11.11)
have
0 (X2) + 0 (X*l X2) + 1
which
(11.10)
qn) Xn
involving only converging series:
(11.12) one
small parameter xj,
(11-13)
+... we can
truncate the
expansion
at
some
order
11. Definitions and General
4
Equations
q
hjXj
f
O(Xq+l)
+
(11.14)
j=O
frequently involve sums of 0 terms, of the form 0 (gi) + abbreviate, we will write this as: 0 (911 92
Our equations will 0 (92) +
..
To
..
11.3 General
Equations
11.3.1 DefinitionS
We consider
second species bifurcation orbit Q (Sect. 6.5). We introduce a numbering, which will be used throughout the present volume:
a
fundamental -
a partial bifurcation of order n, we number the collisions of the bifurcating arc (including the two end collisions) from 0 to n, and the basic arcs
For
from 1 to -
n.
total bifurcation of order n, we number the collisions from 0 to n 1, with collision 0 at the origin, and we number the basic arcs from 1 to n, For
a
-
starting from the origin. Thus in both cases, the basic arc i joins the collisions i 1 and i (except for 1 and 0). arc n in a total bifurcation, which joins the collisions n -
the
-
We define
C In the
=
I
C, the
of
set
n f 11 10'...'n-l} -
of
case
a
internal
collisions,
as
for
a
partial bifurcation,
for
a
total bifurcation.
(11.15)
partial bifurcation, the two end collisions A, the set of basic arcs, as
are
excluded from
C. We define also
n}
A=
(11-16)
.
Here the definition is the
We consider
periodic
now
for
partial and total bifurcations. problem, with /,t small but not
close to the bifurcation orbit Q.
Q,_,
orbit
same
the restricted
(Such
zero, and
orbits exist
by generating orbits, of which bifurcation orbits are a special case.) We use a system of coordinates (X, Y) with the origin in M2 and with fixed directions, related to the coordinates (x, y) and, (X, Y) defined in Sect. 2.2 by a
virtue of the definition 2.9.1 of
X Y
=
=
(x (x
-
-
1)
cost
1)
Sint + y cost
-
ysint
The equations of motion -/.t cost
-IL sin t
-
-
(1
(1
-
-
are
X
=
=
Y
-
-
in the
IL)Xr
1L)Yr
-3 _
-3 -
cost,
(11.17)
Sint.
(X, Y) system I_I(X
lt(Y
_
-
COS
t)P-3
sin t)p
-3 ,
(11-18)
11.3 General
and in the X Y
(X, Y) system
(1
=
M)
-
(1 -p)
=
Equations
cost
(1
-
-
(1-p)(fI+sint)r
sin t-
kp-3
p)( + co*st)r -3 -3
_'Okp-3
,
(11.19)
.
We call p the vector from M2 to M3 with coordinates
(X, Y).
The orbit
by p(t).
is defined
Each collision is
generally replaced by a close approach of M2 by M3. dynamics, we will call encounter such a close the encounters in the same way as the corresponding
now
the tradition of stellar
Following approach. We number collisions. We consider is
encounter i E C. The motion of
an
M3
in the
vicinity of M2
quasi-hyperbolic. We call ei the minimal distance of approach. Consider a point Pi -of the orbit Q. during encounter i, or more
precisely O(ej). (The exact placepoint corresponding to a time tj such that p(ti) ment of.this point is deliberately left unspecified.) Qu is divided by the Pi into pieces. For p --+ 0, each piece tends towards a basic arc of the bifurcation orbit Q. We number the pieces in the same way as the corresponding basic a
=
arcs.
:1
'
The relative
a
velocity of M3 with respect to M2 has coordinates X, collision, there is x 0; from (11.17) we have then 1, y =
X
Ecost
=
=
sint
-
Y. At
Y
,
=
Esint +
(11.20)
cost,
and
. , X
22 y
VFX2
+
6fined
+
2
(11.21)
V
Thus, the'relative velocity modulus at a (, k) system and in the (x, y) system. For bifurcation orbits of types 1, 2, or 3, v is finite, as shown by Figs. 4.1 and 4.3, or by relations (8.3) and (4.8): where
is
v
collision has the
V
v
=
E)(1)
value
v
in the
(11.22)
.
has the
same
give
necessary to
in Sect. 8.2.
same
it
value at all collisions a
by
virtue of
(8.3);
thus it is not
subscript.
For p small but non-zero, the orbit of M3 passes in the essentially the same velocity v.
vicinity of M2
with
The attraction of M2
(Mihalas
and
produces Routly, 1968, p. 174)
a
deflection
by
an
angle Oi, given by
-21L
(11.23)
V2E, We find from
(11.20)
that
Oi also has the
same
systems. Next
we
consider
a
range of values of p
value in the
(X, Y)
and
(x, y)
Definitions and General Equations
11.
0 <
and
'(11.24)
PO
set of orbits
a
Q,.,
defined
,
by
a
function
p(t, p),
tends to Q. In what -follows,
0, Q. simplicity and write p(t). as
jL
-+
we
with the condition
will
generally
that,
omit /Z for
We have
for
0
ei -+
The deflection
(11.25)
/.1-+0.. must also vanish
angle 0
as we
approach the bifurcation
orbit:
for
0
We make
now
(11.26)
/-1-+0.
the fundamental
assumption that, for
M -4
0,
Ei varies
as a
power of p: ei
RoM
=
O(M'i)
(11.23)
Oi
and
()(Pl-vi )
=
(11.27)
-
11.27)
Guillaume 1971, p. 0 < Vi < 1
have then
(11.28)
-
(11.25),and (11.26) (cf.
we
show that the exponent vi, thus
defined,
must
satisfy
98) (11.29)
-
11.3.2 Intermediate Arcs
We will approximate each keplerian orbit, defined as
piece of QA by
keplerian arc, i.e. an arc from a 0 (Septs. 2.9, 2.10). problem 1L Since we are in the vicinity of a bifurcation orbit, the arc must be elliptic (Sect. 6.2.2). It is thus a part of a keplerian elliptical motion around a body M, of
mass
a
a
solution of the
=
1.
Consider the piece i of Q., delimited by Pj_j and Pi. Consider on this piece a point which is far from M2; or more precisely, a time t0i such that
(ti- I + ti) /2; but here again (We could for instance take t0i placement is not important.) We consider now the keplerian orbit which is osculating to Q,, at time t0i. We call it the intermediate orbit i. The corresponding vector description (with origin in M2) will be called pi(t). The
I p(toi) I
=
E) (1).
=
the precise
part of this orbit between times ti-1 and ti will be called the intermediate arc
i. We write
PW
=
Pi(t)
+
(11-30)
Api(t)
The introduction of intermediate orbits will
analysis of
the bifurcations. The real orbit
0A
play
a
fundamental role in the
will not be
compared directly proceed in two steps: the real orbit is first compared to the intermediate orbit, and the intermediate orbit is then compared to the bifurcation orbit. (See Guillaume 1971, p. 72; 1975b, p. 452; to the bifurcation orbit.
Instead,
we
11.3 General
Perko
1977a,
It will be shown in the next section that the real orbit
277.)
p.
7
Equations
position by O(p) only inside the time interval ti-1 < t < ti. Fortunately, O(A) approximation is sufficient for our needs in nearly all cases (the only exception'being the bifurcation 2P1; see Sect. 23.2). In other words, we will be able to replace the study of the true orbit by the study of the sequence of intermediate arcs. Since an intermediate arc is keplerian, it can be described by its four orbital elements. Thus we end up having to consider a finite system of ordinary equations, instead of a set of differential equations. and the intermediate orbit differ in
it turns out that this
11.3.3 Orders of
motion for the true orbit
equations of
The
Api and Ahi
of
Magnitude
are
given by (11.19),
or
in vector
notation
;6
-(1
=
PM
P -
pM13
lp
/-Zp
+ PM
(11.31)
1PI,
where p. is the vector from M2 to Mj:
(-sint) t
Cos
PM
=
(11.32)
equations of motion for the intermediate. orbit 0: equations with p The
are
given by
the
same
=
Pi-Pm
A
Api(t)
1pi is
Api(t0j)
-
pmJ3
-+Pm]
completely defined by =
4,bi(toi)
0,
and the differential equation P
lp
=
the initial values
(11-34)
0
,
Pm,
+
pmJ3
A
1p,
PM
pmJ3
+ Apm
Ap -
JpJ3
(11.35)
or
A i
OW
O(Api)
+
PP
(11.36)
-
JpJ3
magnitude of Api and Abi in the interval t0j. We consider t0i integrate (11.36) twice, starting from t other encounter there is no first the interval t0i < t < tj definition, fi. By in that interval, and the relative velocity v at encounter i is finite. It follows We estimate < t <
now
the order of
ti. We
=
-
that
lp(t)l
=
E)(ti
The last term
will
assume
-
t)
.
(11.37)
in (11.36) becomes large in the vicinity of an encounter, and
for the purpose of
we
dominant, i.e. that the first two terms can be ignored estimating orders of magnitude. We have then
that it is
11. Definitions and General
8
A;6i(t)
=
Integrating,
0[1,(t,
we
t)-2j
_
(11.38)
find
=ft
Albi (t)
Equations
Abi(t')dt'
=
O[ft(tj
=
0 [fz In (ti
-
t)-']
,
.
t
Api (t)
=
ft. iAbi (t)
The last equation
Api (t) We will follows
=
0 (/,t In
ej)
in
t)]
(11.39)
particular
(11.40)
.
neglect logarithmic factors in orders Perko 1976b, p. 399). Thus
of
magnitude,
here and in what
(cf.
Api(t) For t
Ajbi
=
=
=
tj
jP(t)j
00-1) -
(11.41)
,
we
ej,
O(tzlej)
have
Api
,
in the interval tj
Finally,
and
gives
dt'
=
O(Ej)
=
0(ti)
ej < t <
-.
(11.42)
.
ti,
we
have
(11.43)
,
therefore, from (11.36), ignoring again the first
A;bi(t)
=
Integrating,
two
terms,
0(/jft
we
(11.44)
find t
A,bj (t)
=
Abj (tj
-
6j)
+
fti
0(p/ei) -'i
t
Api(t) Thus,
=
we
A,bi(t)
=
Apj(ti
-
Ci)
O(A/C:j)
A similar argument < t <
=
0(p)
(11.45)
.
have ,
can
Api(t) be made
interval ti-1 < t < toi. Therefore
ti-1
Abi(t')dt'
+
=
0(p)
for
toi
< t <
by integrating backward
we
have
Api(t)
=
O(p)
in
tj
.
time,
(11.46) in the
for the whole
piece
ti.
The time tj has been defined with
some
arbitrariness,
'as a
time
during
encounter i. Therefore
Proposition 11.3.1. The intermediate orbit i gives an O(Y) approximation 1 and i, and of the true orbit for the whole piece between the encounters i also during these close encounters. -
11.3 General
9
Equations
the true orbit can be replaced keplerian arcs. This superficially resembles a second species orbit. However, the points at which the intermediate arcs are joined do not coincide with M2 any more. The point corresponding to encounter i lies at a Within
by
0(p) approximation, therefore,
a
sequence of
a
distance
E)(,Ei)
These
take values in the range 0 < vi < 1. be chosen with some latitude, because the two succeswhere vi
=
points
can
can
0(p)
in the ei
vicinity intuitively as follows: the two intermediate arcs make locally an angle of the order of 0, or p'-vi as shown by (11.28); on the other hand, the span of the ei vicinity is of the order of 1-tvi according to (11.27); multiplying, we find that the two arcs deviate from each other by a distance of order 0(p) inside the ei vicinity. sive intermediate
i and i +
arcs
of encounter i. This fact
can
I coincide
11.3.4 More Accurate Estimate of
We compute the second
A i
now
As in Sect. 8.2, and after
time
t
=
an
we
use
Api
accurately; at
an
this will, be needed below to obtain
encounter.
subscripts b and
encounter. Consider the
(Fig. 11.1,
ti
more
matchin relatioq
to order
be understood
a
to
tangent
where the left and
designate quantities before
to the intermediate arc i at
right figures correspond
to the
cases
where M3 passes to the left or to the right of M2, respectively). We define the unit, vector ib on this tangent and the velocity modulus Vb by:
i(ti)
=
Vbib
(11.47)
-
M*
ibf
Fig.
11.1.
We call
Jb
Osculating straight-line
ib
the unit vector
the tangent, and is
db
=
M2
0(ci)
.
db
M2
motion.
orthogonal
to
ib and directed from M2 towards
the distance from M2 to the tangent
(see figure).
There
(11.48)
Finally we consider a point M,, with a uniform motion on the tangent, lating to the intermediate arc. This motion is given by
oscu-
11. Definitions and General
10
p*
db ib + (db tan O)ib
=
Equations
(11.49)
7
where p* is the vector from M2 to M* and the angle 0 is defined (in such a way that it always increases with time). We have
P*
db Vb?,b
=
do
P*(t)
=
The moduli
are
+
0(t,
are
t)2
(11.51)
given by
lpi(t)l
-
Cos
=
__ b L Cos
0
+
APi
A;6i
0(tt)
+
1p,13
Substituting (11.51) t 'i
ft.
(11.52)
and
using (11.46):
(11-53)
z
and
AP*
t)2
_
0(112/P )
+
jp*13
0(t, -
(11.36), substituting (11.30)
We rewrite,
figure
(11.50)
.
_
db
1P*(01
the
Zb -2 (h t os 0 -j-
=
We have for the intermediate
P,(t)
on
integrating,
+
0[(t,
_
we
obtain
t)2tZ/P3]
+
0(tl)
+
0(tZ2/P3)
I
dt
.
(11.54) Using (11.49), (11.52)
and
changing the
variable with
(11.50),
we
find for the
first term
f i,
P* bt
t.
1P*11 A
-
Vbdb P =
-
Vbdb Here
Ob and 00i
f
dt
&b
(ib COS 0
+
ib sin 0) dt Ob
f
UbCOS0+ibsino)do=---L- Ij bsino-ibCOS 01,00i
[j
(1
are
b
Vbdb
+ sin
Ob)
ib
the values of
COS
Ob]
(11.55)
=
able with
(11.50)
ft.'iti0 I
d2b
(t,
and
_
t)
we use
2
COS
01
t t0j. In the tj and t -ir/2. t0j,. there is 00i
corresponding to
last expression we have used the fact that for t For the second term of (11-54), we substitute
(11.22);
we
=
=
(11.52),
-
we
change
the vari-
obtain
do.
(11-56)
0)
(11.57)
There is Vb (ti
-
0
and therefore
=
db (tan Ob
-
tan
11.3 General
tj
-
t
0(db)
=
0(db tan 0)
+
(ti
,
-
t)
2
2
0(db)
=
+
Equations
0(d2b tan2 0)
11
.
(11.58)
(11.56),
After substitution in
0
(P fOb do) Cos
0
expression becomes
that
fOb
+ 0
00i
00i
sin2
(11.59)
do
Cos
The first term reduces to 0 (it). In the second term, we neglect the logarithmic --7r/2 and we obtain again O(IL). singularity which exists for 0 -+ 00i The third term of and
(11.54)
is
The fourth term
becomes, using (11.52)
(11.50), -Ob
fooi (e 0
Cos
d2b
Finally
we
A-bi (to
For t
=
0(112/6?)
(11.60)
obtain
Vbdb
[ib
COS
Ob
-
can
jb (1 +sin
Ob)]
+
O(P)
+
0.(112/ei2)
.
(11.61)
1, near its quantities tj (in the direction of decreasing
be made for the intermediate
at encounter i. We define in the same way
beginning Za7 Va) 3a,
0(A2/d2b)
do
0
Similar computations
t).
0(tz).
as
arc
i+
above the
da, oa. We integrate from to,j+1 to to,j+j, there is 00,j+1 = +7r/2. We obtain
=
Ajbi+, (ti)
[ia
Vada
COS
Oa
+
ja (1
-
sin
Oa)]
+
0(tL)
+
O(M216'? (11.62)
11.3.5
Matching Relations
We consider
now an
P(ti) and, for
Pi(ti)
=
-
Pi+1
(ti)
arcs
i and i + 1.
(11.63)
OW
the intermediate
P(ti) so
-
C, and we effect the matching First, (11.46b) gives
internal encounter i E
between the intermediate
=
are
i + 1
(11.64)
OU0
that
Pi+1(ti)
=
Pi(ti)
+
(11.65)
OGL)
This constitutes the first matching relation. We consider
now
the relations
(11.61)
and
(11.62). Equation (11.46a)
shows that
i?i+l (ti)
=
it follows that
Jbi (ti)
+
O(Pfti)
(11.66)
12
11. Definitions and General
Va
-_
da
=
O(P/'Ei) O(A)
Vb +
db
Therefore
incurring
+
error
al6i+i (ti),
=
ib
--:-:
Ob
O(IL16i) O(P/fi)
ja
+
+
ib
+
O(A/fi) (11.67)
-
replace, a-subscripted by b-subscripted quantities
relative
the absolute
Oa
i
we can a
ia
,
Equations
error
is
plei.
0(p'/Ei2),
A
Ob
b COS
=
Vb db
Since the first term of
(11.62)
in
and is included in the last term. So
ib(l
+
-
sin
Ob)
+
0(p)
+
(11.62), pfti,
is of order we
have
0(p2ft?) .(11.68)
There is
,6i+ 1 (ti)
=
.6 (ti)
and, using (11.61)
-
Abi+ 1 (ti)
and
Jbi (ti)
i (ti)
=
+
A'bi (ti)
-
Abi+ 1 (ti)
(11.69)
(11.68),
-2Ajb
+
Vbdb
O(A)
+
0(t12 Idb2)
(11-70)
This is the second
matching relation (see Guillaume, 1971). Note that we ja Va,- da We note also that ib Vb, db can be taken in an arbitrary point in an ci vicinity of M2: the variations Of ib and Vb inside that vicinity are of order Ei, and the variation of db is of order E?, so that we always have a relative error O(Ei), which when applied to the term of order 1LIEi gives an absolute error 0(ti). might just
as
well have used
The relation
which holds in the
i
(X, k) system
of
axes
with their
M2 and with fixed directions, is also true in a system of rotating with their origin in M2, with Vb, ib, db redefined in that system. Indeed in
origin axes
(11. 70),
-
7
the difference
bi+1 -,bi
is invariant in this
change
of axes; within
a
distance
(Fi of
M2, the relative speed of the two systems is of order 4Ei, so that the variations Of Vb and ib are of order ei and the variation of db is of order q' The second term in the right-hand side of (11.70) is significant if 1_11db > A and lildb > A 2 Id2; we recover precisely the conditions (11.25) and (11.26)! b '
This is
a
confirmation that
tions. When
(11.70)
is indeed the correct relation for bifurca-
away from the bifurcation at 4 both become of order 1 and the relation becomes
that
we
we move
must
now
compute
exactly the
more
node, the terms 2 and meaningless; this means deflection angle, which is not a
small any more. Conversely, in an antinode, the terms 2 and 3 become both of order p and (11.70), coupled with (11.65), shows that the intermediate arcs i and i + I are identical within O(IL): the effect of M2 has become negligible. In
a
partial bifurcation, the boundary conditions
at collisions 0 and
n
must also be considered. In these
vanish at the
O(IL)
and
Pi
(to)
we
=
collisions, the deflection angle does not bifurcation; therefore, for M 54 0, the distance of approach is
have
O(A)
P.(t.)
=
OUZ)
(11.71)
11.3 General
11.3.6 The Case 1A
Until
now
p =
case
(i.e.
in
a
at
a
in this
0 and
case
=
we
have assumed that p
show that the
i4
0. We consider
matching equations
are
now
the
also valid for that
for first and second species orbits), provided that (11.70) is written slightly different way. This extension will be useful in what follows.
ej is
at
zero
node, Oici
node,
a
at
zero
non-zero
at
an
a
place of (11.23). as before;
time tj
in the
time of the collision. The intermediate
0. In
particular,
(ti)
Pi+1
Conversely, Oi
is
non-zero
(11.72)
We define
=
antinode.
antinode. Thus
an
0
=
which takes the
Api
13
0
chapter
we
Equations
Pi (ti)
=
we
case
arcs
of
a
node, tj coincides with the
coincide
now
with the true orbit:
have
P(ti)
(11.73)
matching relation (11.65) is still true. partial bifurcation, there is a collision at each end,
and therefore the first For Pi
a
(to)
=
P(to)
P.(t.)
0
=
P(W
=
=
and
(11.74)
0
and therefore the equations (11M) also are still true. Finally we consider the second matching relation (11.70). For the intermediate orbit is the
A+1 (ti)
(ti)
=
same
an
antinode,
before and after ti, and
kti)
(11.75)
On the other hand, db is non-zero. Therefore (11.70) is still verified. For a node, (11.70) becomes meaningless because pldb has the indeterminate value 0/0. However, we can rewrite the equation by multiplying both sides by db:
db
Oi+j (ti)
We have from Vb
A
-
2Ajb
0i (ti),
+
Vb
0(lidb)
+
O(p2/db)
(11.76)
(11.70)
1041 (ti)
db
-
2jb
-Mti)]. + 0(tL) + O(IL2 Id2)
(11.77)
b
from which
0
( ) P
db
=
0
['0i+1 (to
-
oi (ti)]
(11.78)
(11.77) have been dropped (11.76) can be,written
The last two terms of
(11.29).
and
db
Thus
i+i(ti) 710i(ti)]
2 Ajb Vb
+ 0 (1-idb) + 0
in view of
(11.48), (11.27),
1 P [.bi+l (ti)
-
Jbi'(tol (11.79)
This
new
equation
is still true for y
=
0: all terms vanish.
11. Definitions and General
14
Equations
11.4 General Method Since
we are
vicinity
of
a
considering bifurcation
now are
small but
non-zero
characterized
by
values of it, orbits in the parameters: tt and
two small
specifies how far we are from the two-body problem, specifies how far we are from the bifurcation orbit. Fig. 1.1 essentially represents this two-parameter space; we will consider particularly the cases c and d which represent the generic situation in the 0 is represented by the full vicinity of a bifurcation orbit Q. The value p lines. A value it > 0 is represented by the dashed lines. Const. are approximately parallel In the vicinity of Q, the curves C straight lines (not represented on Fig. 1.1). Thus, JACI generally increases AC. The parameter p
while AC
=
=
with fixed p. are second species orbits, formed by a sequence of T- and S-arcs. (Occasionally they can also be first species orbits.) The branches emanating from bifurcation orbits of various types have when
we move
For p
=
away from Q
0, the solutions
on a curve
are
known:
they
been studied in detail in Sect. 6.2. For IL >
asymptotic
reasonable strategy seems to be to begin by studying the branches of the dashed line characteristics in Figs. 1.1, i.e. the
0,
a
make
full lines of the p = 0 case. Then we progressively vicinity of Q, until at last the branches are joined two We cannot
simply
do this for the characteristic
but fixed value of p, because values of p, but in the limit p
required. AC
We define
=
O(A')
In
JACI
fundamental quantity
=
v
-our
by two. corresponding
to
a
small
interest lies not in small
slightly
a
way toward the
subtler
approach
is
by
(11-80)
a
fixed value of
form of the
The relation
0. Therefore
our
.
We consider then
asymptotic
a
fundamentally -+
approach the
away from Q and
parts of these characteristics which extend
equations
(11.80)
v0(In p)
can
v
and
let p
we
--+
0. We will determine the
and of their solutions in that limit.
also be written
(11.81)
.
0 corresponds to y 0, AC 0 0, i.e. the full lines in Thus, the value v close to the full lines, i.e. value of to A small 1.1. v corresponds points Fig. the distant parts of the dashed characteristics in Fig. 1.1. Increasing values of v correspond to increasingly close parts of the characteristics. Thus, the 0 and strategy defined above will be implemented by starting from v studying progressively increasing values of v. This strategy will be used for bifurcations of types 1 and 2, in Chaps. 12 and 17, respectively. The details differ from one type to another, but =
=
=
the
general
sequence of
results in
method is the cases
same.
We will find in each
for v, such that each
case
specific asymptotic equations.
requires
Each
case
a
case
that there exists
separate
is either
a
treatment and
an
open interval
11.4 General Method
or an
the
isolated value of
For bifurcations of type
v.
following
sequence of
v=O,
O
=
The
0
,
0 <
cases
v
<
we
will find in
Chap.
12
cases:
For bifurcations of type 2, v
1,
15
we
1/3,
will be studied
Chap.
will find in
v
=
1/3
1/3, by
one
(11-82)
v>112.
v=112,
one
<
17 the <
v
following
1/2,
v
=
sequence:
1/2. (11.83)
in the order of the sequence, i.e.
for increasing v. In the first case, v 0, the asymptotic expressions of the variables can be"deduced from the study of bifurcation orbits made in Chap. 6 =
(see the
below Sects. 12.3 and
following procedure
is
17.3). in ea h
of the
following
cases
with
v
>
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 <
12.3.2.2 S-Are. We consider
the
now
case
of
a
m.
Aai+l
=
We call Aa this Yi
=
common
0,
Yi+.
Summing (12.43a) 0
=
G2SiAC
from which Aa Rom
Aai+,,,
Aai+2
=
1/2
(j
from i to
1, 2,
=
27
M
-
1);
(12.58)
.
(12.51) gives (12.59)
-
m
mG3Aa
+
basic
arcs
G3KsiAa
and +
using (12.59),
O(AC2)
+
we
find
O(Aa 2)
(12.60)
extract
we
Q2
(msi
(12.59)
0
the
over
-
value.
<
S-arc, running
We have yi+j i4 0 for the intermediate antinodes therefore, from (12.43b), i+
v
-
K)G3
AC +
(12.43a),
and
.(12-61)
O(AC2)
we can
then compute the successive values of the
Yi:
j
M
msi
Yi+j
-
K
_j -
msi
-
K
G2AC
+
O(AC2)
if
j
is
odd,
G2AC
+
O(AC2)
if
j
is
even.
12.4 The Case 0 < We consider
now
the
<
v
case
ft >
(12-62).
1/2 0, AC 54 0, corresponding
to
v
>
0. We
G(M'). We will try to extend (11.80): each solution obtained in the previous section for v = 0 to the present case v > 0. This corresponds to the asymptotic branches of the families of orbits, AC
recall the fundamental relation
=
for p 54 0, which are close to the branches for p = 0 (see Fig. 1.1). Here we use for the first time the general method described in Sect. 11.4.
Reference is made below to the successive steps. Step 1. We estimate orders of magnitude by extrapolating from the previous case v 0. The equations (12.50), (12.57), (12.62) suggest that yj is of the order of AC inside a first species orbit, a T-arc, or a S-arc; (12.61) suggests that Aaj is of the order of AC in a S-arc. 0: yj at a node On the other hand, some quantities vanish for v (see (12.52)), Aaj in a first species orbit or a T-arc (see (12.48), (12.56)). We cannot at the moment estimate their order of magnitude for v > 0 (this will be remedied below; see Step 8 in Sects. 12.4.1 and 12.4.2). These quantities will become generally non-zero for v > 0; it seems reasonable to assume that they will be small compared to AC if v is sufficiently small, Step 2. This suggests the following change of variables: =
=
Yj
=
AC
Yj*
,
Aai
=
A(7 Xi*
.
(12.63)
28
Quantitative Study of Type
12.
3. It will be convenient. to make
Step
simplify
order to
Yj*
fli-1
Yi (Xi+1
Y10
a
second
change
of variables in
equations:
G2 -sixi
xi
(12.64)
Z;T3
4. Substituting in (12.32), collecting the dominant terms in the members, and dividing by AC or AC2 as appropriate, we obtain
Step +
the
-G2Siyi
=
left-hand
f1i
1
-
+
1
-
Xi)
(1
Ksi)Xi
GjG3 -_
O(PAC-1)
=
+
AC-2
G22
+
O(AC)
+
O(ILAC-1)
0(/_ZAC-1)
F"
,
0(pAC-1)
=
for
i E C
,
a
i E A
,
,
,
partial bifurcation.
(12.65) (12.33)
The additional relation
Ksi)Xi (1 Ksi)gi+l O(PAC-1) + O(Xi2AC) + 0(Xi2+1 AC)
Fj+j
-
Step
Ri-1
(1
+
+
-
case
V
<
-+
0.
(12.67)
2 are
then obtained
by equating the right-hand mem-
zero:
+
Y j_j
Yi (Xi+l
Yo
right-hand members to vanish for A
if
asymptotic equations
Pi
(12.66)
i E C
1
0 <
bers to
-
5. We want all terms in the
This is the
The
becomes
=
0
-
+
,
1
-
(1 +-Ksi)X-i
Xi) F"
=
0
,
0
for
a
i E A
0,
i E C
,
,
(12.68)
partial bifurcation,
and
Fj+j
-,
Ri-I
+
(1
+
Ksi)fCi
(1
-
Ksi)gi+l
=
(12.69)
i E C
0,
equations. We observe that (12.68b) has two distinct 0 corresponds tO-the case where i becomes a Yj 0 corresponds to asymptotic limit; the case Fj 0 0, Xi + Xj+j
We solve
now
these
solutions. The solution node in the
-
=
=
Accordingly, the form of the solution depends on whether the asymptotic orbit belongs to the first species or the second species. Steps 6 to 9 will be executed separately for each species. the
case
where i becomes
an
antinode.
Species Orbit
12.4.1 First
Step
6. The two collisions become
tions
(12.68)
P,
-P'o X1 + X2 +
antinodes; therefore the asymptotic
equa-
become
-
1
=
(1
-
0
-
-
K)Xj
=
0,
Yo
+
P,
-
1
-
(1
+
K)X2
=
0,
(12.70)
12.4 The Case 0 <
These
equations reduce
X1
X2
`
0
=
As before, there is
]PO
__
]Pi.
Y, =.1
+
degeneracy.
a
10
property, which gives
<
112
P1,
=
It
so
(12.71),
-
can
be removed
by using the symmetry
that
(12.72
2
7. For p positive and small, the solution of the system of continues to exist, provided that
Step
(12.65) 1. the
29
to
YO
7
v
right-hand members are small, asymptotic equations (12.68) is
equations
terms in the
error
2. the Jacobian of the
ilon-zero
(implicit
theorem).
function
The first condition is- satisfied if
(12.70),
the equations
1/2.
<
v
If
we
compute the Jacobian of
find that it
vanishes; this is a consequence of the eliminate this We degeneracy. degeneracy by using the symmetry relation reduces The 3 equations for the 3 variables (Yi, Xi, X2): to ] o Pi. system we
=
2RO
-
1
YO(X1
(1
-
+
-
X2)
K)Xi
=
2Y-1
0
-
1
(1
-
+
K)X2
0
(12.73)
0
=
The Jacobian is
-(1
2
JJJ
=
K)
-
2
0
0
-1/2
Therefore: For 0<
v
0
-(l+K) 1/2 1/2,
<
(12.74)
=2.
there exists
for
p > 0
a
solution
of the funda-
mental equations (12.65) close to the solution (12.71a), (12.72) of the asymptotic equations. The error is of the order of the largest term in the right-hand
members of
X1) X2
(12.65):
0(' jq
=
Y701 1 1 =1
+
2
for
Step 8. We X, and X2. X1 +X2
Combining
91
X2
=
=
0(11 IC-2
O(AC)
can
+
O(P AC-2).
(12-75)
(although it is not necessary) obtain (12.65b) and (12.75b) we have
now
finer estimates
From -2
with
-(1
+
+
PAC-2 [1
2
G2
+
(12.66) (with Rj+j
K) K)
O(AC)
+
fli-1),
we
=
O(A AC-2)]
(12.76)
obtain
GlG3,,Ac-2 [1 + O(AC) + O(PAC-2)] 2
G2
G22
1,AC-2 [1
+
O(AC)
+
0('4'AC-2)]
(12.77)
Quantitative Study of Type
12.
30
Step
Finally,
9.
1
physical variables, using (12.64)
go back to the
we can
(12.63):
and
Yo
=
Y1
=
-G2 AC [1 + O(AC) 2 G2 2
Aal
=
Aa2
=
[1 + O(AC)
AC
(K
+
1)
(K
-
1)
6. For
Yj
G2
0,
>
v
G2 G,
O(JUAC-2)]
0(11 AC-2)]
+
/'ZAC-1 [1
+
O(AC)
+
0(/IAC-2)]
PAC-, [1
+
O(AC)
+
0(tIAC-2)]
we
continue to call node
a
value of i for which
(12.79)
0
=
(excluding the two ends of a bifurcating arc) and Yj $ 0. We call also T-arc or S-arc the part
which
nodes, a
T-
or
We
as
one
node and
case
:---
of
Fi+2
==
antinode
a
value of i for
of the orbit between two
bifurcating
arc.
It reduces to
0.
arc an
set of
equations. We consider
0
-
-
-
1
-=
ffi+2
=
Ksi)f(i+l (1 + Ksi)ffi+2
(1 0
=
-
-
This linear system of `
v -+
independent T-arc. The equations are
a
'.Pi+l +fIj 1 Ri+2 + Y41 Xi+1 + ffi+2 Xi+1
end of the
an
defined in Sect. 4.3 for
again obtain for each
first the
Fi
between
S-arc
or
(12.78)
Species Orbit
12.4.2 Second
Step
G,
+
0, `-
0
1
(12.80)
-
equations
-Pi+1
0,
We consider next the
case
of
is =
a
easily solved: 1
(12.81)
-
S-arc. There is for the intermediate antin-
odes
Xi+j We write
Xi+j
+
Xi+j+l
=
Xj+1 =
0 we
(j
M
for the successive
arcs
=
-
-
Yi+,.,-,
(12-82)
(12.83)
Pj+j +Yj 1 + (1 Ksj)X 0 Fi+2+Fi+1-1-(1+Ksj)X=0, +
1)
have then
(-WX
(12.68a) gives
Fi+,,,
-
-
1 +
(1
-
,
Ksi)X
=
-
0
(12.84)
12.4 The Case 0 <
equations and subtracting the
the odd
Adding
(12.68c,d), -1 +
we
(m
even
v
<
1/2
31
equations, and using.
obtain
Ksi)X
-
0
(12.85)
,
from which
M
1Ksi
(12.86)
-
and
j
M
-
Xj+j
m
Ksi
-
m
Yi+j
=
-
Ksi
j
is
odd,
if
j
is
even.
(12.87)
j m
if
Ksi
asymptotic solution for an orbit or Step For 0 < it < 1, the solution of the the second of a bifurcating arc species. if exist to continues v < 1/2 and if the Jacobian IJI of the system (12.65) We must now compute the latter. is not zero. asymptotic equations (12.68) There are 2n variables and 2n equations for a total bifurcation; 2n + 1 variables and 2n + 1 equations fora partial bifurcation.We consider a node ini. The partial derivatives of the expression f appearing on the left-hand 7. We have thus obtained the
(12.68b)
side of
af -
are
Xi+1
+
Xi
,
i
af
Of
'99i+1
.91i
(12.88)
Only the first of these derivatives is non-zero. Therefore the corresponding line in I JI has a single non-zero element. We can then factor out this element and eliminate the corresponding line and column. After this has been done for all nodes, IJI is decomposed into a product of smaller determinants, each of which corresponds to a T- or S-arc. For a T-arc, there are 3 variables Xj+j, Y41i -ki+2, and 3 equations of the form
(12.80b,c,d);
the determinant is
Ksi
-1 +
1 1
0
0 -1
Ksi
=
-2Fi+l
(12.89)
.
Yi+I
Ri + i This is
-
non-zero
in
view
of
(12.81).
-For_a S-arc, formed of m basic arcs, there are 2m 1 variables Xj+j, 1 equations (12.68a) and (12.68b). The Xi+m, and 2m Yi+1 Xi+2 determinant is easily computed by successive eliminations; we obtain -
_
-
i
i
....
f4i
_M
(12.90)
.
j=1
This is
non-zero
Thus
(12-65)
IJI
is
(12.87b). v < 1/2, there exists for p 0 0 a solution of (12.79), (12.81), (12.87). The error is of the order of the
in view of
non-zero.
close to
For
Quantitative Study of Type
12.
32
1
largest term in the right-hand members of (12.65), the following form: -
For
For
a
=
For
O(AC)
O(tZAC-2).
+
(12.91)
T-arc:
fCi+1
=
Xi+2
=
,+J -
that the solution has
node:
a
R, -
so
a
=
0(tZAC-2), O(AC) + OUIAC-2), 1 + O(AC) + O(t,'AC-2). O(AC)
+
(12.92)
S-arc:
li+3
(-W -7-
m
-
+
Ksi
M
-
3 _
Yi+3
.
m
-
3 m
8. We
Step
-.Ksi Ksi
-
O(AC) +
0(tZAC-2),
O(AC)
+
O(tAC-2)
if
j
is
odd,
AC)
+
O(tAC-2)
if
j
is
even.
+0
obtain
can
+
(12-93)
finer estimates for
now
some_variables,
giving
explicitly. We consider first the value Y at a node i. As in Sect. 8.2, we designate by the subscripts b and a (before and after) the quantities relating to the two arcs which are joined in i; in particular we call the dominant term
Mb -
and Tna the values of
(joining
TS node
At
a
to
(12.93):
Xi+1
-
Ma
Ksi
-
O(AC)
(12.65b) gives Pi
=
a
(Ma
for these
T-arc to
-1 =
+
O(AC)
according
and for the T-arc,
fC
m
to
arcs.
S-arc),
a
+
we
have for the
S-arc, according
0(,LAC-2),
(12.94)
(12.92)
O(tZAC-2
+
(12.95)
then
-
Ksi)
GiG3
Z tLA C-2 [1 + O(AC) + 0 (tAC-2)]
(12.96)
2
-
For
a
ST node,
Xi
we
have for the S
-1 =
+
_
?nb +
Ksi
O(AC)
+
arc
O(IIAC-2),
(12.97)
and for the T-arc
fci+l so
=
O(AC)
+
O(ILAC -2),
(12-98)
that
Yi
=
(rnb
+
KSi)
GiG3 2
G2
14AC-2 [I+ O(AC)
+
O(tAC-2)]
(12.99)
12.4 The Case 0 <
-
Finally, for
(12.94)
SS node, the equations
a
(Tnb
Ksi) (m,,
+
[1 + O(AC) consider the values X for
we
Xi+2
+
Combining
-,OAC
Z
-2
-
Ri+2
(1
-
2
O(IIAC-2)]
+
Ksi)
+
_
2
+
2
C 0(fi+2AC)
Yi is given by (12.65c) if the node i by (12.99) if it is a junction of the T-arc
Similarly,
g" we
=
1 if it is
a
2
G2
g"
define
we
G1G3
Ksi)
/ZAC-2
+
O(t,2AC-4)
+
O(t,2AC-4) .(12.102) .
with
a
0(tIAC-1)
that both
so
+
a
cases can
O(t,2AC-4)
be
(12.103)
0 if the node i +'2 is the end of
=
of the T-arc with
junction
+
bifurcating arc, or preceding S-arc. We define:
0 in the first case, g' = 1 in the second case, written as a single formula +
obtain
is the end of
=
91('Mb
(12.101)
tIAC-2
-
2 WCi+1AC)
we
O(X2i+2 AC)
+
2G2
=
O(tAC-2)]
PAc-2
2
2G2
-
Ri
(12.100)
(12.65b) gives, using (12.92),
(12.66),
-
g'
33
(12.97) give
and
[1 + O(AC)
+O(PAC-1) + O(X i +JAC) yi+2 Yi Xi+2 (1 Ksi) 2 +
1/2
2
-
+O(PAC -1)
+
T-arc.
a
with the additional relation
Yi
Xi+1
G1G3
Xi+1
<
Ksi) G, G3 -2 /_IAC 2r G2
-
Tnb + Tna
Next
v
a
a bifurcating arc, following S-arc. Using (12.96),
obtain-
Fi+2
g"(m,,
-
Ksi)
G22
/tAC-2
+
O(IIAC-1)
+
0(/,L2AC-4) (12.104)
Substituting
Xi+1
=
into
[-(I
(12.102), +
Ksi)
+
we
obtain
91(Tnb
+
Ksi)
-
g"(m,,
-
Ksi)] GlG311AC-2 2G2 ,2
+O(IZAC-1) [-(1
Xi+2
-
9.
O(,U2AC-4,)
KSi)
+0(AAC-1) Step
+
Finally,
+
-
we
we can
obtain
+
Ksi)
O(P2AC-4)
(12.63); (12.57), (12.61), (12.62):'
and
91(Tnb
a
.
+
g"(m,,
-
Ksi)] GlG3,,Ac-2 2 2G2 (12.105)
go back to the physical variables, using (12.64) of the equations (12.51), (12.56),
generalization
12.
34
-
For
a
Yi
-
For
a
Yi
-
For
a
Yi
Quantitative Study
of
Type
1
T,5 node: =
-
(Tn,,si
-
K)
G2
PAC-1 [1
+
O(AC)
+
0(tAC-2)] .(12.106)
tZAC-1 [1
+
O(AC)
+
O(PAC-2)]
ST node: =
-(TnbSi
+
K)
G2
SS node:
(TnbSi + K)(m,,si (Mb + Tn,,)Si
-_
-
K) GIG3 G2
ILAC-1
[1 + O(AC) -
Antinode: For
Aai+l
=
a
Aai+2
yj+j
For
a
=
+
0(tZAC-2)]
(12.108)
T-arc:
[(K
si)
+
[I + O(AC)
-
(12.107)
[(K
-
+
si)
-
91(MbSi
+
K)
+
9"(7na8i
-
K)
+
g"(masi
-
K)]
- LyAC-'
K)j
_ l tzAC-1
2G2
0(ttAC-2)] -
91('MbSi
+
2G2
[1 + O(AC) -+ 0(j1AC-2)] = G2SiAC [.1 + O(AC) + O(ILAC -2)]
(12.109)
S-arc:
Aai+j
msi
1_ G2 AC [1 K G3
-
M-j -
Yi+j
7G2AC
K
MSj
+
O(AC)
[1 + O(AC)
G2AC
[I
+
+
+
O(AC)
0(tAC-2)]
O(IIAC-2)] +
0(ttAC-2)]
for
j odd,
for
j
even.
(12.110) Fig. 12.3 shows in log-log plots the variations in order of magnitude of the quantities yj and Aai as functions of AC, given by (12.106) to (12.110) in the interval p 1/2 < AC < 1.
12.4.3 Sides
ofPassage
Incidentally,
we recover
ter 8. As is
easily
seen
the rules for the side of passage established in chapfrom Fig. 12.1, yj has the sign of siacosO, with a
defined in Sect. 8.1. Therefore, from the definition (8.22) of s and the definition (12.31) of G2, a has the sign of -ssjG2ACYi- On the other hand, Ma
equals ka if the point i is P, -ka if it is Q, or sika in general. Similarly, Mb equals -sikb. The above equations show then that for a TS node, 0' has the K); for a ST node, the sign of ssi(K kb); and for a SS sign of ssi(ka of s(K kb)(ka K). We recover the results of Sect. 8.2.1. node, the sign -
-
-
-
12.5 The Case
v
1/2
=
35
AC
AC
1/2
1/2
T, S, E
S
Y,
7
1
71
1
Aa, 1/2
)U1 /2 TE
Ts' Ss\
A
Fig.
values at
Next
obtain: We
as functions of AC. Both scales are logaxithmic. panel, TS and SS represent values of yj at a node, while T and S represent an antinode, inside a T-axc or S-axc respectively.
12.3. Vaxiations of yj and Aaj
In the left
we
o,
=
recover
consider antinodes. For
s; for a S-arc, we have: o, the results of Sect. 8.3.1.
12.5 The Case We
study
that this
now
what
means
AC
=
first species orbit E
sign[ssi (msi
-
K)]
show that when the ways of order
tj/2.
fusion
a
2. This
Step
Al /2y* i
=
Step
Yj* W*
or a
T-arc,
sign[sk (k
-
we
K)].
1/2
happens when the value
v
=
1/2
is reached.
(Remember
O(MI/2).)
=
(12.78), (12.106) to (12.110) of the previous section value v 1/2 is reached, the variables yj and Aaj are al=
The distinction between nodes and antinodes
of the consecutive
arcs.
There is also
first and second species. The S_ and T-arcs lose their apparent on Fig. 12.3.
Yi
=
1. The results
Step
there is
v
a =
=
si
disappears;
fusion between the
identity. This
is also
suggests the following change of variables:
Aaj
=
IL 112X* i
AC
.
3. It will be convenient to make
=
a
sign(cos 0) VIG, G3 Yi
-sign(cos 0)
1
G2
V _G_,G3 W.
Xj*
a
=
P 1/2W*
second
=
-si
(12.111)
.
change
of variables:
sign (Cos 0)
FqG_3 1
Xi
,
(12.112)
36
12.
(Gi
Quantitative Study of Type
positive; see (12.31).) Substituting in (12.32), collecting the dominant terms member, and dividing by appropriate factors, we obtain
and
G3
Step
4.
left-hand
are
Yi +'Yi-l
-
y,(X,+, + Yf0
=
1
W
X,)
-
(1
+ 1
+
Ksi)Xi
=
O(Al/2)
0(/11/2), y 0('41/2)
i E C
=
O(PI/2)
for
=
.
i E
in the
A,
,
(12.113)
partial bifurcation.
a
Step 5. The right-hand members do tend to zero for equations are obtained by equating them to zero!
1L -+ 0. The asymp-
totic
Yi
+
Yi-1
Yi(Xi+1 YO
=
Step
-
+
W
Xi)
-
+ i
Y"
0,
(1
=
+
=
Ksi)Xi
0
explicitly
in
,
(12.114)
partial bifurcation.
a
(see below) suggests
6. The available evidence
cannot be solved
iEA,
0,
i E C
,
for
0
=
general.
Its
study
will be the
that this system of chapters
object
13 and 14.
Step 7. Also it will only be possible to compute the Jacobian after the system has been solved (see below). We will see that it is non-zero in general (but it can vanish in isolated points on the characteristics). Therefore there exists in general, for /-t > 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:
<
we
obtain
thus, the following
asymp-
13.1
-
First species orbit:
O(W-1)
X1
W
YO -
41
Properties
O(W -1)
+
2
O(W-1)
X2
W
YJ
=
2
O(W-1)
+
(13-11)
Node:
O(W-1)
Yj
-(13.12)
T-arc:
Xi+1
=
O(W-1)
Xi+2:--
O(W-1)
Yi+1
W +
O(W-1) (13-13)
-
S-arc:
Xi+3
(-1)3 -
m
-
Ksi
m
Y
rn
j m
These results
Xi
=
-
O(W-1),
W + 3
-W +
O(W-1)
if
j
is
odd,
W +
O(W-1)
if
j
is
even.
-.Ksi Ksi
-
can
strenghtened.
be
At
an
Yj
=
always
(13-15)
O(W-1)
antinode, =
We have
O(W)
Therefore, from (13.1b and (13-12) Yj
(13.14)
,
we
-
Xi
have
+
we
obtain for
E)(W)
Xj+j
a
node i
(13.16)
.
always
(13.17)
Off)
13.1.2 Variational
for W -+I oo
Equations
some equations which will be needed in the next two chapters, again the limit W -+ oo. We consider the case where in this limit the bifurcating arc or orbit becomes a sequence of T and S arcs (i.e. we ignore the particular case of a first species orbit).. In a total bifurcation, for each given free branch, we select one root ed branch for which the origin is a node. (There is always at least one such branch; otherwise all nodes would be in Q and the orbit would consist only of T arcs, which is impossible.)
We derive here We consider
We introduce
some new
notations. We call N the number of
orbit.
from 'the
arcs
which
call the
origin, Starting UN. Each Uc, arc is either a T-arc or a Ua; 0 S-arc. We call io, ii.... 7ial iN the positions of the nodes, with io of number basic The from to extends that the and iN arc i,,,. so i,,-l U,, n, arcs in arc Uc, is m,, i,, i,, 1 make up the bifurcating successive arcs U1, U2,. .
arc or
.
...
)
we
)
=
...
)
=
=
-
-
-
13. Partial Bifurcation of
42
Rom the two initial values
(13.1b),
Type,
1
YO and X1, using alternatively (13.1a) and
compute successively Y1, X2, Y2,
we can
....
We compute
now
the
arbitrary infinitesimal variations dYO and dXj and we compute the corresponding variations dYj, dX2, (In general the perturbed values no longer correspond to a bifurcating corresponding
variational
equations:
we assume
....
are.) (13.1)
From
dYj As
-dYj-j
=
a
have
we
(1
+
+
Ksi) dXj
preparatory step,
we
from i to i +
extending dXi+,. From (13.18) arc,
-dYj
dYi+l
-dXi+l
dXi+2 i + 1 is
-
2,
as a
dXi+2
dYj
=
We do
a
Proceeding
Ksi) dXi+,)
-
(13.20)
Off),
and
-
dXi+l we
[1 + O(W-2)]
-
[1 + O(W-2)]
2dXj+j
similar computation for same way as for the
(13.21)
[1 + O(W-2)]
S-arc, extending from i T-arc, we obtain generally
-
dYi+m
-dYj [I
dYi,,, where
we
(13.23)
+
(m
dXi+l
+
(13.24)
Ksi) dXi+l
-
We can regroup (13.21), (13.22), (13.24) into. the initial and final variations in arc U,,,:
O(W-2 ) dYi,,: 1)
m)
have
O(W-2 ) dYj
m,
M.
[1 + O(W-2)].
have used the abbreviation
dXi+,,,
dXj
1:!
to i +
-
=
we
(13.22)
a
-
particular
write
have
in the
=
we
we can
as
=
In
(13-18)
particular T
.
(0<2j<m-1), dyi+2j=dYi[]-(2j)dXi+,[] -dYj + (2j + 1 (1:! 2j + Ksi) dXi+,[] dyi+2j+l < < m O(W-2 dYj dXi+l (2 2j 1) dXi+2j < + + dYj O(W-2 2j (I m) dXi+l ) dXi+2j+l where
-
(13.19)
y;i+17(-dYj + (1
Finally, using (13.18a) again,
dYi+2
a
KSj) dXi+l
O(W-2 ) dYj
=
dYj
+
function of its initial variations dYj and
antinode; therefore, from (13-17), Yj+j
an
the last relation
-dXj
compute the variations inside
1
+
=
obtain
we
(1
+
dXi+l
,
1) m- dXi,,,
dYi.,
have introduced
+ u,, .
dXi,,
-,
+1
general equations relating
4-1
(13.25)
U
-2
for
a
m, +
for
a
Ksi,,:
T arc, S are.
(13.26)
equals -k,,,si. (see Sect. 12.4.3). Using Definition 8.2.1,
m,,,
if U,, is
a
uce < 0
if Vo, is
an
abnormal S-arc
We compute now the initial variations in the next dYi.. Since i,, is a node, we have from (13.16):
=
dYi.-, []
y2
U,+,. =
We
already
E)(W-1).
Fron
(13.28)
YZ
equations (13.25b) and (13.28) give the initial
functions of the initial variations of U,. We we
arc
Yi.
11 dXi.-,+,
+
i.
The
(13.27)
obtain
we
dXi.+,
find that
T-arc.
or a
know
(13.18b),
we
S-arc,
normal
u,,, > 0
43
Properties
13.1
variations of
iterate these
can
U,+, as equations and
obtain
dYi.
(-1)M1U2U3 -
Y,
2
dXj.-,jL
(-1)M1U2U3 -
y2 y2 i2
The orders of
ayic' =
ayo
Uce
...
L
2 r2 ilyi
dYo [] Ua
Yi
magnitude
The variations
+
i2
dXj
i. -1
Ua
UIU2
Y2 y2
'9Yi-
...
y2
dXj
(13.29)
i.,
E)(W2(ce
ax,
Oxi"'+i -
ax,
O(Wa
strongly amplified after each
are-
Uce
Y, 2
are
E) Wa
=
a YO
diYfo
...
...
il
O(W2(a -1))
'9Xi"'+1
UlU2 2 2 Y,i-1y, i
-2
1
...
...
+
(13-30) node.
13.1.3 Jacobian
equations (13.1)
The fundamental
fj
=
Yj
f2
=
X2
f2i-I f2i
=
=
-
+
Yj
f2n
+
Y,, Yn
-
X1
Xi+1
f2n-1 =
W
=
+
-
+
Xi
-
+
Y,,-l
kept Yn
W
-
0,
(1
+
Ksi)Xi
=
0,
0,
Yi -
=
be written
0,
1
W
-
(1
+
Ks,,)X,,
=
0,
(13-31)
-
We have eliminated have
K)Xj =
Yi-I
+
0
(1
can
YO, substituting YO 0. equation Yn
and the last
=
=
0 in the first
equation, but
we
44
13. Partial Bifurcation of
For
given value of W, this
Type
a
is
a
1
system of 2n equations for 2n variables.
The Jacobian is
09(fl, 19(X1, Y1 X2,
f2n)
7
-(1
Xn, Yn)
-,
K)
1 1
Y12
-(1
+
K)
1
-(1
+
Ksn)
1 1
(13-32) which reduces to
Iii
Nfi, 09(X1 Y1 X2) -
=
7
-
7
13.1.4 Relation with
We show
now
-,
f2n-1)
....
Yn-1, Xn)
(13-33)
*
Stability
that there exists
an
intimate relation between the Jacobian I J
and the
stability of the orbit (Sect. 2.8). In general, a second species orbit is "infinitely unstable" (116non and Guyot 1970, p. 364). For It small but non-zero, the stability index z takes large positive or negative values. However, when a family of second species orbits is followed, the stability index occasionally jumps from +00 to -00, or vice versa. For M > 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 <
+1
+1
< K
13.1. Chaxacteristics for 1P2 bifurcations.
The branches
sign
W
W
W
are
either of
one
T-arc
easily identified. (.13.9)
(under IP2) (branches 2)
-of W. Table 6.8
shows that their
shows that the or
bifurcating
two basic S-arcs
sign
arc can
(branches
is the
s
consist
11).
From
13. Partial Bifurcation of
48
equations of Sect. 13.1.1,
the
Type
1
easily derive that the first case corresponds W, while the second asymptotic to Y, the branches asymptotic to Y, 0. We can then label we
to the branches of the characteristic
corresponds
case
the branches
We
dY2/dX,
(13.39)
0. From 2
YJ
W
the
recover
junctions established
in Sect. 8.4.1
principle.
Stability and Jacobian. A critical bifurcating
13.2.1.1
W
2
possibe only if
This is
=
=
(Fig. 13.1).
from Broucke's
to
to
=
and
(13.41)
V2(1
K2)
-
-1 < K < 1. It
we
find that this
corresponds happens for
arc
(13.42)
.
corresponds then
to the extremum in
(Fig.13.1b). Using Proposition 13.1.1,
find thus that the Jacobian is
we
generally
non-
zero.
13.2.2
n
3
=
We have two
1
(1
-
(1
-
equations 1
K)Y2
+
2Y,
+
K)Y1
+
2Y2
+
-
K2
Y, 1-K 2 -
-
Y2
2W
=
0,
2W
=
0.
Subtracting, multiplying by YY2,
(Y1 Thus
Y2)(YIY2
-
we
+ K
1)
-
=
and
(13.43)
dividing by
1 +
K,
we
obtain
(13.44)
0.
have two distinct families. The first
corresponds
to
Y,
=
Y2, which
gives the characteristic 2W
=
(3
The second W
=
K)Y1
-
1 +
-
(13.45)
Y,
corresponds
Y,
K2
1 +
to
Y1Y2
1
-
K and has the characteristic
K
(13.46)
Y,
again hyperbolas, having as asymptotes the lines YJ 0, K). A simple analysis shows that, depending on the value of K, the characteristics have one of the shapes represented on Fig. 13.2. For K < 1, the two curves intersect at the point These
YI
=
Y,
curves are
W, Y1
=
=
2W/(3
VI--K
and this point is
,
an
-
W
=
2-\/1--K
(13.47)
,
extremum in W for the second
family.
13.2 Small Values of
Y1
Y1
+21
n
49
+21 +3
+3 +12
-
V
W
W
-1
K < -1
Y,
+ 3
-
3
Y,
-12
-12
< K <
+1
+ 21
W
W
+1
Fig.
+ 21
< K <
+ 12 +
+3 < K
+3
13.2. Characteristics for 1P3 bifurcations.
The branches
are
listed in Table 6.8
(under M). They
are
identified
on
values of Y, and Y2 and
comparing by looking at.,the asymptotic them with the expressions of Sect. 13.1.1.For K > 1, the two hyperbolas do not intersect, and the junctions are completely determined; we recover the junctions established in Sect. 8.4.1 on the basis of Broucke's principle (Table 8.4 to 8.11). For K < 1, the two hyperbolas intersect, and the situation is more complicated. The characteristics of Fig. 13.2 are based on the asymptotic equations (13.1), which are.correct only within O(yl/2). So, for p :A 0, the true characteristics are slightly different from those represented here. This may be sufficient to change the junctions between the four branches at the intersection point. We remark however that in the first family, corresponding to Y, Y2, the bifurcating arc is symmetric: Y, X3. By virtue of RestricY2, X, tion 7.3.1, the whole orbit is then symmetric, and we can, as in Chap. 7, use the fact that the symmetry property is invariant along a family. In the second. family, the bifurcating arc is asymmetric: Y, 54 Y2, except at the intersection. Therefore the branch 3 is necessarily joined to the branch 111. We recover
Fig.
13.2
=
=
the results of Sect. 7.3-1.1'.
satisfied, i.e. if the complement is not symmetric, quantitative approach is unable to establish the junctions for K < 1. likely that this difficulty corresponds to a real fact, namely that the
If Restriction 7.3.1 is not
then the It
seems
13. Partial Bifurcation of
50
Type
1
junctions do depend on what happens outside of the bifurcating arc, i.e. in the complement, through higher-order perturbations exerted at the two ends of the bifurcating arc. Stability and Jacobian. A critical bifurcating are corresponds 0. From (13.39) we find for the first family (Y1 Y2)
13.2.2.1 to
dY3/dX, (3
Using
-
=
=
K)Y14
4(1
-
(13.45),
also
1 -K V_
Y,
-
K)Y12
+
(1
-
K 2)(1
-
K)
=
0
(13.48)
.
find the two solutions
we
W
,
2V1_-K,
=
(13.49)
and
E1-:K
J(13
Y,
-
2
W
K
sign(3
=
-
K)VI(l
-
K2)(3
-
K)
(13.50)
.
only if K < 1; it corresponds to the point of interfamily. The second solution exists only if 1 < K < I +3 < K; it corresponds to the extremum in K along the family. For the second family, we obtain
The first solution exists section with the other or
Y14 Using
-
2(1
-
K)Y12
(13.46),
also
This solution and is also
we
-
(1
+
n
=
+
(1
-
(1
+
(1
(13-51)
0.
extremum in K
(13.49),
which exists for K < 1.
of intersection with the other
point along the family.
we
find that the JacObian is
generally
family,
non-zero.
4
We have three
2Y,
=
to the
Using Proposition 13.1.1', 13.2.3
K)2
find twice the solution
corresponds
an
-
-
equations K
1
K)Y2
+
K)Y1
+
2Y2
+
K)Y2
+
2Y3
+
Y,
(1
K)Y3
+
K
1
Y3
2W 1 +
2W
=
-
0, K2 -
Y2 =
2W
=
0,
(13-52)
0.
analytical solution. Figs. -4, -2, 4, i.e. the solutions 0, 2, 4. Solid lines represent the characteristics for n 3 (dashed lines) and of (13.52). Characteristics are also represented for n This system
does
not
13.3 to 13.7 show the
seem
to be amenable to an
numerically computed
characteristics for K
=
=
=
n
=
2
(dotted lines).
are listed in Table 6.8 (under IP4). They are identified by asymptotic values of Y1, Y2, Y3 (obtained from the numerical computation): an asymptotic behaviour O(W-1) corresponds to a node, while an asymptotic behaviour O(W) corresponds to an antinode (Sect. 13.1.1).
The branches
looking
at the
13.3 Positional Method
51
+211
8
K
-4
Y1 6
4
+31 2 -112 1111 -13
+ 121
0 -5
5
0
10 W
Fig. M3. Characteristics shown: 1P3 bifurcations
for IP4
bifurcations, for
(dashed lines),
-4 (solid lines). (dotted line).,
K
IP2 bifurcation
=
Also
Computations for other values of K indicate that the figure remains the qualitatively inside each of the five intervals K < -3, -3 < K < -1, , K < +1, +1 < K < +3, +3 < K. A proof of this fact will be given in
same
-1
Section 13.3. The
junctions
are now
fully determined,
for all values of K.
By contrast,
of symmetry and of Broucke's principle left some junctions undetermined for n = 4 (Sect. 8.4.1). So we have here a first instance where the
the
use
quantitative approach allows
us
to make progress.
13.3 Positional Method 13.3.1
In
Principle
principle
the numerical
computation
can
be used to determine the junc-
tions for any value of n (see for instance Sect. 13.3.3.5 where some characteristics are computed for n = 6). But in practice this approach cannot
pursued very far. The computations become time-consuming when n increases; they must be done for more and more values of K; and numerical problems arise because some branches lie very close to each other (see below). 4 suggest a new approach. We However, the resultsl obtained up to n observe on Figs. 13.3 to 13.7 that two characteristics which correspond to be
=
52
13. Partial Bifurcation of
Type
1
+211
K
-2
4
Yl +31 2
+121
-112
+ 13
0 0
-2
4
2
6 w
Fig.
13-4. Characteristics for lP4
bifurcations,
for K
-2.
+211
K
+31
0
2
Yl
+121 + 13
+112 + 1111
0
2
3 w
Fig.
13.5. Characteristics for lP4
bifurcations, for
K
=
0.
13.3 Positional Method
+ 31'
Yi
+211
-112 13 1
1210- 6'
-
2
0
-4
w
Fig.
13.6. Characteristics for IP4
bifurcations, for
K
2.
-31
6
1
K
1
11
4
1
1
Yi 4 +211
-
112 13
-
-121
0 -8
-6
-4
2
0.
-2
w
Fig.
13.7. Characteristics for 1P4
bifurcations, for
K
=
4.
53
54
13. Partial Bifurcation of
Type
1
different values of n never intersect. It is easy to prove that this is true generally. Let n and n' be the orders for the two characteristics, with n < n. We have then, for the first characteristic: Yn 0 from (13.1c), and for the second characteristic: Yn 54 0 from (13.1b). But Yn is a definite, single-valued function of W and Y1. It follows that no point of the (W, Y1) plane can belong =
to both characteristics.
Suppose
now
that
we
already know the shape of the characteristics for
orders up to n 1, and we are trying to determine the junctions for order n. The previous characteristics divide the (W, Yi) plane into several regions; and -
each characteristic for order In other
n
must lie
entirely within
one
of these
regions.
words,
Proposition region.
Two branches
13.3.1.
can
be
We
can
do better than that. Instead of
we can
just
as
joined only if they lie
in the
same
defining an orbit by W and Y1, symmetrically by W and Yn-1, and compute the other values backwards. The characteristics up to n 1 again divide the plane (W, Yn-1) into several regions, and two branches can be joined only if their characteristics in that plane are in the same region. Using the symmetry E (see Sect. 13. 1), we can reduce the consideration of the (W, Yn- 1) plane to the consideration of the (W, Y1) plane, and we obtain well define it
-
Proposition
Two branches
can be joined only if the two symmetriregion of the (W, Yi) plane, for the same value odd and for the opposite value- of K if n is even.
13.3.2.'
cal branches lie in the
of K if n
is
These two
same
propositions, taken together,
constitute
a
powerful
criterion for
the determination of the junctions.
13.3.2 Branch Order.
To
apply
As
we
the
will
asymptotic
method,
we
need to determine the
now, this can be done branches for W -+ *oo.
see
region inwhich a branch lies. by studying the relative position of the
13.3.2.1 Variations. We return here to the full set of
the Xi and the Yi. For
equations (13.1) for
0 given large value of I W I, using the condition Yo and starting from any given value of X1, we can compute the values of Y1, X2, Y2, successively by applying (13.1a) and (13.1b) alternatively. We consider the value of X, which corresponds to some particular branch of order n (so that the computation ends in Y,, :7= 0). We apply now a small variation dX1, and we compute the orders of magnitude of the corresponding variations of Y1, X2, (There are two small parameters in this problem: dX1 and W-1.) We can use the computations made in Sect. 13.1.2, setting 0. We obtain in particular dY0 ...'
=
a
=
13.3 Positional Method
dYi,,,
UlU2
Ua
...
[1
_
y2 Yi2 i
dXj
*
Z
Y
*
*
2
1
S
magnitude
of this quantity
(13.54)
sign
(')
ign
sign (Ul U2
=
dXj
where ( is defined set
(13.53)
0 (W2(a-1))
=
dXj and its
O(W-2)]
-I
We will need the order of d Yi.
+
(U,,..., U,,),
(-W
Ua)
(13 .55)
the total number of T-arcs and abnormal S-arcs in the
as
by (13.26).
shown
as
...
We have referred the variations to dXj. On the
acteristics, however, Yj two variables
(1
dYj Therefore
(13.55)
55
related
are
-
K)dXj
(13.54)
(13.1a)
is used. From
figures showing
with i
=
1
we
the char-
find that these
by
(13.56)
.
is still true if
we
replace dXj by dYj. On the other hand,
becomes
(_ Yi
sign
(-1)(sign(l
dYj
-
K)
(13.57)
.
13.3.2.2 Relative Positions of Two Branches with Initial Common
Arcs. We consider
now
two branches for which the
arcs
U,
to
U.
are
the
same, but the continuation is different: either the arcs U,,+, are different in the two branches, or the arc U,,+, does not exist in one of the two branches
(the bifurcating
arc
ends in
i,,)..
asymptotic expressions of Xj,...' Xj. and Yj,..., Yi. given in Sect. 13.1.1 are the same for the two branches; therefore these quantities differ by O(W-1) between the two branches. For JWJ 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. The
From.
(13.13)
and
(13.14)
we
have
o(W-1 Xi.
,
W-
m,,, +
Ksi.
+
O(W-1)
if U,, is
a
T-arc,
if U,,, is
a
S-arc,
O(W-1) W
Xi"'+1
-
m,,,+,
We have m,,
=.
-
-
generally (see s i,,,
k,,,
.
Ksj, Sect.
+
O(W-1)
(13.58)
if
U,+j
is
a
T-arc,
if
U,,,+,
is
a
S-arc.
(13.59)
12.4.3)
(13-60)
13. Partial Bifurcation of
56
Type
1
We find then from Definition 8.2.1 that the
U,,,
is
normal
a
m,,+,
and the if it is
S-arc, negative
quantity
7n,,,+,
-
We consider first the
This is
Ksi,,,
=
an
Ksj is positive if Similarly, we have
positive if U,,+, is
is
a
normal
S-arc, negative
I
case
first that U,, is
W > 0. T-arc. If
a
U,+, exists,
it must be
a
S-arc. We
(13.1b)
obtain from
Yi.
m,, +
(13.61)
abnormal S-arc.
Suppose
an
quantity
abnormal S-arc.
si.k,,,+,
--=
an
if it is
(m,,,+,
-
Ksi
)W-1 [I
+
O(W-2)]
(13.62)
increasing function of m,+,. Therefore, the order of the values of given by the order of the two values of m,+, in
for the two branches is
Yi.
the sequence
(13.63)
1,3,5....
case where the bifurcating arc ends in i,,. 0. In that case Yi,,, 0. The symbolically by m,,+, expression (13.62) is positive if U,,+, is normal, negative if it is abnormal. Thus, this case should be positioned in the sequence (13.63) so that abnormal U,,+, arcs (if any) are on its left and normal arcs are on its right. For instance if 1 < Ksi. < 3, the sequence is
We should also consider the
We represent this
case
=
=
'1; 0; 3,5,7,... Suppose
Yi. This
next that
(m,,
+
U,,, is
S-arc. If
a
Ksjj(m,,,+j M"' + M,,,+,
=
[m,,,,
+
Ksi.
(m,, m,
+
Ksi,,, )2
+Tn,+,
again an increasing function U,,+, is a T-arc, we have
Yic, This is
is
a
S-arc,
we
obtain
[1 + O(W-2)]
(13-65)
also be written
This is If
U,,+,
Ksi.) W-1
-
=
can
Yi.
(13.64)
=
(m,,
equal
the values of
+
Ksi.)W,-' [1
W-1
[1 + O(W-2)]
(13.66)
of m,,+i.
O(W-2)]
(13.67)
(13.66) for m,,,+, -+ +oo. Therefore, the order of the for two branches is given by the order of the two values
to the limit of
Yi,, following
of m,,,+, in the
1,3,5....
+
]
sequence
(13-68)
; 2.
symbol 2 represents the case of a T-arc. 0. The expresFinally, if the bifurcating arc ends in i,, we have Yi,, sion (13.65) is positive if U,, and U,,,+, are both normal, negative if one of The
=
them is abnormal. So:
13.3 Positional Method
-
If the U,,,
(13.68) are on
its
1,3;
normal, this case should be positioned in the sequence U,,+, arcs (if any) are on its left and, normal arcs right. For instance if 3 < Ksi. < 5: is
arc
so
57
that abnormal
2
5,7,...;
0;
abnormal, this
If the U,,
arc
sequence
(13.68):
is
1,3,5....
(13.69)
.
2; 0
;
should be
positioned
at the
right
of the
(13.70)
.
the order of
Having found
case
the
values of
Yj.,
it is
a
simple
matter to
determine the order of the values.of Yj for the two branches with the
of
help
(13-57),
13.3.2.3 Relative Positions of Two Branches with Different First
Arcs. The method of the previous section does branches differ to
is
a
a
=
0).
Y,
Mi
1
-
=
If the first =
case we
have from
we
mi
Y,
already
In that
S-arc,
in their first
K
-
arc
W +
is
W +
a
compare
(this
directly
can
be taken to
correspond
the values of Y1. If the first
arc
(13.14)
O(W-1)
T-arc,
arc
not work when the two
we
1 +
have from
Off-1)
mi
-
K
W +
O(W-1)
(1 3-71)
(13.13) (13.72)
equal to the limit of (13.71) for mi -+ +oo. If K < 1, the denominator (13.71) is always positive, and Yj is an increasing function of mi; the order
This is in
is
1,3,5,...;
2
(13.73)
If, 1 <,K, then for mi < K (abnormal arcs), Yj is smaller than 1, and is a decreasing function of mi; for mi > K (normal arcs), Yj is larger than 1, and is again a decreasing function of mi. Thus, the order is: for 1 < K < 3 1; 2;
....
7,5,3,
(13.74)
for 3 < K < 5
3, 1; 2; and
....
9,775,
(13.75)
so on.
previous two sections, we have supposed 0; -we must now consider the opposite case W < 0. We notice that all equations used to determine the order of the branches, such as (13.62) for instance, are odd functions of W. Therefore, in the case W < 0, all sequences such as (1 -63) are simply inverted. This is also a consequence of the symmetry V of the figure in the (W, Yj) plane (Sect. 13. 1). 13.3.2.4 The Case W < 0. In the
W >
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 <
3,
5. The first arc can be 1, 3, 5, or 2. +oo, and the branches tip to n According to (13.74), the corresponding packets are arranged for increasing
W
-+
=
by Fig. 13.8, left column. (In this in Figs. 13.1 to 13.7, and packets of first order, second order, etc. have been separated into different columns, with their filiation indicated by line segments.) Next we consider the packet associated with a first arc 1. This arc is abnormal, therefore the order of the second arcs is given by (13.70) and the corresponding second-order packets are positioned as shown in Fig. 13.8. The first arc 3 is normal, and the next arc is also always normal, so the values of Y3 increase along the sequence 0, 0, sign(dYi., /dYj) 1, so that the order is 1, 2. In (13.57) we have C of shown the values 13.8. reversed for Yj as Finally, if the first arc is in'Fig. the of increase Y2 along sequence 1, 0, 3 as shown by (13.64). 2, the values In (13.57) we have +1, so that the order is the same 1, sign(dYi. /dYj) for the values of Y1. In the same way we find the order of the third-order packets, etc. Yj in the order 1, 2, 5,
3. This is shown
increases from bottom to top,
figure, Yj
as
=
=
-
,
=
For W with the
We
because of the symmetry V, we have the same branches replaced by -, and the ordering in Yj is inverted.
-+ -oo,
sign apply now +
the convention
(13.5,):
we
consider
0. We remark also that the branch labelled
+
1
on
only the half-plane Yj Fig. 13.8 corresponds
>
to
59
13.3 Positional Method
+3 + 311
+31 +32 +5 +23
+,2 + 2111
+211 + 212 +2 1 + 1 + 1211
+ 13 + 131 + 1121
+112 +113 + 11
11 11" +1112
11111 ", +
+
2
a
Fig.
3
13.8. Order of the branches for 1 < K <
upwards.)
=
3, W
5
4
+oo
,
n:! 5
-
(Y1
increases
60
13. Partial Bifurcation of
Type
1
single basic arc, and therefore to Y, - 0. (Incidentally, that branch does 1 there are only two branches not participate in a true bifurcation: for n + 1 and 1 smoothly joined, and therefore only one family (see Sect. 6.8).) a
-
Therefore
we
full picture of the position of the branches in the halffollows: (i) we take that part of the sequence of branches
obtain
a
plane Y, > 0 as in Fig. 13.8 which lies above
+
1, and
we
copy it
on
the
right (W large
and
positive); (ii) we take the part of the sequence of branches in Fig. 13.8 which lies below + 1, we invert it, and we copy it onthe left (W large and negative). Fig.
13.9 shows the result.
Y1 +3
11 1112 1
+311
+'3 1 +32
+5
113 112 1121 131 -13 -121 1211 -12 -
+
-
23
+2
-
+2111
-
+211 +212
-
+21
0 0
Fig.
13.9. Position of the branches in the
W
half-plane Y1
> 0 for 1< K <
3, n:! 5
13.3.3 Results
systematically to determine the junctions. The comhand up to n 6, A program was also written to by putations this allowed a check of the results branch order the automatically; compute and also a computation for higher values of n. We
use now
the method
were
13.3.3.1 and
n
=
done
=
2. In the
11, which
are
half-plane Y,
>
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-<
13.10. Junctions for
n
< 3
+3
.
+2 +21 +3 +12 + 111 + 11
-3 Ill 11 -12
< K <
+1
-
-----7
+3 < K
+2 +21
61
13. Partial Bifurcation of
62
13.3.3.2 cases
n
=
3. The
Type
1
junctions are represented on Fig. 13.10 for the various They are determined in three steps.
with respect to K.
1. We determine first the
3 in the halfposition of the branches up to n 0, as explained above (see example of Fig. 13.9). 2 schematically, as a We represent the already known junction for n solid line. This junction divides the half-plane into two regions. We consider now the branches for n 3 and we use, Proposition 13.3. 1. In the two cases +1 < K < +3 and +3 < K, there are two branches in each 3 are immediately established. region; therefore the junctions for n They are represented as dashed lines. For K < -1, all four branches are in the same region., We have here the first example of a situation which we will frequently encounter, and which we call a trident. Four branches lie in the same region. Two of them, H, and H2, are made of symmetric orbits (in the present case: 111 and + 3) (we use Restriction 7.3.1). The two other branches, H3 and H4, are made of asymmetric orbits, but are changed into each other by
plane Y,
2.
=
>
=
3.
=
=
-
the symmetry (in the present case: as in Chap. 7 and conclude that H,
+
21 and
can
be
+
12).
Then
only joined
to
we can reason
H2, and H
to
H4. Consider two orbits 93 and Q4, symmetrical of each other, belonging to H3 and H4, respectively. We let these two orbits move towards each other until they meet. The orbit Q thus reached is symmetric. This symmetric orbit belongs to a family of symmetric orbits, which can only be the family HH2. Thus the two families HH2 and H3H4 intersect in a common orbit Q. Moreover, the two symmetrical orbits H3 and- H4 have the same value of W; it follows that Q is an extremum in W for the family H3H4..
(W, Y1) plane,
In the
shown in
Finally, for
-
the four branches of the trident
13.10. The
Fig.
common
1 < K < + 1,
we
trident is folded towards the
orbit Q is
have
again
a
Thus, all junctions =
3. We
recover
disposed a
as
dot.
trident. Here the stick of the
right.
In the present case of partial bifurcations of type only when n is odd (see Sect. 7.3.1.1).
n
are
represented by
have been established
1, tridlents
by the positional
can
exist
method for
the results obtained in Sect. 13.2.2.
4 are determined in the same way. junctionsfor n corresponding to opposite values of K are shown side by side, because these cases are associated in Proposition 13.3.2. First the position of the branches along the two sides is determined. The 3 are drawn schematically as solid lines. already known junctions up to n 4 Proposition 13.3.1 then immediately determines the new junctions for n (dashed lines) in several regions which contain only two branches. In the other regions, we use Proposition 13.3.2. Thus, for -1 < K < +1, + 211 is joined to + 31; therefore + 112 must be joined to + 13 (the change of sign 13.3.3.3
Fig.
n
=
4. The
=
13.11 shows the results. Cases
=
=
13.3 Positional Method
+2 +21
+211 +31 +3 + 12 + 121 + 13 + 112 +
+ +
-1
11 -112 -
-
-
-
-
-
-
il l
1111
-
-
-
-
-
-
<
+2 - 211 +21 +3 +31 +121 + 12 - 13
K < +1
ill 1111 11 -112 13 -121 -12
11 112
-
-
-
-
-
-
1111 13 -
-
Fig.
13. 11. Junctions for
n
< 4
-
-
-
-
-
-
-
-
-
-
-
-
-
+21.
+1
+2 211 +21 +3 +31 +121 +12 -
-31 -3 ill 1111 11 112 -13 121 -12
<
K < +3
-
-
-
-
-
K < -3
-
-
-3 < K < -1
-
+3 - 31 +2 - 211
-
-
+ 2 -
-
-
-
-
-
- 211
-
-
+21
+3 < K
63
64
13. Partial Bifurcation of
of K leaves
us
in the
same
Type
case).
1111 must then also be
1
The two
remaining branches
121 and
+
For -3 < K < -1, the branches
+
joined. joined; therefore, for +1 < K < +3, the branches + 13 and + 121 are joined. Using the symmetry V, we find that 13 and 121 are joined. K the K and in -3 treated the cases are < < +3 same way. Finally, All junctions have been established. We recover the results obtained numerically in Figs. 13.3 to 13.7. 121
+
+
31 and
are
-
13.3.3.4
n
increasing
=
5. The
(here
K
-
junctions are shown in Figs. 13.12 to 13.14, in order of no particular reason to associate opposite values
there is
K).
of
+ 2
1;
+ 211
11 112
+ 2
-
-
+ 211
1121
-
-
11
-113 1112
-113 -
-
-
-
-
-
-
- 2111
-
1
ill
-
11111
1111 13 131
-
1 1
-
1
-
1112
+21 +212 +23 +5 +32 + 3 +311 +31 + 121 +1211 + 12
Fig.
-3
13.12. Junctions for
-
-
-
-
-
1 1
11111
Most junctions for n plication of Propositions
- 2111
-
.
-
-
-
-
-
-
-3 < K <
n
-
ill
-
K <
-
-
-
+21 +212 +23 +5 +32 + 3 +311 +31 +121 + 1211 + 12 - 131 + 13
-1
< 5
5 can be determined by the straightforward ap13.3.1 and 13.3.2. New tridents appear; they are resolved in the same way as in Sect. 13.3.3.2. For -1 < K < +1, there are two regions containing 6 branches each, and a, different reasoning is required. =
We consider first the upper region. It contains only two branches of symmetric orbits: + 212 and + 5; these branches must be joined. The branch + 23
joined to + 2111, because the symmetrical branches + 32 and + belong to the same region. In the same way, it cannot be joined Therefore the branch + 23 must be joined to + 32 (incidentally, it
cannot be
1112 do not to
+
311.
13.3 Positional Method
+ 2
+21
ill -1112 1111 -
+ 21 1
1 -
+2111 +212 +23 +5 + 32 +311 +31 + 3 + 12 + 121 + 1211 + 131 + 13 + 112 + 1121 + 113 + 1112
-
-
-
-
-
-
-
-
-
-
-
11111
11 -113 I l2 -1121 131 -13 -121 1211 12
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
'
-
-
+3 - 311 +31 +32 - - 5 +23 +2 - 2111 +211 -
-
)
1
+21
-
+ +
+
Fig.
< K <
+1
13.13. Junctions for n:!
+1
5
212
-
+
-1
-
I
(continuation).
< K <
+3
65
13. Partial Bifurcation of
66
-311 -31 -32 -3
-------
Type
-5 -311 -31 -32 -3
111
-1112 11 ill, -
1
-1112 -
-
-
-
-
-
-
-
-5
113 1 1
+ 2
11
+2 +23 +2111 + 2 11
-
+23 112 1121 131 13 121 -1211 12
+2111 + 211
-
-
-
-
-
-
-
-
-
-
-
-
-
-
212
1
)
1
-
-/
112 -1121 131 -13 -121 1211 12 -
-
+
21
-
-
-
-
-
-
- 212
1
1
n::'it 5
+ 21
-
+3 < K < +5
13.14. Junctions for
-
-
-
Fig.
-
+5 < K
(continuation).
13.3 Positional Method
67
part of a trident). The two remaining branches + 2111 and + 311 are then joined. A, similar reasoning determines the junctions in the lower region. is
Thus all 13.3.3.5
n
have been determined.
junctions =
6. The
junctions are shown in Figs. 13.15 to 13.18. As in cases corresponding to opposite values of K are
Sect. 13.3.3.3, the related shown side by side.
Many junctions for
n
6
=
are
found by the
of
use
Propositions 13.3.1
and 13.3.2. For -3 < K < -1 and +1 < K < +3, in fact, all junctions 4 are found. In the other cases, however, we find that there exists a group of branches for which the
positional
method is unable to establish the
junctions.
+1, this group is: + 2112, + 213, + 33, + 312; for +3 < K < +5 and for +5 < K: 1113; for -5 < K < -3 and for 11121, 321, 33, K < -5: + 33, + 3111, + 12111, + 123. For -1 < K <
-
-
-
-
positional method: the junctions between the 4 branches change inside the given interval of K. -5.66 Fig. 13.19 shows the numerically computed characteristics for K about at -5.68. The junctions between the branches change and K The positional method does not distinguish between -5.669369. K these values of K, for which the normal and abnormal arcs are the same; therefore it cannot predict the junctions. The characteristics were computed for a number of other values of K There is
a
good
reason
for this breakdown of the
=
=
=
-
..
inside the intervals K < -5 and -5 < K < -3; the results indicate that there is no other change, i.e. the junctions remain as shown by Fig. 13.19, left,
-3, and by Fig. 13.19, right, for K < -5.669369.... Using Proposition 13.3.2, we find that there is a similar change in the junctions in the case +5 < K, at K ='+5.669369.... Finally, in the interval -1 < K < +1, there is a change in the junctions at the value K 0, as shown by Fig. 13.20. Computations for other values 1 < K < + 1, i.e. there is no other change in the interval that of K indicate K < 0, and by -1 for shown < remain the junctions as by Fig. 13.20, left, for -5.669369
...
< K <
=
-
Fig. 13.20, right, for 0 < K < 1. The fact that the change happens precisely
at K
=
0
can
be
explained.
Consider the symmetry defined at the end of Sect. 13.1 for n even. In the particular case K = 0, this symmetry consists simply in inverting the sequences of the
Yi and the Xi. A solution of (13.1)
can
be invariant under
that symmetry. This is indeed the case for the solutions of the branches + 2112 and + 33. The two other branches, + 213 and + 312, are changed into
each other
by
Thus, for
the symmetry.
and all four branches
come
to
K
a common
-=
0,
we
have
solution 0
a
trident situation, As
(Fig. 13.20, center).
K 0 0, the four branches become joined two by two. All junctions of branches have been thus determined up to n = 6. Results are in Tables 13.1 to 13.10. The format is similar to that of Tables 8.4 to 8.11.
soon as
All
cases
left undecided in these tables
are now
solved.
68
1j.
Partial Bifurcation of
Type
1
1 1
1
1 1
1
1 1
1
1
(
+2 + 21 + 211 + 2 111 +21111 +2112 +213 +2121 +212 +23 +231 +51 +5 +32 +321 +33 +312 +3111 + 31 1 + 31 + 3 + 12 + 121 + 1211 + 12111 + 1212 + 123 + 15 +132 +1311 +131 +13 +112 +1121 +11211 +1131 ,+ 113 +1112 +11121 +1113 + 11112 + 111111 + 11111 + 1111 1 + 1 +
-1
Fig.
13.15. Junctions for
n
< K
< 6
.
+1
13.3 Positional Method
-112 -11211 -1121 -113 -1131 -11121 11 1 2 -1113
-
-
-
-
-
-
-
-
+2 +211 - 2112
-
-
-
-
-
-
-
- 21111
+21 +213 +212 +2121 +231 +23 +5 -i 51 +321 +32 +33 +3 +3111 +311 +312 +31 +121 +1212
1 1
1 1
+1211 +12111 +12 -
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
- 123
+15 + 132 +131 - 1311 +13
111111 -
-
-
--
7--
13.16. Junctions for n:!
-
-
-
-
-
-
-
11 -113 -
-
-1131 -112 11211
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-1121 -131 -1311 -13 -132 -15 123 -1212 -121 -12111 -1211 12 -
(continuation).
+3111 +31 +312 +33 - 321 +32 +5 +51 +231 +23 +2 +2111 - 21111 +211 - 2112 +213 +2121 +212
1
1 1
1 1
1 1
+21
-
+1 6
-
11111
-3 < K < -1
Fig.
-
+2111 -
1 1 1
-11112 lil l
-
-
-
111111
+3 +311
-
-1112 -11121 -1113 -11112 1111
69
< K <
+3
13. Partial Bifurcation of
70
11 1 12 11211 -1121 -113 1131 -11121 1112 -
-
-
-
-
-
-
-
-
-
-
-
-1113 111111 iiiii 11112 1111 13 132 13 1
-
---------
-
1
-
1
1
-
-
-
-1311
-
-
-
-
-
-
-
-
Type
+2 + 211 - 2112
+2111 +21111 +213 +21 +212 +2121 +231 +23 + 5 +51 +321 + 32 + 3 +33 +3111 +311 + 312 +31 + 121 +1212 + 1211 +12111 +123 +12 - 15
1
-311 -3111 -31 -312 -33 -321 -32 -3 ill 1112 -11121 1113 -11112 lill 111111 11111 -113 1131 11
1
1
-
-
-
-
-
-
-
-
-
-
+ 5
-
-
-
-
-
-
-
-
-
1 1
112 11211 -
1121 131 -1311 -13 -132 -15 -123 1212 -121 12111 1211 12
-
-
-
-
-
-
-
-
-
-
-
1
1
7
-
+ 21
-
-5 < K <
Fig.
-3
13.17. Junctions for n:!
+3 < K <
6
(continuation).
- 51 +2 +231 +23 +2111 +21111 +211 - 2112 +213 +2121 + 212
+5
13.3 Positional Method
11 112 11211 -1121 113 -1131 -11121 1112
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
---
-
-
-
-
-51 -5 311 + 2 -3111 + 211 -31 - 2112 -312 -33 -321 -32 3 + 2111 +21111 111 1112 +213 -11121 +21 -1113 +212 11112 + 2121 lill +231 +23 -111111 iiiii +5 113 +51 -1131 +321 11 + 32 + 3 +33 +3111 +311 112 + 312 11211 +31 + 121 1121 + 1212 131 + 1 21 1 1311 + 12111 -13 + 123 132 + 12 15 123 -1212 -121 -12111 -1211 12
71
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
1
-
-
-
-
-
-
-
-
---
+ 2
-
ill 1113
-
-
111,111 11111
-
1111 2 1111 13 132 -131 -1311 15
-
-
-
-
1
1
1
1
1
-
-
-
-
-
-
-
-
-
-
-
-
-
+231 +23 +2111 +21111 + 211 - 2112 +213 ''+ 2121 + 212
-
-
-
-
+ 21
-
+5 < K
K < -5
Fig.
13.18. Junctions for n:!
6
(continuation).
13. Partial Bifurcation of
72
2.5
1
K
Y,
Type
1
2.5
+33
-5.66
1
K
Yl
+33
-5.68
+3111 2
2
1. 5
1.5
+ 3111
+ 12111
+12111 +123
1
8.5
8
9
8.5
8
1
w
w
Fig.
13.19. Some characteristics of lP6 bifurcations for K
=
-5.66 and K-5.68.
f- 112
+213
2
+3.3
+
.8 K
K
-0.01
0 I
.7
2.5
2.6
I
K
+0.01 I
2.8
2.7 w
Fig.
13.20. Some characteristics of 1P6 bifurcations for
K-0-01, 0, +0.01.
312
13.3 Positional Method
Nothing prevents
in
numerical computation. 13.3.3.6
the two
> 6. The
n
cases
73
principle the solution of higher values of n, using However, the amount. of work grows exponentially.
positional
method has also been
-3 < K < -1 and +1 < K <
+3,
applied for n > 6 in explore how far
in order to
one can go. For n = 7, all junctions can still be determined. We do not show the schematic pictures of the junctions in the (W, Yi) plane, which are too
However, the results are included in Tables 13.4 and 13.7. 8, in each of the two cases -3 < K < -1 and +1 < K < +3, there
voluminous. For
n
=
.appear two groups of 4 branches which cannot be resolved
by the positional
method.
bifurcation, type 1: branch junctions determined by the quanOther branches axe starting point in P and K < -5.669369
Table 13.1. Partial titative
joined
study,
as
f6r
in Table 13.3.
IP6++---A
1P6 ..... A -
15
+33
-
1311
+3111 +
123
+
12111
bifurcation, type 1: branch junction determined by the quanti< K < -5. Other branches starting point in P and -5.669369
Table 13.2. Partial tative
for
are
in Table 13.3.
study, joined as
1P6 ..... A -
15
-
1311
...
13. Partial Bifurcation of
74
Type
1
junctions determined by
Table 13.3. Partial
1: branch
titative
and -5 < K < -3.
bifurcation, type study, for starting point in P
lP2+A + -
5
2
+
11
+212
lP3++S +
lP5 .... S
lP6 ..... A +
IP6++--+A +312
15
-1311
+
1212
-
131
-
1113
+2112
-
11111
-
Mill
-
3
11211
-1131
IP5 .... A
lP6 ..... A
+32
+51
+23
+321
lP6++---A
IP5++--A
+231
+33
-11121 IP3++A +21 +
12
lP4++I+A
+311 +1211
-
+123
-132
+3111
-
-
13 lill
+2111 -
-
IP4++-A *
31
*
121
+211 -
112
1112
-
113
-
1121
-
+2121
11112
+
12111
+213
+211111
the quan-
13.3 Positional Method
junctions determined by the
Table 13.4. Partial
1: branch
titative
and -3 < K < -1.
bifurcation, type study, for starting point in P
IP2+A
1P6 ..... A
IP7 ...... S 7
+
2
+33
+
-
11
+213
+232
lP3++S
+
15
+132 3
IP3++A +21 +
12
IN ... A +
13
+123
IP7 ...... S +313 +
12121
+
151
+21112
-
1111111
-
11311
1P7 ...... A
IP7 ...... A
+52
+3121
IP6++++-A
+
+51
+331
+3112
+3'21
+2131
+
l12112
+231
+133
+
2113
+2121
+1312
+21121
+1311
+
1321
-
112111
-11112
+1231
-
111211
lP6++--+A
IP7 ...... A
1P7 ------ A
+312
+511
+31111
+3211
+
25
+
1213
IP4++-A *
31
*
121
+211 -
112
+
1212
+2112 -
11211
121111
+2311
+211111
+21211
-
11131
+13111
-
115
111112
-
1123
11113
-
1132
111121
-
11212
1P5 .... S
-1131 +
5
-
11121
+212
lP6++---A +
131
+3111 +
12111
IP5 .... A +21111
+32 +23 lP5++--A +311 +
1211 2111 1112 113
1121
-
1113
75
quan-
13. Partial Bifurcation of
76
Type
1
Table 13.5. Partial
1: branch
titative
and -1 < K < 0.
bifurcation, type study, for starting point in P
IP2+A
IP5 .... S
junctions determined by
lP6 ..... A
+
the quan-
15
+132 +2 +
11
+5
+51
+212
+231
+131
+33
+1311 +12111
IP3++S
11111
+213
IP5 .... A
+312
+
+
ill
+2112 +32
lP3++A
+23 +21 +
+123 +1212
+3
+311
12
+2111
IM ... A
+1211
+1131 +
11121
+321 +2121
+11211 +
111111
+3111 +21111
+1113 +
11112
+1121
+31 +211
+113 +1112
+13 +112
+121 +
lill
bifurcation, type 1: branch junctions determined by the quanstarting point in P and 0 < K < 1. Other branches are joined as
Table 13.6. Partial
titative
study,
in Table 13.5.
IP6 ..... A
+33 .L.
12 -in
+213 -
n-I
in
for
13.3 Positional Method
Table 13.7. Partial bifurcation, type 1: branch junctions determined study, for starting point in P and 1 < K < 3.
by
the quan
titative
IP2+A
lP5+--+S
+
2
+212
-
11
-
IP3++S
131
IP5+--+A
lP6 ----- A
lP7 ...... A
+21111
+3111 1
-
l13l
-11113
-
15
lP7+-- ... A
I
-123 + -
3
-1211
ill
-
1121
-132 -
-
lP3+-A +21 -
12
lP5+---A
-
113
lP4 ... A
V6 .....A
+31
+51
+
lill
lP4+--A
+231 +33 +312
+211 -
112
-
IP5 .... S
iiiiii
IP6+++--A +3111 -11112
+
5
-1113 -
lP5 .... A +32
+23
+213
-
1112
2T121
+2112
+311 -
11121
lP6+--++A
+
lP5+++-A
13111 121111
lP7 ------ A
+313
+2131 '
+
1111111
21121
+21211 lP7 ...... A +52
-
11212
-
1312
+25
-121.12
lP7 ...... A
lP7 ------ S
+511
+21112
+321
13 121
7
-
-
+232
-
-
d2lll
-
lP7 ...... S
+2111
+2113
1212
11211
-
+2311
-11311
+331
-
151
+3121
-
12121
+321.1
+31.12
-
12111
-
1321
-
1231
lP7 ------ A
-111211 -11131 -
-1311
1P7 ------ A
-111112 IP7 ...... A
+211111 -
1123
-
133
-
1213
-
1132
lill2i
77
-
13. Paxtial Bifurcation of
78
Type
1
Table 13-8. Partial bifurcation, type 1: branch junctions determined study, for starting point in P and 3 < K < 5.
by
the quan-
titative
1P5 .... S
lP2+A +
2
51
+213
+
ilill
-1131
+2121
IP5 .... A
-3111
+2112
-
111111
3
-311
ill
-113
lP6+++--A
lP3+-A
IP5+++-A
-33
+21
-32 -1112
12
-11211 -1311 -12111
-
-
lP6+--++A
5
+
lP3++S
lP6 ..... A
321
-1113 -
11121
lP6+----A +231 +
21111 __
lP4 ... A
IP5+--+S
-31
+212
-312 -
11112
131
-
is
-123 -132 -1212
IP4+--A
lP5+--+A
+211
-1211 -
-
112
-
13
-
121
T12 1
-
05+-"A +23 +2111
bifurcation, type 1: branch junctions determined by the quanstudy, for starting point in P and 5 < K < 5.669369.... Other branches joined as in Table 13.8.
Table 13.9. Paxtial titative axe
lP6 ..... A
IP5 .... S -
-
5
-51
11.111
-1131
Table 13. 10. titative
joined
study,
as
IP5 .... S -
-
5
hill
Paxtialbifurcation, type 1: branch junctions determined by the quanfor staxting point in P and 5.669369 < K. Other branches are
in Table 13-8.
lP6 ..... A
lP6+++--A
-51
-33
-
1131
-1113 -
J21
-11121
14. Total Bifurcation of
The
equations (12.114) Yj
+
Yj_j
Yi(Xi+l
W
-
Xi)
+
-
(1
+ 1
in the
are
Ksi)Xi
+
0
=
case
of
a
1
total bifurcation
0
(14.1)
1
where i is to be taken modulo
These equations form
=
Type
and takes all values from 1 to
n
system of 2n equations for the 2n
a
n.
+ I variables
to Yn-1, X, to X, As in the case of partial bifurcations (Sect. 13.1), expect one-parameter families of solutions, which correspond to ordinary
W, YO we
one-parameter families of orbits. As in Sect. 8.5. 1, case
K > 0, and the
14.1
we can use
YO > 0. The
case
isomorphisms to consider only the origin is taken in P.
the- two
Properties
properties are essentially the same as for a partial bifurcation (Sect. 13j). regard to property 5, the bifurcation orbit corres ponds to a cycle of period n of the mapping. The
In
14.1.1 Jacobian
equations'(14.1)
The fundamental
f, f2
=
==
Yj X2
f2i-1 f2i
=
+
Yo'- W
+
Yj
X1
+
Xi+1
f2n
=
Xn+1
+
Xi
+ +
-
-
1 +
Y,,-, Xn
-
=
Yi -
K)Xj
W
(1
0,
Vn-
+
Ksi)Xi
=
0
K)X,,,
=
0
0,
-
I +
=
be written
0''
=
T,
Yj_j -W
+
Y,,
(1
-
1
can
=
M. Hénon: LNPm 65, pp. 79 - 91, 2001 © Springer-Verlag Berlin Heidelberg 2001
(1 0,
+
14. Total Bifurcation of
80
f2n+1
-_
f2n+2
:-`
YO
-
X1
Yn
0
-:--:
Xn+1
-
Type
1
1
(14.2)
0
7--
equations ensure periodicity. given value'of W, this is a system
The last two For
a
of -2n + 2
equations for
2n + 2
variables. The Jacobian is
19(fl, O(y0, X1
7
Y1,
f2n+2) Xn, Yn, Xn+1)
X2,
K)
1
Y;2
-(1
+
K)
1
-(1
+
K)
1
Yny
(14.3) decompose with respect to the last two lines, obtaining fourn x n deterare equal to 1; the third is obtained by removing the second column and the second from the last column; the fourth is obtained by removing the first and last columns. The result is We
minants. Two of them
JJJ
=
2 --
49(fl,
-
-
-,
f2n)
Nfl, 19(X1iY1)X2)
f2n)
-
19(YOiY1.iX2,---'Xn,Xn+j
...
1
Xn, Yn)
"
(14.4) 14.1.2 Relation with
Here
again
Jacobian For
a
we
JJJ
Stability
will show that there exists
an
intimate relation between the
and the
total
stability of the orbit. bifurcation, the stability index (Sect. 2.8)
can
be
computed,
since the whole orbit is known. We consider the system of equations (14.2), for a given value of W, with the last two equations deleted. We have then a
system of 2n equations for 2n + 2 variables. Starting from given values of YO and X, and applying the equations one by one, we obtain successively Y1,
X2,
....
we can
Yn, Xn+1
-
In the
For the whole
orbit,
dYn
dXn+l
way,
stability index
we
starting from variations dYO and dXj, dY, dXn+f. ...'
have then
ayn
(9yn.
01"0 19xn+l
19xn+l
ayo
ax,
I
The
same
compute successively dYj, dX2,
ax,
)
(dXj dYo
is the trace of the matrix divided
(14-5) by
2:
14.1
('9Y
1 Z
=
09YO
2
Properties
81
(14.6)
ogxl
We have the variational equations
af,
dYo
5-Y-0
19f2n ayo
+
dYo
9x.+i
+... + 0
In order to compute
equations for
dX,,,+,
9Xn+l
dXn+l
gYn/0YO,
2n + 1 variables
09(fl,
=
-
-
-
set
(14-7)
0
dXj
0. The
then be solved to
remaining system of give
...
to
compute
1
(14.8)
i
(9Xn+,/,OX,,
we
set
dYO
0,.and
we
find
19(fl, f2n) Xni Yn) 19(Xl) Y1, X2, 49(fl, f2n) Xn Yn) Xn+l) 19(yl X2,
aXn+l
-
09X1
-
-
(14.9)
-,
i
We obtain the
2n
1
...
7
i
ay0
5
stability
index
aul,
19(fl, 571((LYO Y1, X2, f2n) 09(fl, Xn, Yn, Xn+l) 19(yl, X2 f2n)
49 (XI 7 Y1 2 X2, -Xn,
Z
=
f2n) Xn7 Xn+l) 09(YO Y1 X2) 19(h) f2n) Xn, Yn, Xn+1 19(yl) X2,
ayn
Similarly,
we
can
0,
=
2
Tn)
-
-
f2n) Xn, Xn+1
(14 10) .
-,
i ....
The denominator reduces to 1, and z
=
1
comparing with (14.4),
we
111.
obtain'
(14.11)
-
2
We have thus shown
Proposition if and only if is
z
=
bifurcation of type 1, the Jacobian vanishes of the first kind (the stability index
14.1.1. In
a
total
the orbit is
a
critical orbit
1).
14-1.3 Asymptotic Behaviour for
JWJ
-+ oo,
general case, the bifurcation orbit becomes in this limit a sequence of T and S arcs, and we can use the computations of Sect. 13.1.2. (The particular case of the first species orbit E will be considered in Sect. 14.2.1.) Rom In the
(13.29),
with
a
=
N, i,,
=
n we
obtain for the
stability index
14. Total Bifurcation of
82
UlU2 Z
=
2
...
Yi2 Yi22
There is
z
UN
...
=
11
y2
+
where
C
=
E)(W2N):
-2
A
(14.12)
the orbit is
sign
unstable. The Jacobian
strongly
never
is
(-I)
defined,
is
O(W
I
iN
vanishes in.this limit. The
sign(z)
Type
(14.13) as
in Sect.
13.3.2-1,
as
the total number of T-arcs and
abnormal S-arcs.
14.2 Small Values of
n
We will represent the characteristics in the (W, a point of this plane is not sufficient to define
Yo, X,
needed. As
are
YO) plane. an
In the present case,
orbit: the three values W,
consequence, the intersection of two characteristics
a
necessarily, correspond to a true intersection of the figures be it can merely a projection effect. families; half the Yo > 0 is represented, since here again the figure is Only upper the origin because of the symmetry E' (13.4). to with respect symmetric on
the
does not
two
14.2.1
Rom
n
(13.6)
2YO
2
=
2Y,
+
have
we
K
1 +
2W
Y,
=
2Y,
0,
+
2YO
K
1 +
YO
2W
=
0,
(14.14) from which
W
=
we
2YO
obtain Yo
Y1, and
K
1 +
=
2Yo
X,
'
Depending on the value represented on Fig. 14.1. It
of is
=
-1+K 2Yo
(14.15)
'
-
K, the characteristic has one of the shapes hyperbola; its asymptotes are the straight
a
W/2 (dashed line). (under IT2) shows thai the bifurcating orbit can consist either of species orbit (branches t E) or two basic S-arcs (branches t 11). Rom
lines Yo
0 and Yo
=
=
Table 6.4 a
first
the
equations of Sect. 13.1.1,
we
find that the first
asymptotic to Yo the branches asymptotic to Yo
corresponds
to
established in Sect. 8.5.1 from Broucke's For 0 :!
Fig. 14.1), YO
0. We
corresponds
to the
while the second recover
case
the junctions
principle.
1, the characteristic has
K <
case
W/2,
branches of the characteristic
an
extremurn in W
(dot
on
with the coordinates
v/-1---K2 =
2
1
W=2V1-K2.
(14.16)
14.2 Small Values of
+ E
YO
YO
+
83
n
/E
+
W
W
0
K <
Fig. 14.1. Characteristics
2
T'0-2
For 0 < K <
for IT2 bifurcations.
K
1 +
(14.17)
2 Y04
1, there
On the branch
is
1 at the extremum.
z
E one has
+
species orbit is stable. This 1994, Chap. VII). pp. 112 to
Yo
-+
+oo, and therefore
z
-*
1-: the first
agrees with the numerical results of Bruno
The total bifurcation 1T2
( 1971,
stability index from (14.6)
We obtain the
z
1
1
was
(1976;
already studied quantitatively by Guillaume
119). Computation:
14.2.2 Numerical
Method
2
4, the system of 4 equations (13.6) does not seem to be explicitly (For n 4, the symmetric orbits can be determined.) No equivalent of the positional method used for partial bifurcations (Sect. 13-3) was found in the present case. A program was written to solve the equations numerically. The computation of a given branch is begun at a large value of W. An iterative relaxation method, based on the asymptotic decomposition of the orbit into T- and S-arcs and on the asymptotic equations of Sect. 13.1.1, works well in that case. For a given value of W, the steps are as follows: For
n
solvable.
1. We take
=
as
initial
2. For each arc,
we
approximation Yj
=
0 in
nodes, 11Yj
0 in antinodes.
(antinodes) and (14.1b), the value
compute the values of the internal Yj
the values of the Xi by solving the equations (14.1). In of Yj from the previous iteration is first substituted; we have then
system of 2m
equations for 2m 1 unknowns, which 3. For each node, we recompute Yj, using (14.1b). 4. We go back to step 2, until the solution has converged. -
1
The branch is followed towards
-
decreasing IWI. Usually
at
is
a
linear
easily solved.
some
point the
iteration does not converge anymore, because the decomposition into T- and S-arcs ceases to be a good approximation. We shift then to a trial-and-error
14. Total Bifurcation of
84
Type
I
shooting method, which works well for moderate values of W where the branches are well separated: starting from approximate values of YO and X1, we compute successively Y1, X2, Y, X,,+,, with (14.1). We compute also the variations, as functions of dYO and dX1, obtaining the 4 elements of the matrix in (14.5). This allows a correction of the initial values by the NewtonYo, X.+1 Xi). Raphson method, until a periodic orbit is obtained (Y. It allows also a computation of the stability index. ...'
=
=
14.2.3
n
=
4
show results for K
Figs. 14.2 to 14.4 represented for n
2
=
0.3, 2.2, 4. Characteristics
are
also
(dotted lines). +-112+-211-
+ -1111-
+E 0.3
K 2
YO 4
3
n2 5
0
2
1
1
3
4 T
Fig.
14.2. Chaxacteristics for 1T4
shown: M bifurcation
The branches
identified
on
the
are
bifurcations, for
K
=
+211 +31 + 112 + 13 5 +121 + 11
0.3
(solid lines).
listed in Table 6.4
figures
(under M). They
since the numerical
are automatically procedure explicitly uses their
asymptotic behaviour. Computations for. other values of K indicate that the figure < K < same qualitatively inside each of the three intervals 0 1, 3 < K. The junctions are thus fully determined. ,
Also
(dotted line).
remains the 1 < K <
3,
14.2 Small Values of
+ -112-
n
+ -1111-
4
YO
K
2.2 + E
3 +-2112
04
03
+211
-121 13 -112
+31
1
0
-
-6
-4
-2
0
2
6
4 w
Fig.
14.3. Characteristics for IN
bifurcations,
for K
=
2.2.
+
+E
y
K
0
4
4
3
2
-121 13 -112 -31
14
03
+211
-
Fig.
0
-5
14.4. Characteristics for 1T4
0
bifurcations,
5
for K
4.
85
14. Total Bifurcation of
86
Several intersections cated
are
Type
1
significant
Figs.
on
14.2 to
dots and labels. The orbits 112 and Q3 have
by
as
14.4; they
are
indi-
coordinates respec-
tively
(3
W=
K)VL2K
K
+
K
2K
-
YO
1+K
X,
_-K) //_2(1 (14.18)
and
W
(3
=
-
+ K L+
K)
YO
2
=
F+2
K
+ K 1:+::K::
X,
2
(14.19) (n2
exists
n
4
for 0 :!:'i:: K < 1 -) Q2 and
Q3 correspond to an intersection of a family described twice. The intersection is also an 4 family. The stability index at the intersection extremum in W for the n is z 4 family, z -1 for the n 2 family. +1 for the n The orbits Q4 and n5 exist for all K. They are respectively in =
only family with
a n
2
=
=
=
=
3+K
W
=
YO
=
v _2
v'Kv12-
1 + =
=
X1
1
-
,r2-(,
VKY +
(14.20)
VY)
and
W
sign(l
=
-
K)
3+K
YO
27
V2__
X,
1 +
\/211
-
VK (14.21)
They Correspond
intersection of two
4 families, which is also an Moreover, one of the families is made of asymmetric orbits and the other of symmetric orbits. This is another instance of the trident Situation already found in the case of partial bifurcation (Sects. 13.2.2 and 13.3.3.2). The stability index at the intersection to
an
n
extremum in W for one of the two families.
is
z
=
+1 for both families.
14.2.4
n
Numerical
Showing
=
6
computation
was
also used to determine the junctions for n = 6. (W, YO) plane would not be useful because the
characteristics in 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. The computations were made for several values of K. They indicate that the figure remains the same qualitatively inside each of the three intervals 0 :! K < 1, 1 < K < 3, 3 < K < 5. For 5 < K, however, some junctions 5.2612 change at the value K ; this is illustrated by Fig. 14.5. =
...
14.3 Conclusions for
+-1131-
YO
K
1
87
+-1131-
+ -111111-
5.25
Type
K
5.27
+
3
2 1111 +231
+
+21111 +231
5.8
Fig.
6
5.8
6
W
14.5. Some characteristics of IT6 bifurcations for K5.25 and K
14.3 Conclusions for
Type
=
-5.27.
I
study of bifurcations of type 1, begun in Chap. 12, is now complete. All 6. Results are junctions of branches have been determined at least up to n 14.1 Tables in and to 14.5 for in Tables 13.1 to 13-10 for partial bifurcations, The
=
total bifurcations. These tables
are
in the
same
format
as
Tables 8.4 to 8.11
and 8.14 to can be verified that there is agreement for series of tables. In addition, all cases up to both all branches which appear in in the tables of Chap. 8 are now solved. = undecided left been which had 6 n 8.17 of Volume I. It
This shows that the
quantitative approach is much
more
powerf l
than the
qualitative one. Nothing prevents in principle the solution of higher values of n, using numerical computation. However, the amount of work grows exponentially. ,
14. Total Bifurcation of
88
Type
bifurcation, type 1, quantitative study, for 0:!! K < 1.
Table 14. 1. Total the
1
n
2 to 6: branch
IT6 ....... 5
IT6 ....... 1
M ... 1
=
junctions determined by
+-2211+-1311-
-RI9-
+SI
"r
10
"r
+
1212
+
+
1T4 ..... 1
+-211+
13 -
IT4 ..... 3
+31 +
-112-
1T4 ..... A
+211 +
-1111-
+112 +
1
_Z10-
+2121
+1131
+-11112-
+-21111-
+1311
IT6 ....... 3
IT6 ....... A
+-114-
+321
+33
+231
+-411-
+-1212-
+-11211-
+312
+3111
+-1131-
1113
+-1122-
+
+
132
+
123
+21111 +
-111111-
+
11112
+
11121
+
11211
+12111
121
Table 14.2. Total the
bifurcation, type 1, quantitative study, for 1 < K < 3.
n
2 to 6: branch
junctions determined by
IT2 ... 1
IT6 ....... I
1T6 ....... 3
IN ....... A
t E
+-312-
+-411-
+213
+-231-
+-11211-
+-2121-
IN ....... 3
+3111
-
11211
-1113
-
12111
-
11
IT4 ..... 3
+31
+-114-
+-112-
+33
lT4++--+A
1T6 ....... A
+211
+321
+
-1111-
+312
-
112
+-1212-
-
121
+
+
-1122-1131-
+231
IT6 ....... A
IT6 ------- 1
-11121
+-21111-
IT6 ....... 5
IT6 ------- A
+-213-
-
-
132 -
+2121
-
+-11112-
+21111
123
-
-1311
+
-111111-
14.3 Conclusions for
bifurcation, type 1, quantitative study, f6r 3 < K < 5.
Table 14.3. Total the
lT2+++l +
E
11 M ..... 3
n
2 to 6: branch
IT6 -------1
+-312-
-321
+-231-
1131
-312
+-21111-
-
1T6 .......3 +-114-
-3111
IT4++--+l
IT6 .......5
+-211-
+51
-
+-1311-
-.1212
IT6 ....... 5
--11112-
1T6 .......3
+211
-33 1113
+
-1111-
-
-
112
+-411-
-
121
+-11211-
11112
1T6 -------A
-11121
-
IT4++--+A
15
+-2211-
-
-132-
1'311
+231 -
132
-
123
+-213-
+21111
+2121
+
1T6 ....... A --1212-
-1122-
+213 +-2121-
11211
-
12111
89
junctions determined by
IT6 ....... A
--112-
13
1
1T6.......1
-31
-
Type
-111111-
14. Total Bifurcation of
90
Type
bifurcation, type 1, n quantitative study, for 5 < K < 5.26.
Table 14.4. Total the
-
-
2 to 6: branch
1T6 ------- 1
E
--312-
-321
+-231-
11
-1131
-312
+-21111-
31
IT6 ....... 3
-
15
-
1212
-
1 1
+
-3111
lT4++--+l
1T6 ....... 5
+-211-
11112
-11121
--114-
--112-
13
+211
-33 -
1113 -
112
+-411-
121
+-11211-
-132-
-1311
IT6 ....... 3
-1111--
1T6 ------- A +-1131-
1T6 ....... 5
-51 --11112-
IT4++--+A
+
-2 211-
+-1311-
-
-
junctions determined by
..
IT6 ....... A
IT4 ..... 3 -
=
IT6 ....... 1
IT2 ... 1 +
1
+231 +21111 +
-111111-
+-213-
-
132
+2121
-
123
1T6 ....... A --1212-
-112211211
12111 +213 +-2121-
14.3 Conclusions for
bifurcation, type 1, n < K. quantitative study, for 5.26
Table 14.5. Total the
-
E
IT6 ....... I -
11
IN ..... 3
-312-
-1131 1T6 ....... 3
-31
--114-
--112-
-3111
IT4++--+l
1T6 ....... 5
+-211-
-51
1T6 ------- I
-321
+-231-
-
13
-
15
-11112
-
1212
+-2211-
+211 1111-112 -
121
-33 1113 +-411+-11211-
1T6 ------- A
+-1311+-1131IT6 ....... 5
-
lT4++--+A
+-21111-
-312
--11112-
1T6 ....... 3
11121
+
-111111-
-132-
+231
1311
+21111
+-213-
-
132
+2121
-
123
1T6 ....... A
--1212-12111 -
-1122-
-
11211
+213 +-2121-
91
junctions determined by
IT6 ....... A
-
-
1
...
M ... I +
2 to 6: branch
Type
15. The Newton
Approach
After the present monograph was completed (with the exception of the present chapter), there appeared a book by Bruno (1998, 2000). Chapter 1 of that book presents a general geometrical framework, based on the fundamental concept of Newton polyhedra, for the study of algebraic, ordinary differential, and partial differential equations in the vicinity of a singular point. Chapter 2 applies that framework to the case of a system of nonlinear equations. This work is in principle directly applicable to the problem which we have considered in the preceding chapters. We have a system of nonlinear equations (12.32), relating the variables IL, AC, yi, Aaj, and we wish to investigate the solutions of that system for small values of these variables, i.e. in the vicinity of the origin. If the origin corresponds to a bifurcation, more than one family passes through it (see Definition 6.0.2) and the origin is a singular point. We solved this problem in Chapters 12 to 14, using a somewhat intuitive, heuristic approach. Here we present a more rigorous approach, based on the use of Newton polyhedra as developed by Bruno. We ,will call it the Newton approach. Unfortunately, we will find that this approach is riot applicable to the general case, and that in practice it can only be used for small values of n. However, the results of the preceding chapters will be confirmed and put on firmer ground. In the present Chapter, a reference such as "Bruno 1.6" will mean: Bruno (2000), Chap. 1, Section 6.
15.1 Partial Bifurcation of
Variables and
Type
1:
Equations
We will describe the Newton
approach mostly
in the
case
1Pn, i.e.
a
partial
bifurcation of type 1 in which the bifurcating arc is made of n basic arcs (see Sect. 6.2.2); the case of a total bifurcation of type 1 will be briefly considered in
Sect. 15.8. There is n
The
> 2
(Sect. 6.2.2.1): (15-1)
.
equations which
we
the additional relations
have to consider
(12.33).
M. Hénon: LNPm 65, pp. 93 - 129, 2001 © Springer-Verlag Berlin Heidelberg 2001
are
(12.32),
and
occasionally'also
We eliminate the two variables yo and y.,
15. The Newton
94
Approach
using the last two equations (12.32). We n AC, yj (i 1, 1), Aaj (i 1, =
si
=
(-1)i
.
.
.
,
constants.)
are
It will be convenient to
the uniform notations xi, X2,
by
X1
=--
A
X6
`
Y2
X2n-1
To avoid
X2
)
X3
7
X2i+1 X2n
=
are
we
X2i+2
Y1
x5
,
these variables
write =
Aa2
,
Yi
(15.2)
Aan
X2n+1
variables will be written with
a
n by Bruno. subscript B when
have
(15-3)
2n + 1.
G2X2
=
X4
f2
=
X4(X5
f2i-I
-
-
X2i+2
+
X3)
G3(1
+
(12.32)
K)X3
-
Gixl
X2i +
-
+O(X2) 2
+
G2SiX2
+
O(X2i+l) 2
X2i+2(X2i+3
X2i+l) +O(XlX2i+3) 0
=
-X2n +
G28nX2
+0(Xj)
+
+
O(X2) 2
+
become
O(Xl)
O(XlX3)
=
G3(i
+
=
O(X2) 2
=
O(X2) 3
+
+
Ksi)X2i+l i
+
+
O(XIX4)
+
0, i
+
+
G3(1
Gjxi
+
=
f2n-1
=
designate we
2n + 1 variables. The number of variables is called
fl
f2i
X4
,
Aaj
The fundamental equations
=
Aal
Yn-1
=
in Bruno:
as
...,
=
=
confusion, Bruno'
needed; thus nB
AAC
=
Aan-1
=
Thus there
thus left with 2n + 1 variables: p, n). (We recall that G1, G2, G3, K,
are
-
O(XlX5)
O(XlX2i+l) 2,...,n
-
+
=
0
0,
O(Xl)
+
2,...,n
=
-
1,
O(XlX2i+2)
1,
KSn)X2n+l
O(X22n+l)
0
=
(15.4) The constants
G1>0,
G1, G2, G3
G2$0,
We have thus reduced
are
defined in
(12.31);
we
G3>0our
problem
(15.5) to the form considered
(5-1), (6-1), (8-1)).
The number of equations is called
will
MB. This number is
designate MB =: 2n
There is MB
it -
by
recall that
m
by Bruno (2, by Bruno; here we
(15-6)
1.
2; therefore we expect that the solutions' of the system nB equations (15.4) will lie on two-dimensional manifolds. One of the two dimensions corresponds to the variation of p, while the other is the parameter along a family of periodic orbits, which can be taken to be AC. The most studied case in the literature has been the particular case 7nB =
-
of
nB
-
1
(Bruno 2.8),
for which the solutions lie
on
one-dimensional manifolds.
Indeed, most examples in Bruno's book fall into this Pase, the only exception being the comparatively simple Example 4.1 in Chapter 2 which has nB 3, =
95
15.2 Method of Solution
MB
1. The present
=
problem might therefore more general case.
be of theoretical interest
as a
non-trivial example of the
15.1.1 Additional Relations
the system of equations becomes degenerate, as Will be seen, and it is then necessary to substitute one or more additional relations (12.33) to so,me of the equations (15.4). (The total number of equations remains constant.) Therefore we write also these relations in Bruno's variables. Their 1. For definiteness we describe them with indices from 2n to number is n
In
some cases
-
3n
-
2:
f2n
=
16 +
+O(Xl) f2n-l+i
=
h
+
O(X2) 5
G3(1
-
=
G3(1
K)X5
0
Ksi)X2i+l
+
+
O(X22i+3)
G3 (1
+
KSn-l)X2n-1
O(X22n-1)
+
particular =
+
+
O(X2i+l) 2
-X2n-2 +
+O(xl) In the
O(X32)
+
GO
+
-X2i + X2i+4 +
=
+O(Xl) f3n-2
K)X3
GO
case n
K)X3
=
+
O(X22n+l)
2, there is
G3(1
+
=
+
i
0
=
+
G3(1 =
G3 (1
Ksi)X2i+3
-
-
2,
KSn-l)X2n+l
(15-7)
0
O(xi)
+
-
2,...,n
single additional
a
K)x5
+
+
O(x 2) 3
relation +
O(X2) 5
=
0.
(15-8)
15.2 Method of Solution The method consists in
looking for solutions where
powers of a parameter. More precisely, the system (15.4) of the form (Bruno 2.2)
pressed xi
as
bi-rPi (1
=
o(l))
+
i
,
=
1,...,nB
all variables we
can
be
ex-
consider solutions of
(15.9)
,
with
bii4O, ,r
is
,r -
i
pji4O,
parameter, and
a
+oo
MAO)
.
consider the limit
(15-11)
The parameter use
we
1,...,nB
.
damental role in here. We
=
v
which
Chap.
12
was
defined in
(11.80)
and which
can
be related to the
quantities
the first two relations
equations (15.9)
in
(11-80).
(15.2)
We obtain
and
we
played
a
fun-
pi introduced
substitute the first two
15. The Newton
96
b2T P2 (1
+
o(1))
=
Approach
0[blTP'v(i
+
o(i))]
(15.12)
which reduces to P2
()(,rplv)
=
from which
we
(15.13)
have
P2
V
(15.14)
=
Pi
(15.9) represents a one-parameter family of solutions. In the most studied (Bruno 2.8) MB 1, the solutions form one-dimensional manifolds, nB curves, passing through the origin. Then the expressions (15.9) provide a
case or
=
-
parametric representation of one of these curves. In our case, there is MB 2, and (15.9) represents a curve lying on one of the two-dimensional nB manifolds passing through the origin. Bruno considers the general case where the pi can be positive or negative, i.e. in the limit some variables tend to zero while others tend to infinity. In our problem, all variables tend to zero; there is =
-
Pi < 0
The
(6.5)
is
simply (see also Bruno 2.6, between
(6.6)):
JP:
=
(15-15)
nB
=
of the problem (Bruno 1.6)
cone
and
K
i
,
p, <
0,...,Pn,,
<
0}
(15.16)
-
now one particular expression fi(X) in (15.4). -For simplic-, ity, we drop temporarily the subscript i and write simply f (X). This expression is a power series with respect to the variables. Each term is of
We consider
.
Xq2 the form Xqj 2 1 X
=
(Xj
I ....
It will be convenient to introduce the abbreviations:
XnB)l Q
=
qnB)l XQ
(ql,.
=
XqlXq2 1 2
a nB-dimensional space RnB with coordinates qj, qnB ...' Each term in the equation corresponds to a given point Q in PnB We call S
We consider
,
.
the set of these points for the equation f (X) = 0; S is called the support of f (X). The equation can then be written in a compact way (Bruno 2.8)
f (X)
"`
fQXQ
=
=
(15.17)
0.
QES
We
fQ
assume
that similar terms have been
differ from
We also introduce P consider
a
XQ where B
(P, Q)
=
monomial XQ. If
BQ-r(P,Q) (I
+
(bl,..., bnB) =
collected, and that
all coefficients
zero.
p1q, +
-
-
-
(pj,...,PnB ), we
substitute
and
we
(15.9),
consider
o(1)) and
given P. We
(15-18)
(P, Q)
+ PnBqnB
a
it becomes
'
is the scalar
product
(15.19)
15.2 Method of Solution
Each term of power series of
T.
97
f (X) is of the form (15.18); f (X) is thus expressed as a Clearly, as -r - +oo, terms with the largest exponents
dominate. We define sup (P,
cp
Q)
for
Q
We call dominant the terms of
(P, Q) For
T
=
(15.20)
E S
f (X) for
which
(15.21)
cp
+oo, non-dominant terms become negligible. If
-*
f (X), we designate by
dominant terms in
we
obtain the truncated
(X).
P, which truncated equation
the order
AX)
we
keep only
function Correspondingly, or
we
have the
(15.22)
0
=
the
truncation to
In that equation, we can factor out an equation for the bi:
T(PQ)
TIP. In the limit
T -4
oc),
we
obtain
f (B)
(15.23)
0.
=
These considerations
are
illuminated
by
a
nice
geometrical interpretation
in the space RnB We consider the set of points S. Its convex hull is called the Newton polyhedron, and is designated by 11'. For a given vector P, the .
equation
(PI Q)
=
(15.24)
C
defines
a hyperplane orthogonal to P. If c cp, we obtain the supporting hyperplane, which touches the Newton polyhedron. Intuitively, this plane may be defined as follows: we start with a plane (15.24) at a large distance in the direction of the vector P, and we let it slide in the direction of -P until it touches the Newton polyhedron. The intersection of the supporting hyperplane with S is called a boundary subset and designated by Sp. Each boundary subset corresponds to a face of the Newton polyhedron, which is the convex hull of Sp. The ,term "face" is used here in a general, multi-dimensional sense: it can be a vertex of the polyhedron (its dimension isd 0), an edge (d 1), a face in the usual sense =
(d
=
2),
and
=
so on.
Consider
given boundary subset, which we call S. The set of the boundary subset Sp coincides with S' is called the the boundary subset S'. The normal cones form a partition
now a
vectors P such that their
normal
cone
of
of the set of all vectors P.
(15.9) which satisfy one of the following geometrical approach. of points S. It defines the Newton
In order to find the solutions of the form
equations (15.4),
we
can
First from
f (X)
we
construct the set
polyhedron
r. We find the faces of IF. Next
We find the associated function
f (X).
We
therefore
use
the
we
consider each face in turn.
boundary subset, and the corresponding truncated write down also the set of equalities and inequalities which
Approach
15. The Newton
98
define the normal
of the
cone
boundary subset, and
the
found that
boundary
(P, Q)
from the above definitions:
(i) f (B)
0 and
(ii)
(They
lower value for all other
a
set of solutions of the form
a
subset.
derive
immediately
must have the same value for all
(15.9):
points.)
points of
We have thus
it consists of all solutions such
cone. belongs 0 corresponds to a vertex of the Newton polyhedron; the corresponding boundary subset contains only one point. But in that case, the truncated equation (15.22) contains only one term and cannot be satisfied (see Bruno 2.8).- (In other words, there must be at least two dominant terms.) Therefore, we only need to consider the faces with. a dimension. =
A face of dimension d
P
to the normal
=
d > 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 <
V
<
(15.128)
2
Conversely, for any value of v in that interval, we. can find positive values of A, and A2 which satisfy (15-127). Thus, the present case dad should correspond to one of the cases 0 < v < 1/2, considered in Sect. 12.4.2. Indeed, the equations (15.123) are identical (except for the error terms) with the equations (12.110a), (12.108) corre-
spon4ing to the case of two S-arcs. It is possible to recover also the error terms in these equations. We describe the computations in detail in the present case dad; similar computations can be done in the other
n
cases.
We go back to the whole equations: (15.38), (15.26) for i = = 2. We apply the change of variables (15.115). We obtain
91(wl'..., W5)
=
W2W4
92(Wl,
=
W2
....
W5)
-
(W3
G2
-
G3(1
+
wi)
+
=
-
=
-
O(WO
+
O(W2iW5)
+
=
0,
G,
+O(W2W4W5) O[W5 (W3 W1)] -W2W4 + G2 + G3 (1 93 (W1, -W5) +
K)wl
-
1, (15.44) for
0
+
K)W3
+
O(W5)
+
O(W23W5)
=
0
(15.129) (cf.
2.8, (8.4)).
Bruno
We effect Wi
V)i
=
-
01
order 0
(cf.
new
i
+ zi,
=
change
(15-130)
1, 2,3, is
E)(1). On the other hand the (with respect to the zi) disappear
=
02W4
h2
=
02 (Z3
h3
=
-02W4
Bruno W41 W5
=
Z1)
-
-
+ Z2 (03
Z2W4 +
-
-
K)zi
01)
GO
+
+
0i
=
i
=
1, 2, 3, and also
small quantities. Terms of and we obtain
0*5) +
E)(1),
zi are
+ Z2 (Z3
K)Z3
-
=
Z1)
O(WO
0 + =
,
O(W4W5)
+
=
O(W4)
we can
V)i[1
Coming
+
+
write
O(W4)
O(WO
O(WO
=
0
,
(15-131)
0
2.8, between (8.8) and (8.9)). small quantities. Solving these equations for the
are
Z1, Z2, Z3
Therefore
G3 (1
+ Z2W4 +
(Bruno 2.8, (8.8))
of variables
=
hi
Wi
the
ipi given by (15.118). There
with the
03
now
zi,
we
obtain
(15-132)
-
+
O(W5)]
back to the xi,
we
-
have
(15-133)
121
15.7 Power 'kansformations
G2
2
GjG3(1
X4
W5
X2
W4W5
X1
-
K 2)
2G2 G2
X5
GO
W4 W5
W5
K)
+
X3
)
AC)
W5
W4
G3 (1
_
k)
W5[1
+
O(W4)
+
O(W5)]
+ 0 (W4) + 0 (W5)]
(15.134)
+ 0 (W4) + 0 (W5)]
[1
back to my notations,
Going
[1
-
we
obtain
before
as
tIAC-2
=
(15.135)
and
G2
Aal
YJ
GO
K)
-
GjG3(1
-
K 2)
LAC-'
=
2G2 G2
Aa2
GO
We have
now
[1 + O(AC)
AC
=
[1 + O(AC)
[1 + O(AC)
_AC
K)
+
O(11AC-2)]
+
0(/_1AC -2)]
O(tAC-2)]
+
recovered exactly the results
+
,
(15-136)
(12.110a), (12.108), including
the
terms.
error
15.7.5 Case caA We take
# The
(E3, E4, E5, -N1,
=
-N 2).
(15.137)
equations of the change of variables X1
2
W4W5
=
X2
,
=
W5
X3
,
=
are
WIW4W5
,
X4
W2W5
,
W3W4W5
X5
-
(15-138) The truncated equations become
G2
91
=
W2
2
=
W2(W3
3
=
(1
-
-
-
=
0
,
wl)
K)wl
This system has
W2=G2i
+
+
G,
(1
+
=
0,
K)W3
=
0
(15-139)
-
unique solution
a
(K
Wi
+
1)GI
W3
2G2
(K
-
1)Gl.
(15.140)
2G2'
The determinant of the Jacobian is 0
.
-W2
1
-
K
W3
0
1 -
W1
(15.141)
2G2 54 0
1 + K
0
The solution is not
2W2
W2
a
critical
point.
It is
acceptable
since wi
j4 0,
i
=
1, 2, 3.
15. The Newton
122
Going X1
=
X4
=
This is with
Approach
back to the variables xi, 2
W4W5
X2=W5)
G2W5
X5
X3-
(K
=
we
-
1)Gj
2G2
obtain
(K
+
1)GI
2G2
-4-t)
(15.142)
-W4W5
system with 2 parameters W4 et w5. Going back to my notations we obtain the relations between these two parameters and the
a
(15.2),
fundamental parameters p, AC: W5
'61C
=
W4
,
These relations
(K
Aal
=
Aa2
=
the
1)Gj
+
(K
(15.143)
.
same as
in the
previous
yj
G2AC,
pAC-1,
2G2
=
case
dad. We have also
'
1)G,.MAC-1
(15.144)
2G2
There
N'
are
ILAC-2
=
two vectors
are
=
(-1, 0, -1, 0, -1)
N2
,
=
(-2, -1, -1, -1, -1).
(15.145)
Therefore
pj=-Aj-2A2, P4
-A2
=
P5
7
P2=-A2) -Al
=
-
A2
P3=-Al-A2;
-
(15.146)
The elimination of parasitic solutions gives
A, From
and
> 0
(15.147)
-
(15.14) A2
,
V
A2
0,
>
(15.148)
=
A,
as
+
2A2
in the
0 <
case
dad
we
obtain
1 V
<
(15.149)
2
Thus the present
caA should also correspond to one of the cases Indeed, the equations (15.144) are identical (except terms) with the equations (12.109) corresponding to the case of case
considered in Sect. 12.4.2. for the a
error
T-arc.
15.7.6 Case cda We take
,8
=
(E2, E4, E5, -N1, -N 2).
123
15.7 Power Transformations
change of variables
The equations of the 2 =W 5
XI
X2
W1W5
=
X3
i
are
W2W5
X4
W4W5
7
X5
i
==
W3W5
(15.151) equations become
The truncated
1
=
W2
92
=
W2W3 +
?3
=
-W2 +
This
-
G2WI
=
0)
G1
=
0,
G2WI
is similar to
case
G3(1
+
acc
+
K)W3
=
(Sect. 15.7.2):
(15-152)
0
the system has
no
solution.
15.7.7 Case bab
We take
#
=
(E3, E4, E5, -N1,
change of variables
of the
The'equations 2
X2
X1 =W 5
(15.153)
-N 2).
W4W5
=
W1W5
X3
7
are
W2W5
X4
,
X5
W3W5
-
(15-154) The truncated
equations become
GO
I
=
W2 +
92
=
W2(W3
93
=
-W2 +
-
-
wi)
K)w,
0,
G,
0,
+
G3(1
+
K)W3
0
=
IKI
This system has two solutions if
IG I G3 (K
W2
> 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,
to:
v
1/2.
=
then P2i+2
=
pi/2 for
any i E C.
P2i+2 <
have then P2i+2 :! p, /2, P2i+1 < p, /2 from (16.9) and (16.25). If pi/2, then both terms 1 and 2 in (16.2) are negligible in comparison
to term
3; this
Proof:
we
Y2i+1
write
we can
X2i+1
liM
=
(16.26)
Tpj/2
-r-++oo
^/2i+l is finite we
impossible. 0
is
(16.25),
In view of
Dividing (16.2) by
or zero.
-rP, and
taking
the limit
+oo,
have
r -4
+oo,
have
b2i+2 (^f2i+3
-
-y2i+,)
rp, /2 and
Dividing (16.3) by b2i+2
(16.27) We
-
and
b2i
+
G,
+
G2sjb2
(16.28)
form
simplify
this system
b2i+2
sjVGjG3 Yi
a
by
(16.27)
.
the limit
taking
G3(1
+
0
=
+
Ksi)-y2i+l
=
0
we
(16.28)
.
system of equations for the
-Si
V/G3
Xi
to
1
b2
,
bj and (12.112):
constants
change of variables similar
a
'Y2i+l
i
-r
-
Z2
4,1G, G3
7j.
W
(16.29) equations become
The
Yi(Xi+l Yj
+
Yi-,
+
Xj)
+ 1
W
-
-
0
=
(1
+
Ksi)Xi
We have thus recovered the
Sect. 12.5 in the
This
0
(16-30)
.
equations (12.114) which
were
obtained in
1/2.
case v
The Case
16.5
=
pj/2
P2 <
corresponds to v > 1/2. This case is similar to the previous one. We again 72j+1 by (16.26), and we obtain again (16.27). Instead of (16.28),
define we
have
now
b2i+2 We
-
b2i +. G3 (1
apply again
the
Yi(Xi+l Yj
+
+
Yi-,
Xi)
-
(1
+
Ksi)-y2i+l
change
+ 1 +
(16-31)
.
(16.29)
and
we
obtain the
equations
0
=
Ksi)Xi
case v
0
of variables
=
We have thus recovered the
Sect. 12.6 in the
=
>
1/2.
0
(16-32)
.
equations (12.119)
which
were
obtained in
Proving General
Results
16.6 The Case P2 >
Pj/2
16.
136
This
corresponds
<
1/2.
(16.9)
and
to
We know from
-r
(16.25)
X2i+1
liM
#2i+l
v
P2) P2i+1
<-
P2
-
We write
X2i+2
liM
#2i+2
7-P2
++oo
that P2i+2
7---*+oo
(16.33)
7-P2
There is :-=
0
(16.34)
similarly for 02i+2. Dividing (16.3) by
and we
if P2i+1 IN) if P2i+1 < P2)
b2i+1
#2i+l
-rP2 and
taking
the limit
+oo,
-r
have
#2i+2
02i
-
+
G2sjb2
Dividing (16.2) by
02i+2(#2i+3
-r
G3(1
2p2 and
#2i+l)
-
+
=
Ksi)#2i+l
+
taking the
limit
=
(16.35)
0.
T -+
+oo,
we
have
(16-36)
0
an encounter i for which P2i+2 < P2 , and antinode* an 0 for a node*, #2i+2 = b2i+2 which for encounter P2. There is #2i+2 P2i+2 for an antinode*. (It will be shown below that these definitions are natural
We will call node*
=
=
tt > 0 of the definitions of nodes and antinodes introduced case of generating orbits. We use asterisks to distinguish
generalizations for in Sect. 8-.1 in the
them.) We call also are* the part of the bifurcation orbit or bifurcating arc extending from one node* (or end of the bifurcating arc) to the next. (Again this is a natural generalization of the definition of an arc introduced in Sect. 4.1 in the
case
of
generating orbits.)
16.6.1 No Arcs*
special
We consider first the
happen
in the
P2i+2
=
case
of
a
where there
case
total bifurcation with
are no
no
arcs*. This
can
only
nodes*. Then
(16-37)
P2
for all encounters. Assume first that for
(16.2) can sum an
negligible
is
continue and
some
we
arbitrary value, G3 (1
same
+
(16.4)
P2. Then the term 3 in
every other value of
over
the terms 1 and 2
are
i, starting from
eliminated. The terms 3 and 4
are
order and must balance:
Ksi) (b2i+ 1
+G3 (1
-
-
the additional relation
all of the
i there is P2i+1 > P1
we obtain P2i+3 = P2i+1 > pi p2, b2i+3 = b2i+1. We obtain P2i+2j+l = P2i+l, b2i+2j+l = b2i+1 for all j. If we
and
-
+
b2i+5
Ksi) (b2i+3
+
+
-
-
b2i+7
+
-
-
0
,
(16.38),
16.6 The Case P2 > pi /2
137
which reduces to
G3nb2i+l
(16.39)
0
`
This contradicts the fundamental
assumption (15.10).
Therefore P2i+1 P1 P2 for all i. Since P2 > pl/2, there is P2i+1 < pj/2 < P2 and the term 4 in (16.3) is negligible. The terms 1, 2, and 3 are of -
the
order. Therefore
same
b2i+2 We
can use
from"i
b2i
-
G2.9jb2
+
---:
(16.40)
0
this relation to compute the successive values of the
b2i+2, starting
2. We obtain
=
b2i+2
for i odd
b4 b4
=
-
G2 b2
for i
3, 5, 2, 4,
even
(16.41
We write 7241
'r2i+1
liM
=
Dividing (16.2) by
b2i+2('Y2i+3 this
Dividing
(16.42)
++oo TPI-P2
,r
-
-rP, and
72j+j)
+
taking
Gib,
=
0
the limit
r -4
+oo,
we
have then
(16.43)
.
by b2i+2 given by (16.41), and summing
over
all values of i,
we
obtain
Gibln 2b4
+
Gjbjn
2(b4
-
G2b2)
(16.44)
01
from which
b4'-' G22b2 and
more
b2i+2 We
(16.45)
generally, using (16.41) again: =
can
-(-l)i
G2 b2
(16.46)
2
then compute the successive values of the 72j+1 from
(16.43),
obtaining 73
2Gibi
7241
73
-
G2 b2
for i
odd,
for i
even.
(16.47)
Summing once more the additional relation (16.4) over'every other value of i, dividing by -rPI-P2' taking the limit r --+ +oo, and using (16.47), we obtain G3 (1 .+ K)n 2
from which
GA)
2Gibj
+
G3(1
-
2
K)n
73
=
0
(16.48)
Proving General Results
16.
138
(1
73
and
Gib,
K)
+
(16.49)
G2b2
generally, using (16.47) again:
more
[K
^/2i+l
These values P2i+1
(16-50)
G2b2
non-zero; therefore
are
P1
"
Gib,
(-l)']
-
and
P2
-
b2i+1
---:
(16.51)
^/2i+l
for all i.
asymptotic coefficients have been determined. It can be remarked that they repeat with a period 2. Therefore we have here a total bifurcation M. We go back to the physical variables. From (15.2), (15.9), (16-37), (16.46) All
we
obtain
G2
-(-l)'-AC
Yj
Rom
2
(16.52)
.
(15.2), (15.9), (16.51), (16.50)
Aaj
=
[K
-
(-1)']
G, G2
/tAC-1
we
obtain
(16.53)
.
We have thus recovered the main terms of the in the
species orbit
0 <
case
v
<
expressions (12.78) for
a
first
1/2.
16.6.2 Arcs* We consider
now
the
where there
case
are
arcs*. The whole bifurcation orbit
bifurcating arc can them be decomposed into a sequence of arcs*, which we analyse individually. We consider an arc* made of m basic arcs, extending from encounter i to encounter i + m. All encounters inside the arc* axe 4ntinodes*; therefore or
#2i+2j+2 where the
limit,
we
=
#
b2i+2j+2 0
are
defined
by (16.33). Dividing (16.3) by
-
02i+2j
+
G2 si+j b2
+
G3 (1
+
Ksi+j)P2i+2j+l 1,
(16.36)
)62i+2j+3 Finally,
we
taking
the
-
=
0
7
-
-
-'M
-
=
0
,
(16.55)
have
02i+2j+l
=
0
j
,
the two ends of the arc*
#2i+2
TP2 and
obtain
#2i+2j+2 From
(16-54)
1
j
0
02i+2m+2
=
0
=
are
-
1'...'M
nodes*,
-
1.
.(16.56)
and
(16.57)
pi/2,
16.6 The Case P2 >
In the
case
the
case
i
0
=
of
02
=
partial bifurcation, these equations can be made to cover also 'end of the arc* is also an end of the bifurcating arc (i.e. + m n) by defining
a
where
or/and
139
i
one
=
02n+2
0
=
(16.58)
0
The equations (16.55), (16.56), and (16.57) form a system of 2m + 1 linear #2i+2m+2. This system is easily equations for the 2m + 1 variables 02i+2 solved by elimination.
S-,krcs*.
16.6.2.1
For
odd,
m
obtain
we
-G2sjb2
02i+2j+l
Z 3_(m
-)M
Ksi)
-
-jG2sib2 M
#2i+2j+2
Ks.1
-
(M
m
We call this
Ksi
-
j
for
j odd
even
S-arc*. All values in the
a
2,4,..., m 1, 3,5,.
we
=
P2
P2i+2j+2
=
P2
,
We go back to the obtain
b2i+2j+l
=
#2i+2j+l
b2i+2j+2
=
02i+2j+2
physical
G28t
Aai+j
G3 (M
-
Ksi)
-jG28i m
Yi+j
Ksi
-
(m
m
-
Ksi
case
0 <
v
<
=
We call this P2i+4
=
0 a
m
82i+5
,
are
non-zero;
determined:
1 ....
I
M
I
(16.60)
(15-2), (15.9), (16.60), (16.59)
for
j
even,
(16.61) for j odd.
AC
expressions (12.110) for
a
S-arc
b2i+4
P2
=
=
2,
we
82i+4
0
=
#2i+4
The asymptotic coefficients of the point; we know only that P2i+5 < P2
,
obtain =
G2sjb2
(16.62)
We have
T-arc*
P2i+3 < P2
=
sides
2.
1/2.
16.6.2.2 T-Arcs*. For
,82i+3
are
-
j=l'...'m-l.
We have thus recovered the main terms of the in the
1;
,
AC
j)G2Si
-
i
,
variables. Rom
AC
m
right-hand
therefore the asymptotic exponents and coefficients P2i+2j+l
-
(16.59)
j)62sib2
-
for
-
=
(16-63)
G2sjb2
two basic
arcs are
not determined at this
(16.64)
A better characterization of the T-arcs* will be obtained in Sect. 16.6.2.4. For
0 for
m even
2, 4,. j possible. =
and .
.'
larger than 2, the resolution of the system gives 02i+2j+2 2; this contradicts (16.54). Therefore this case is not
m
-
140
Proving General
16.
Results
16.6.2.3 Nodes*. We consider
arc* made Of Mb basic arcs,
now a
node* i. If the previous arc* is (16.59a) with j = m
G2sib2
02i+1
+
Ksi)
This value is non-zero; therefore P2i+I P2 and If the previous arc* is a T-arc, we have from P2i+l < P2
il the
Similarly, from
82i+1
)
(16.59a)
with
=
following 1 j
b2i+1
=
#2i+l
(16.62) (16.66)
0 arc* is
S-arc* made of m,, basic arcs,
a
have
(16.67)
-
G3(Ma
we
=
G2sib2
0243
S-
(16.65)
-
G3 (Mb
a
have from
we
-
Ksi)
from which P243 = P2 and b2i+3 = 02i+3If the following arc* is a T-arc*, we have P2i+3 < P2
Thus, for
(16.2)
a
SS node*
of the
are
8243
1
same
=
0
(16.68)
-
(joining
S-arc* to
a
order and their
S-arc*),
a
the terms 1 and 2 in
has the asymptotic value
sum
Ma)G2sib2b2i+2 rP2+P2i+2 G3 (Mb + Ksi) (ma Ksi)
(Mb
+
(16.69)
-
This we
sum
does not vanish. Therefore it must be balanced
by
the term 3, and
have P242
For
a
=
P1
-
b2i+2
P2
For
a
P2i+2 We
(Mb
ST node*, the term 1 in
2. The terms 2 and 3 must
P2i+2
G, G3 bi (Mb
=
P1
-
TS node*, ==
can
P1
-
and
we
-
Ksi)
negligible compared
GlG3b,(Mb
to the term
+
Ksi)
(16.71)
G2sib2 same
(16.70)
have
-
obtain in the
b2i+2
is
+
way
GlG3bi(m,,
-
Ksi)
(16.72)
-
G2sib2
go back to the physical variables, using (15.2), (15.9), (16.70) We recover the main terms of the expressions (12.108), (12.107),
The last case never
we
P2
(16.72). (12.106) for SS, ST,
to
balance,
b2i+2
P2
(16.2)
Ksi) (ma Ma)G2sib2
+
-
case
and TS nodes in the
would be
a
TT node*. It
case can
0 <
be
v
<
1/2.
proved, however,
arises:
Proposition
16.6.1.
There cannot be two T-arcs* in succession.
that this
'
16.6 The Case P2 > 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 <
v
<
1/2.
16.7 Conclusions
rederiving all basic results for a bifurcation of Chap. 12, in a completely independent way, ustype 1, previously In particular, we have made no recourse fundamental the equations. ing only to the qualitative analysis of bifurcations which was presented in Volume I; on the contrary, we have rederived here the existence and properties of nodes, antinodes, S-arcs, and T-arcs. The present approach is strongly dependent on the particular form of the system of fundamental equations in our problem; it is an ad hoc method. It cannot be generalized to other problems in a straightforward manner. In other words, there is no guarantee.that a similar method could be devised in another given problem. We have thus succeeded in
obtained in
16.8
143
No TT Node'
Appendix:
16.8
No TT Node*
Appendix:
We prove here Proposition 16.6.1. We recall that this takes 16.6, i.e. we are considering the case
place
inside Sec-
tion
P2 >
pi
Suppose of two
(16-84)
.2
that the
bifurcation orbit
T-arcs*. We call this
or more
16.8.1 Partial
bifurcating T-sequence.
or
a
arc
contains
a
sequence
T-Sequence
We consider first the
case
of a partial T-sequence, made of 1 T-arcs*, extending (with 1'2: 2). Each T-arc* is made of two
from encounter i to encounter i + 21
1 and i + 2j. basic arcs, numbered i + 2j The inside encounters of odd rank with respect to i + 21 1, are antinodes*; from (16.63) we have -
,
i,
i.e. i +
1,
3,
i +
-
b2i+4j
P2
P2i+4j
=
The. inside encounters of
nodes*;
G2sib2
even
(16-85)
j
,
rank,
i.e. i +
2,
i +
i + 21
4,
-
2,
are
therefore
For the arcs,
have from
we
P2i+4j-l < P2
(16.86)
j
P2i+4j+2 < P2,
(16-87)
j
P2i+4j+l < P2
7
end of the
Finally, each
(16.64)
partial T-sequence
is either
a
junction with
a
S-arc* or, in the case of a partial bifurcation, an end of the bifurcating arc. We will consider the case where both ends of the sequence are junctions with a
S-arc*; the
other
cases can
We have then from P2i+2
=
P1
-
be treated in
a
similar way.
(16.71)and (16.72)
P2
P2i+41+2
P1
-
(16.88.)
P2
We define
P*
=
(16-89)
P2i+41+1)
MaX(P2i+3,
There is P2
>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*
-
=
<
Y
-
and therefore P2i+4j'+2
negligible,
Gib,
+
If
P1
=
cases
(16.94)
-
"
or
pi
1, Similarly, the final encounter i + 2j is the end of the T-sequence if j a following T-arc* if j" < 1. In the second case, we have P2i+4j', +2 p* and
there is -
-b2i+4j"+2b2i+4j"+l and in both
cases we
P1
P2i+4j" +2
-
+
Gib,
have
P*
(16-96)
-
We consider three separate 16.8.1.1
p*
>
P2i+4j' +2 < From
pl/2. P*
cases
for
We have from
p*.
(16.94) (16-97)
-
(16.4) applied
P2i+4j'+6
(16-95)
0
to the first
T-arc*,
we
<- P*
(16-98)
because otherwise the term 2 would be the
again (16.4)
to the
P2i+4j+2
have then
following T-arcs*,
P*
we
single dominant generally
term.
Applying
obtain
(16.99)
3
We write
02i+4j+2 Rom
(16.94)
lini -++00
and
12i+4j+2 'rp.
(16.96)
we
have
(16.100)
Appendix:
16.8
02i+4j'+2
::--
02i+4j"+2
0
Dividing (16.2) by
-r
2p
*
#2i+4j+2(b2i+4j+3
and
-
7- -*
+oo,
we
letting
b2i+4j+l)
The term 3 has vanished.
---:
No TT Node*
(16-101)
0
+oo,
-r
we
obtain
j=j'+1'...'j"-1. (16.102)
0
`
145
Dividing (16.4)
for i +
2j
-
I
by 7-P*
and
letting
obtain
-)62i+4j-2 + #2i+4j+2 + GO +G3 (I + Ksi)b2i+4j+l
=
Using (16.92),
we can
-#2i+4j '2
0
rewrite this
02i+4j+2
+
-
+
Ksi)b2i+4j-l +
,
11
...
(16.103)
1
as
2G3b2i+4j+l
0
+
(16.104) multiply
We we
add also
this
now
(16.102)
j' + 1, j", equation by b2i+4j+l we sum it for j and we use ..,j"-l, j'+l,. j (16.92) and (16.101). .
,
for
.
.
,
=
We obtain
2i+4j+l
2G3
b2
But the one
b2i+4j+l
term in the
pi/2
P2i+4j'+2 Rom
(16.105)
0
according
to
(15.10),
We have thus reached
an
impossibility.
are non-zero
sum.
pjL/2.
p*
16.8.1.2
=
We have
from,(16.94) (16.106)
.
(16.4) applied successively
P2i+4j+2 ! pi / 2 We write
and there is at least
to the
T-arcs*,
we
have then
(16.107)
,
now
X2i+4j+2
lim
02i+4j+2
(16.10'8)
rpj12
++00
2j'. If f 0, it is the beginning (16-84, we have P242 < pi/2, and
We consider the initial encounter i +
of the
T-sequence.
therefore
=
(16.88)
and
Ifj' > 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*.
<
g'
(16.4) applied
-
4 are also P2i+4j'+6 > 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
<
We define then
+
Gib in =0,
-
(16.129)
2
pi/2. Reasoning P1
P2i+4j+2
we
as
as
P*
for all
82i+4j+2
as
in
as
in the
previous
case,
we can
show that
(16.130)
j
(16.120)
and
we
continue
as
in Sect.
16.8.1.3;
obtain
Gi. bi which is
n -
2
=
(16.131)
0,
impossible.
This completes the
proof of Proposition
16.6-1. N
Quantitative Study
17.
We
begin
now
of
Type
2
to which
quantitative analysis of bifurcations of type 2,
the
the remainder of the present volume- will be devoted.
17.1 New Notations We introduce
a new
Tf,
notation
T9
for the
which
T.-arcs,
will,simplify the
presen
tation for type 2. Definition 17.1.1. A T-arc will be a
1,
>
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
<
v
p >
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 <
U
(_,61AC)1/2X,?N
=
and
a
IAC)112XII*
0
changes of
We need different
1/3
<
v
variables for
u
(17-90)
u'
and
161
in the
cases
of
a
S-arc
T-arc.
Step 3. It will be convenient to make a second change of variables orderto simplify the constant coefficients in the equations:
xj* Z! z
revx!
=
a2 VY -
_Z
Ve'VY Xf/
X!I* x 2*
1
VY
=
)
z
37rI(a
1)Vy
-
V2
Xi
;
Y
37rI(a
-
1)Vy
Xi
V2
9-7r 2J2 a(a
Yi
-
1)Vy2
2V4
equations, the sign is + for a Te arc, Xi" are always positive. Note also that, by
In the first two
Xi' (17.91f),
way,
-
for
a
Ti
virtue of
arc; in that
(17.88a)
and
there is
sign(yi) where
(17.91)
Yi
Y
Y
and
in
=
(17.92)
o,
is the relative side of passage, defined by (17.1). 4. Substituting in the fundamental equations (17.63) to
ai'
(17.67), Step and left-hand the in the terms dominant by the members, dividing collecting obtain: of we AC, appropriate powers -
For
a
T-arc:
xill Xil
o(Acl/2)
x,1
-
2 _
1
_
Z,
Z,(Xf +,Xii) i 2
Note: since AC
=O(AC,/-1AC-1) O(AC1/2, /,ZAC-3/2) can
be positive
O(IAC11/2), O(PJACJ-3/2 ), for
symbols -
For
a
the intended
simplicity;
,
x/1 i+
-
j
x'+j i
Yi+j
-
-
zi+j
=
we
meaning
=
O(PAC-3)
(X,+l i +
+
j
O(AC, A)
in
principle
we
should write
will omit the absolute value is clear in any
=
1".-, M
1'..
,
-
For a TT node
a
i+j )
=
J= 1....
there is
X11
_
O(AC, PAC-') Zi+j (X ?I+i + x4i Yi+j-1
zi+j + 1
Yi
negative,
or
etc. However
case.
S-arc:
Yi+j ( 1;Z+J+l
-
(17.93)
=
11
1
M
-
1
M
3
a'T-arc to
O(PAC-5/2)
sign because
(Proposition 4.3.2).
(17.96) (17.97)
(joining =
(17.95)
O(AC, PAC-2)
1M.
X11) i
(17.94)
a
T-arc):
;
two T-arcs in succession must have
(17.98) opposite signs
162
-
Quantitative Study of Type
17.
For
TS node:
a
Y,Xfl -
For
O(AC1/2'PAC-5/2)
=
For
O(AC1/2' tl'AC-5/2)
=
Z)
-
Y,
<
V
0.
(17.103)
-
3 are
then obtained
by equating the right-hand mem-
zero.
Step
6. We solve
show that for
a
-+
if
case
asymptotic equations
Yj
right-hand members to vanish for tL
1
0 <
bers to
arc:
(17.102)
5. We want all terms in the
This is the
As
bifurcating
a
O(tAC-2)
=
Step
(17
O(AAC
For the two ends of
The
(17.100)
SS node:
a
Yi (X -
(17.99)
ST node:
a
Y,Xi+1 -
2
=
a
now
asymptotic equations. (17.98)
these
to
(17.102)
in all cases,
node,
(17-104)
0
consequence, here
again
we
obtain for each
T
are
or
S
an
independent
can separately. equations, For a T-arc, the system (17.93) has three solutions:
be solved
which
set of
.
Zi
=
0
Zi
=
0
Zi
=
Xil
,
X!
,
==
Xf, '
=
+1 ;
=
Xi"
=
_1 ;
Z
-1
X/i
,
=
X/I i
=
(17.105)
0
By continuity with the previous For a S-arc, (17.96) gives
case v
=
0,
we
choose the first solution.
(17-106)
Zi+j
(17.94) gives X
-
for
an
X" i+
i
antinode
=
(17.107)
0
(17.95), (17.97), (17.107), plus (17.104) system of linear equations for the
X +j+i Step
=
XI/
i+j
x +j,
M =
-
2
-3,
7. For 0 < IL <
continues to exist if the
Yi+j
for the two ends of the arc, form x'Ij, yi+j which is easily solved: i+
_j(M
-
j)
(17.108)
-
1, the solution described by (17.93)
error
a
terms in the
right-hand
side
are
to
(17.102)
small and if
17.4 The Case 0 <
JJJ
the Jacobian is satisfied if
There
of the
asymptotic equations
is not
v
<
1/3
163
The first condition
zero.
compute the Jacobian. 4n variables and 4n equations for a total bifurcation; 4n + 1
v
are
1/3.
<
We must
now
variables and 4n + 1 equations for a partial bifurcation. We consider first the case of a TT node. The partial derivatives of the expression f appearing on the left-hand side of
Of Oyi
-
X!
+
In the solution zero.
(17-98)
can
Yi
axi to
Therefore the line in
element. We
Of
Of
X!/'
(17.104)
are
-
axi,+1
(17-109)
Yi
(17.108), only the first of these derivatives is nonJJJ corresponding to (17.98) has a single non-zero
then factor out this element and eliminate the
corresponding
line and column.,
Similarly,
(17.99) to (17.101), (17.102), we find that
for the other kinds of nodes, corresponding to
and for the ends of
a bifurcating arc corresponding only the partial derivative af 1,9yi is non-zero and the determinant can be simplified. After this has been done for all nodes, JJJ is decomposed into a product of smaller determinants, each of which corresponds to a T- or S-arc. For a T-arc, there are 3 variables Xj', Xil zi, and the 3 equations (17.93); the
to
Jacobian is
which
_1
1
0
2Xj'
0
-1
Zi
Zi
X!21 + X!, Z
(17.110)
equals -4 for the solution (17.105). a S-arc, the determinant has 4m
For
(17.95), (17.96), (17.97), (17.94) zi+,,. (17.96) gives only one
-1 lines and
for the variables
can
eliminate the
nant of 3m
variables
-
=
t+j
Z
derivative:
columns, given by
x11
i+ 1) Zi+1
Of lo9zi+j
I
line and column. There remains
X +21
Yi+1)
2
1, and
=
-Yi+j
(17.94),
yi+j. For
af
the derivatives
I
-
Oyi+j
xi+j+l
-
X/I
i+j
=
the
are
Of
-
-
we
determi-
corresponding columns, defined by (17.95), (17.97), (17.94) for a
1 lines and
x'i+j, x'i'+j,
Of OMI
non-zero
x +,,
0,
=
4M
Yi+j
Z+j+l
(17.111) We can factorize yi+j (which is determinant takes then the form
non-zero
according
to
(17.108b)).
The
Quantitative Study
17.
164
of
Type
2
(17.112)
This determinant is
easily computed by
successive
eliminations;
its value is
-2m.
IJI
is non-zero. For 0 < v < 1/3, there exists for M 54 0 a solufundamental equations (17.93) to (17.102) close to the solution (17.104) to (17.108) of the asymptotic equations. The error is of the order
Thus
of
tion
of the
the
largest
term in the
O(AC1/2' pAC-3) X,
members of
right-hand
(17.93)
to
(17.102),
i.e.
Thus the solution has the form
O(AC1/2' IIAC-3),
1 +
=
.
(17.113)
etc.
Steps
8 and 9., We refine
also go back to the physical that for a TT node, Y,
=
(17.101) Y,
For
a
=
O(MAC-512), shows that for
hi a
0(tAC-3),
TS node,
we
now
SS
error
estimates for
=
to
some
variables. We
(17.91). (17.98)
O(ItAC -1/2).
shows
(17.114)
node,
O(ttAC-1)
hi
have
the
variables, using (17.88)
E)(AC1/2 )
uj"
(17.115)
.
and
u'i+l
O(AC); (17.63) gives
then
hi For
a
=
O(PAC-1/2),
Y,
=
0(tZAC-5/2).
ST node, the result is the same. a T are, we have Xj' + Xj" 2; therefore u
For
=
(17.116) + 0
E)(ACl/2,),
and
(17.66) gives Aaj
(17.65)
=
O(tzAC-1)
(17.117)
.
under consideration of
VYAC +
U 2
==
(17.103)
O(AC2),
reduces to
(17.118)
from which
U'j Rom
=
V/--VYAC[l + O(AC)]
(17.64)
U'j'
-
U'j
=
we
.
(17.119)
have
0(tZAC')
,
(17.120)
17.4 The Case 0 <
<
v
1/3
165
and therefore also
.\/--VYAC[l + O(AC)]
0 In
(17.121)
.
and (17.121), the sign is + for equations are identical with (17.76). For a S-arc, we have first from (17.96)
(17.119)
a
T' arc,
for
T'
a
arc.
These
two
a2 VY
Aa
VY
Ac[i
+
(17.122)
O(AQ
(17-84). On the other hand, the error terms in (17.94) (17.97), and also in (17.115), (17.116),.(17.102) for the ends, are all included in O(AC, pAC-1). Therefore, going back to physical variables, we
which is identical with to
have
U
=
.I+j
(m
2j
-
37rI(a
2)
+
1)Vy
-
2V2
AC[l
+
O(AC pAC-3)]
Y
u".
=
(m
37rI(a
2j)
-
-
1)Vy
2V2
AC[I
+
O(AC, PAC-3)]
Y
j=11
hi+j i These
=
=
...
7
M
-j(M
equations
presence of
an
j)
-
0 ......M
(17.123)
1
1)V2 Y
97r 2J2 a(a 4
2vy
AC2[j+0(AC,11AC-3)j, (17.124)
-
similar to
are
additional
error
(17.86)
and
(17.87); they
differ
Finally, we can compute exactly the dominant term of hi the'help of (17.63), (17.119), (17.121), (17.123). We obtain -
At
a
1 V
with For
in
a
node with
TT node:
hi
-
only by the
term.
a
a
+
TS
t4(_VYAC) -1/2 [1
sign for or
a
+
O(AC1/2)]
TIT' node and
a
-
(17.125)
sign for
a
TiTI node.
ST node:
2tt(_VYAC) -1/2[1 + 0(AC1/2)j
hi
(17.126)
V
with -
For
a
a
hi
+
sign for
T'S
a
or
ST' node, and
a
-
sign for
a
TiS
or
ST' node.
SS node: 4v =
W(Ma
+
Mb) (a
_tAC-1 [1 -
1) Vy
+
O(AC, IAC-3)] (17.127)
where Mb and Ma
are
the values of
m
for the two S-arcs.
166
Quantitative Study
17.
Substituting
(17.66),
of
into
Type
2
obtain also the dominant term of Aai for
we
a
T-arc:
a(g'
Aai
+
g")sign(vy_)
,AC-1 [1
37rIVy
+
O(AC1/2)]
(17.128)
where
1
if the previous if the previous
0
if the previous
2 9
9
2
if the
1
if the
0
if the
aside
(Sect. 17.2.3)
is
a
arc
is
a
arc
following following following
The previous or following arc cating arc. Note that g' + g"
S-arc, T-arc, does not exist;
arc
(17.129)
S-arc, T-arc,
arc
is
a
arc
is
a
arc
does not exist.
(17-130)
does not exist in the 0
=
only
in the
case
of
an
end of
a
bifur-
2P1, which has been
case
set
and will be considered in Sect. 23.2.
The variations in order of
magnitude of the physical quantities hi, Aai, AC, given by (17.119) to (17.128), are shown below in Fig. 17.2. Only the right part of each panel, corresponding to the interval 1/3 < AC < 1, should be considered at the present stage; the left parts, it corresponding to AC:! t,113, will be described in the following sections.
u'i
and
u'i'
functions of
as
17.5 The Case
study
We
to AC
now
what
1/3:
v
happens
Transition 2.1
when
we
reach the value
v
=
1/3, corresponding
E)(/_tl/3).
=
In the
case
of type
1,
we saw
in Sect. 12.5 that when the value
v
=
1/2
is
reached, joined by two, and the story ends there. In the branches case of type 2, the situation is more complex. As will be found below, some all
branches on
to
To
are
joined
1/3,
>
v
two
are
at
v
=
1/3,
while other branches traverse this
and meet another branch
distinguish
This is intended
now
v
=
value,
go
1/2.
these two cases, we introduce the notion of transition. kind of subdivision in a bifurcation. Generally, when a may
change (see
the bifurcation is not yet ended. We call thus transition 2.1 (type 2, first transition 2.2 the case v = 1/2. We start
at
as a
reached, the exponents
transition is
only later,
the
study of transition
transition)
2. L We
use
below the
Fig. 17.2),
but
1/3,
and
case v
=
again the general method
of Sect. 11.4.
Step v
=
1/3, a
1. The results of the
the orders Of
previous section show that when magnitude are as follows:
we arrive
at
T-arc:
Aai
=
0(y 2/3)
U'i, U'i'
=
0
(17.131)
17.5 The Case
-
For
hi+j
167
Transition 2.1
0 (M 1/3)
0(,12/3)
=
rh
-
(17.132)
1
TT, TS, ST nodes: hi
-
1/3:
S-arc:
a
Aai+j, ui+j, ui+,
-
v
=
O(A5/6)
(17-133)
0(p 2/3).
(17.134)
SS node:
hi
=
that, the hi are now of the same orderinside a S-arc and at fusion of consecutive S-arcs takes place: they form a single will call a R-region. (An isolated S-arc will also be called which we region, We remark
a
SS node. A
a
R-region.)
This fusion is the
essence
of transition 2.1. Two
cases
must be
distinguished.
(i) a
In the
case
of a total
bifurcation, and if only S-arcs
total transition 2.1. The whole orbit obtained for
which will be called
a
v
are
=
present,
1/3
is
a
we
have
R-region,
R-orbit.
In the other cases, i.e. for a partial bifurcation and/or if T-arcs are present, we have a partial transition 2.1. One or more finite sequences of S-arcs may be present. Each sequence terminates on both sides either at a
(ii)
junction, with a T-arc, or in one end of a bifurcating arc. Each such sequence is a R-region, which will be called a R-arc. These new concepts will. play a fundamental role, and it is essential to understand them. We illustrate them with a few examples (Fig. 17.1). Full lines represent schematically the original bifurcation orbit or bifurcating arc, with their division into S- and T-arcs. Dashed lines represent the R-regions. S
S
S
S
R S
S
S'
S
R
S
T S
T
S
S
T
R
R T
T R
S
S
a
Fig.
17.1. Fusion of S-arcs into
b: total 2.1.
S
S
b
R-regions. bifurcation, partial transition 2.1.
C
a: c:
total bifurcation, total transition 2.1. partial bifurcation, partial transition
Quantitative Study of Type
17.
168
In
case
a,
fore this is
a
have
we
total
a
2
bifurcation, and only S-arcs are present; thereR-region is a R-orbit, which coincides
total transition 2.1. The
with the whole bifurcation orbit. In we
case
have
happen In
a
b, we still have a total bifurcation, but T-arcs are present; therefore partial transition 2.1. The R-regions are R-arcs. In this case, it can
that
tion 2.1.
T-arcs
S-arcs
no
present.
are
partial bifurcation, and therefore a partial transiThe R-regions are again R-arcs. In this case, it can happen that no
case
c,
have
we
present,
are
a
S-arcs.
or no
it should be well understood that there is
no one-to-one corparticular, and bifurcation between total/partial total/partial transition. respondence In case b, we have a total bifurcation and at the same time a partial transi-
In
tion 2. 1.
A bifurcation orbit
regions.
We define
more.
T-arcs,
or
a
a
or
A of basic
a
as an
encounter
T-arc and
a
arcs
6.2.1 and
is
formed of T-arcs and R-
now
which form the
appropriate junction between
R-antinode
as an
any two
encounter
We define also
We
one
introduced for second
use a
letter T
R-region.
We
a
use a
or
species
designate
P to
a
R-region as the number short designation of the
P> ; for instance: 2. 1P3.
or
10'...'A
A,
-
-
1} 1}
the set
we
define
for
a
R-arc,
for
a
R-orbit.
of basic
arcs
of
the set
of internal
collisions of
(17-135) a
R-region,
as
(17-136)
Al
11,
2. The estimates in the
Step
a
at the
transition. We define the order of
A
=
lying
R-arc, and
6.2.2).
By analogy with Sect. 11.3-1, R-region, as
A
arc
notation similar to the
a
(Sects.
partial
form 2. 1
a
bifurcating
R-region.
We introduce
bifurcations total
R-node
between
inside
lying
or a
The former division into nodes and antinodes is not
previous step suggest the following changes
of variables. -
For
a
T-arc: I
-CIAC,Y)112Xf*
U
U l
=
(_r1AC)112X11*
Aai
=
ACZi*
.
(17-137) We
use
here powers of AC rather than p in order to show the continuity neighbouring cases v < 1/3 and v > 1/3. These relations are
with the
identical with For
a
U
(17.90)
and
(17.88b).
R-region: JA 1/3x
U '
=
A113X 1*
Aa
M 1/3z
(17-138)
17.5 The Case
-
=
1/3:
169
Transition 2A
For all encounters:
hi -
v
t12/3 yi*
=
(17.139)
;
For AC:
AC
/.1
=
1/3 W*
(17.140)
.
E)(1). according to the definition of v, w* Step 3..It will be convenient to make a second change of variables:
Note that
=
x1* i
V, VY yx
E
'
2VY
zi
VY
a2v,Ll
Z! z
z
a =
37rI(a
-
1)
V2
2
Y!
-LiZ
Z* 2
-Ljx
x z*
Zi*
evY xi
xi
(17.141)
Y
W*
a-yi
37rIVy(a
-
1)
with 1/3
(17.142)
L, av
for a Ti arc; in that equations, the sign is + for a Te arc, relations first The 3 are identical with are positive. always way, Xj' the corresponding relations in (17.91). Here again, we have from (17.139),
In the first two
-
Xj"
and
(17.141g), (17.142),
sign(yi)
=
u,,
and
(17.1)(17.143)
.
Note also that
sign(w) Step
4.
=
-sign(c'AC)
Substituting
(17.144)
.
in the fundamental
equations (17.63)
to
(17.67),
we
obtain: -
For
T-arc:
a
Xi" Xi' 0 (A 1/6) Z,(Xi, + XiI) O(Pl/6) =
These
equations
0(p'/'). -
For
a
are
The T-arcs
2 -
-
zi
=
O(Y' /3) (17.145)
.
identical with are
i
(17.93),
in the
essentially unaffected by
particular
case
AC
transition 2.1.
R-region:
Yi+j (X Z+j+l x
Xi
=
-
i+j
Z,+j +
X
Z+j
W
=
-
-
x
"
i+j )-l=
Zi+j
=
O(P1 /3)''
O(P'
O(AI/3)
yi+j-yi+j-1-zi+j(X ?I+i
jEj, E
E +X
A If
i+j)
0 (A, /3),
j
E
(17.146)
17.
170,
-
For
.
-
For
Quantitative Study
For
y,(Xil+l
2
Xill)
+
=
O(pl/6)
(17.147)
TR R-node:
a
OW /6)
=
(17.148)
RT R-node:
a
YiXi+1 -
Type
TT R-node:
a
yix? ' -
of
=
OOL1 16)
For the end of Yi
=
(17.149)
bifurcating
a
arc:
O(A 1/3).
(17.150)
0. The asympStep 5. The right-hand members do tend to zero for p them obtained to are zero. by equating equations Step 6. We solve the asymptotic equations. For a T-arc, the system (17.145) becomes identical with the system (17.93). It has three solutions; by continuity with the previous case v < 1/3, we again choose the solution
totic
Zi
0
Xi'
,
=
Xi'
1
(17.151)
.
For the R-nodes, each of the equations (17.147) to (17.149) has tions; by continuity we choose in all cases the solution
two solu-
(17.152)
Yj =0
For
a
Zi+j We
R-region, =
-W
j
Yi+j
-
-
x4j
(17.153)
-=
-
+
49+3) W
Yi+j-1 +
addition, Yi
(17.146c)
.
Yi+j (X 'I+j+f
In
obtain from
then eliminate the zi+j variables and
can
X11 i+
we
=
0
-
1
01
-
,
j
=
0
E
a
A, i
E
A
(17.154)
-
R-arc there is from
(17.150)
and
(17.152) (17.155)
-
The available evidence
obtain the system
E
W(X/i+j +X11i+j )=O,
at the two ends of
Yi+h
j
we
(see below) suggests
that this system cannot be solved
explicitly in general. It will be studied in detail in Chaps. 19 and 20. Step 7. We must compute the Jacobian. We consider first the R-nodes. As in Sect. 17.4, Step 7, we find that only the partial derivative Of /Oyi is non-zero in (17.147) to (17.149), and the determinant can be simplified. JJJ is decomposed into a product of smaller determinants, each of which corresponds to a T-arc or to a R-region. It is then sufficient to verify that each of these determinants is For
a
non-zero.
T-arc, the determinant is again given by (17.110) and equals -4. R-region, we can eliminate the lines corresponding to the equations
For
a
17.6 The Case
1/3
<
<
v
1/2
171
(17.146c) and the columns corresponding to the variables zi+j; we are left with the determinant of the system (17.154), which we call R-Jacobian. It will only be possible to compute that determinant after the system has been '
solved
(see Chaps,
however it
20).
19 and
We will find that it is
vanish in isolated
can
Therefore there exists in
points
hi For
a
error
.
close to the asymp-
estimates for
some
vari'ables. We
We have for the R-nodes
0(/_,1/6).
=
non-zero;
the characteristics.
general, for IL > 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
-
<
(17.160)
'
v
T-arc
< or
1/2 R-regions, by continuity, for
V
>
1/3,
and
same sequences of intermediate arcs. We use once more the general method of Sect. 11.4.
for the
1. The results obtained for T-arcs in the two
Step for 0 <
v
<
1/3 ((17.117), (17-119), (17.121))
and for
Y
previous sections, =
1/3 ((17.157),
Quantitative Study of Type
17.
172
2
(17.158), suggest that these arcs are not affected by transition 2.1, and that v > 1/3 we continue to have u'j, u'j' O(AC1/2 ) and Aaj small For R-regions, on the other hand, the study of small values of ft in the case v 1/3 (Sects. 19.2, 20.2) shows. that the, following two properties hold for
=
=
generally:
(i) w can tend (ii) There is: Yi
=
The -
towards 0
O(w'/')
only
X'j, X'j'
only exceptions
case
case
corresponds
2. 1PI
to
the
positive side:
O(W-1/')
=
for
w -+
of order
e'AC
1) (Sect. 19.2.1),
0; therefore there
>
has been
bifurcating arc, and we have temporarily excluded (Sect. 17.2.3).
The
2. M
case
0+.
0+
(R-orbit
of order
a
1) (Sect. 20.2.1).
bifurcation 2T1, also excluded. These properties will be proved in Sect.
for
are no
coincides with the
-
w -+
(17.161)
are:
(R-arc
The
on
w
-4
0-. But this
T-arcs. The R-arc
bifurcation 2PI, which But this
corresponds
to
a
we
Going back to the physical variables change (17.161) into hi
=
0(/.Z1/2AC1/2)
The results for that in we
a
v
<
U 'U I
1/3 (see (17.122))
R-region, Aaj
is not affected
=
Step
of
(17.137)
to
O(IL1/2AC-1/2) and for
by
v
(17.141), (17-162)
1/3 (see (17.159)) suggest 2.1, and that for v > 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
<
v
<
1/2
2
VY
zi
VY
a2 VY
Z
173
change of variables:
second
a
1/3
2 y
Zi
-
vL2
VY
Y.
(17.167) with
L2
sign(a
The factor same we
signs
sign(yi)
1)
-
(17-168)
(17.166), (17.167g), (17.168), o,
=
x , x ', yj have the (17.141). In particular, (17.1):
has been introduced here
so
and
that
2
*1
in
(17.169)
.
Substituting
4.
1/2
corresponding quantities defined
the
as
have from
Step
[37rIaV] 2v
sign(a'- 1)
=
in the fundamental
equations (17.63)
to
(17.67),
we
obtain: -
For
T-arc:
a
Xi12 O(AC1/2) + O(Al/2AC-1) Z,(Xi, Xill)
Xil I
_
X1i
=
_
1
_
Z,
=
0(/11/2AC-1/2) (17.170)
=
(p 1/3). equations are identical with (17.145) if we substitute AC This shows again that the T-arcs are not affected by transition 2.1. Fora R-region: These
-
it
O(Al /2AC-1/2)
Yi+j (X Z+j+l
xi+j )
x
0 (/t-1/2 (AC3/2)
i+j
1
,
=
O(PAC-2)
Z,+j + 1
=
2(Yi+j
Yi+j-l)
7
-
-
j
zi+j(x4j
j
j
E
E
E
+
It xi+j)
=
O(Al/2AC-i/2)
j
E
(17.171) -
For
a
TT R-node:
Y,(Xi,+l -
For
a
For
a
(17.172)
('17-173)
RT R-node: =
O(A112 AC-1)
For the end of Y,
O(P1/2 AC-')
O(P 1/2 AC- 1)
y,Xi,+l -
=
TR R-node:
yixi" -
Xill)
+
=
a
bifurcating
0(pl/2AC-1/2)
(171 .174) are:
(17.175)
174
Step to
Quantitative Study of Type
17.
5.
Examining the right-hand members,
for p
zero
1/3
<
v
2
<
we
they do
find that
tend
0 if
-+
1/2.
The asymptotic
(17.176)
are obtained by equating them to zero. asymptotic equations. For a T-arc, the system is once more identical with the previous cases (17.93) and (17.145), and by continuity with the previous -cases, we again choose the solution
Zi
equations
6. We solve the
Step
X!
0,
=
=
XY
(17.177)
1.
=
Similarly, for the R-nodes, each of the equations (17.172) to (17.174) solutions; by continuity we choose in all cases the solution
also
has two Yj
For
(17.178)
0.
=
R-region, (17.171c) gives
a
(17.179)
Zi+j
(17.171b) gives X '
=
X
(17.180)
We call xi this
Yi+j (xi+j+l
Yi+j In
-
=
Xj+j)
-
Yi+j-1 + Xi+j
addition, Yi
-
(17.171a)
value.
common
1
=
=
0
0
Yi+fi
=
0
j
,
at the two ends of
j
,
a
E
and
(17.171d)
become
E
.4
(17.181)
R-arc there is from
(17.175)
and
(17.178) (17.182)
-
The available evidence
(see below) suggests
that this system cannot be solved
explicitly in general. It will be studied in detail in Chap. 18. Step 7. We must compute the Jacobian. As in the previous cases (Step 7 in Sect 17.5), we find that the R-nodes can be eliminated; I JI is decomposed into -a product of smaller determinants, each of which corresponds to a Tarc or to a R-region; the determinant corresponding to a T-arc is non-zero. For a R-region, we can eliminate the lines corresponding to the equations (M171c) and the columns corresponding to the variables zi+j; we can also eliminate the lines corresponding to the equations (17.171b) and replace the two columns for x'i+j and x'i'+j by a single column for xj+j. We are left with the determinant of the system (17.181). It will only be possible to compute that determinant after the system has been solved. We will find that it vanishes (Sect. 18.1.3).
Therefore in the interval to the
asymptotic solution,
O(A-1/2AC3/2, P,
1/2
(17.176),
for
the distance
AC-1)
.
>
being
0, there
exists
a
never
solution close
of the order of
(17.183)
17.6 The Case
The first of these- two
error
8 and 9. We terms.
some error
hi For
can now
(17.172)
<
v
<
1/2
to
v
v approaches the value approaches the value 1/2. For
go back to the
(17.174)
physical variables,
show that
we
and refine
have for all R-nodes
O(IjAC-1/2).
=
175
terms becomes 0 (1) when
1/3, while the second becomes 0(1) when 1/3 < v < 1/2, both error terms are o(l). Steps
1/3
(17.184)
T-arc, (17.66) gives then
a
Aaj
(17.65)
=
O(ttAC-1)
(17.185)
reduces to
VYAC +
U 2
O(tAC-1)
=
(17-186)
from which.
U
Aqj + O(IIAC-2)].
VY
(17-187)
(17.64) gives u '
u
O(IIAC-1)
=
(17.188)
and
V/-.vY'AC AC[j + 0(/-JAC -2)].
Ui
For
a
R-region,
a2 VY
Aai+j
have from
we
VY
Since the system
AC[l
+
(17.181)
(17.180)
(17.171c)
O(tAC-2)]. is not
(17-190)
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 by
u +pu"j i+3
=
-L2Xi+3 t11/2(_61AC)-1/2
[1 2
hj+j
vL2
+
0(,_1-1/2AC3/2,M1/2AC-1)]
j
E
j
E
y,+j/_11/2 (-C:,Ac) 1/2 [1
+
00171/2AC3/2' IL 1/2w-l)i
(17.191)
(17.187), (17-189),,(17.190) is the same as in the pre1/3 (see (17.119), (17.121), (17.122)) and v 1/3 ((17.158),
The main term in vious cases
v
-,:
=
(17-159)). Finally, we can compute exactly the dominant term of hi in help of'(17.63), (17.187), (17.189), (17,191). We obtain
R-node
a
with the -
for
a
TT R-node:
hi V
with
a
+
It(-VYAC)-1/2[j + O(AC1/2'PAC-2
sign
for
a
TIP R-node and
a
-
sign
for
(17.192) a
T'Te R-node.
176
For
-
Quantitative Study
17.
a
TR
or
of
Type
2
RT R-node:
21z(_VYAC) -1/2[l + O(ACI/2' IL 1/2AC-1 A
hi
(17-193)
V
with
+
a
sign for
a
TIR
or
RT' R-node, and
a
-
sign for
TIR
a
or
RTI
R-node.
Substituting
into
(17.66),
obtain also the dominant term of
we
Aai for
a
T-arc:
a(g'
Aai.-
+
g")sign(vy) ,AC-1 [I
+
37rIVY
O(AC1/2' /_11/2AC-1)]
(17-194)
where if the previous if the previous
are
is
1
are
is
0
if the previous
arc
2 9
9
2
if the
1
if the
0
if the
following following following
a.R-arc, a T-arc, does not exist;
(17-195)
arc
is
a
arc
is
a
arc
does not exist.
These equations are identical with replaced by "R-arc" everywhere.
R-arc, T-arc,
(17-196)
(17.125), (17.126), (17.128),
with "S-arc"
Fig. 17.2 shows in log-log plots the variations in order of magnitude of the quantities hi, Aai, u and u ' as functions of AC, given by (17.184) to (17.191) in the interval p1/2 < AC < tZl/3 These curves join smoothly at AC 111/3 =
.
with those found before in Sect. 17.4 for the interval /-z 1/3 < AC < 1. We remark that this figure is more complex than the corresponding
Fig.
12.3 for type 1.
17.7 The Case We
study
to AC
=
=
1/2, Aai
A
what
=
1/2:
Transition 2.2
happens when
we
reach the value
v
=
1/2, corresponding
E)(/,11/2). 1. The results of the
Step v
now
v
previous
the orders of magnitude =
O(ILI/2)
U 'U ' 71
general fusion takes place:
z
are
=
section show that when
in all
cases
0 (IL 1/4)
all T-arcs and
we
arrive at
(R-nodes, T-arcs, R-regions): hi
=
0(/,13/4).
R-regions fuse
into
a
(17-197) single region.
This is also apparent on Fig. 17.2. We will call this transition 2.2 (type 2, second transition). This transition affects the totality of the bifurcation orbit or
bifurcating
T
arcs.
We
arc, at the difference of transition 2.1 which did not affect the
distinguish a total transition 2.2, corresponding to a total bifurcaa partial transition 2.2, corresponding to a partial bifurcation. Here
tion, and
17.7 The Case
=
1/2:
Transition 2.2
A1/2A
1/3
1/2
1/3
113
hi
A
R
S
S
177
AC
AC
AC
1/2A
v
1/3
IT
T
1/4
S
1/3
R
R
1/2 2/3
R
3/4
SS
TT, TR
N
Aaj
T
5/6
U41
I
TT, TS
IL
Fig. 17.2. Variations of hi, Aaj, u and 0 as functions of AC. Both scales axe logarithmic. Dots correspond to transitions. In the left panel, TT, TS, SS, TR represent values of hi at a node or a R-node, while S and R represent values at an antinode or a R-antinode, inside a S-arc or R-region respectively. z
there is
correspondence between total/partial bifurcation and the situation is simpler than in the case of transition
one-to-one
a
%
total/partial transition; 2.1 (Sect. 17.5). 2. We make
Step Aai
=
p 1/2Z* i
U
/.t3/4y*i
hi
Xj'*
=
=
ttl/2W*
Xj"*
_a L3 zi
so as
U zl
to obtain
=
quantities 0(l):
/11/4 xi,
I 1*
(17.198)
to make
-L3xi
2
2
=
P 114XI* i
convenient
-L3XI,
variables
=
2
C
3. It will be
Step
Zi
changes of
a
second
Y* i
,
change of 2
vL3
variables:
Yi
L2
3W,
W*
(17.199)
VY
VY
with
L3
sign(a
The factor
1)
-
sign(a
-
[37rIal
1)
2v
1/4
has been introduced
(17.200) so
that
Xj', Xj", Yj
have the
same
x , x ', yj defined in (17.141) and also x , xY, yj defined in (17.167). signs In particular, we have from (17.198d), (17.199c), (17.200), and (17.1): as
sign(Yi) Step obtain
4.
=
uz
(17.201)
.
Substituting
in the fundamental
equations (17.63)
to
(17-67),
we
Quantitative Study
17.
178
of
2
Type
i E C 0(/11/4), X! i E A, X!, 0(/1-1/4), iEA, Xi12 Zi_ W O(t,1/4)' 2(Yi Yi-1) Zi(Xi'+ Xj") O(p 1/4),
XiI)
y,(Xi,+l
1
=
=
_
Z
11
=
_
-
In
_
addition, for the ends of
a
bifurcating
(17.202)
i E A
=
-
arc,
0(/11/4).
y0, y
(17.203) members tend to
for p
0. The
asymptotic right-hand Step equations are obtained by equating them to zero. Step 6. We solve the asymptotic equations. (17.202b) gives 5. The
Xi'
(17.204)
Xi'
=
We call Xi this
common
Yi(Xi+i Xi) 1 Xi2 Zi_W=O, -
=
-
value. The system reduces to
0
-
Yi-1
ZiXi
-
YO We
=
can
Y
=
0
,
iEA, i E A
0,
=
and at the two ends of
i E C
,
_
Yi
zero
a
bifurcating
(17.205)
,
arc,
(17.206)
-
eliminate Zi between
Y4i-Y4,_,_X(X i i
2
(17.205b)
_W)=O,
and
(17.205c), obtaining (17.207)
i E A
However, the available evidence (see below) suggests that the system formed
by (17.205a)
and
(17.207)
studied in detail in
Chaps.
cannot be solved
explicitly
in
general.
It will be
21 and 22.
7. We must compute the Jacobian. This will only be possible after the system has been solved. We will find that it is generally non-zero; however
Step
it
can
vanish in isolated points
on
the characteristics.
general, for It > 0, a solution close to the asympsolution, being of the order of til/4. Since the 9. system (17.205) is not exactly solvable, it is not Possible Step give explicit expressions for the physical variables. If a particular solution that system is known, then the physical variables can be computed with Therefore there exists in the distance
totic to
of
(17.198)
(17.199):
and
a
Aai
2
2
L3
VY
-L3XiY 1/4[l +00_1 1/4)],
u'j, uj" hi AC
ZA112 [1 + o(MI/4)],
2
vL3
Y
L23 VY
,,3/4[l + O(ttl/4)]' W/J/2.
(17.208)
17.8 The Case
17.$ The Case The detailed
v
>
study of the
1/2
v
>
1/2
Does Not Exist
179
Does Not Exist
case v
=
1/2
which will be made in
and 22 shows that all branches
Chaps.
21
two. Thus the program
are joined by completed. At the difference of type 1, Sect. 12.6, there is no case v > 1/2, because W never vanishes (see (21.7), (22.5)). Thus, for type 2, v 1/2 is the highest of AC E) value to (Ml/2). possible v, corresponding
two
outlined in Sect. 11.4 is
=
=
1/3
18. The Case
<
In the previous chapter, three mental equations could not be
v
<
cases
1/2
have been found for which the funda-
explicitly solved:'
1/3 (transition 2.1), the equations (17.154) of a R-region. 1/3 < v < 1/2, the,equations (17.181), again for a R-region. v 1/2 (transition 2.2), the general equations (17.205).
1. For
v
2, For 3. For
=
=
study now these cases in more detail. We begin with the second case, simplest, and also because the resultsvill be needed in the first
We
which is the and third
cases.
18.1 R-Arc To simplify the notations, we renumber the encounters and the basic,arcs starting from the origin of the R-arc. The, equations are then Yi
Yi-1 + Xi
-
=
i
0,
Yi(Xi+i-Xi)-1=0, YO
=
0
This is
Yj!
,
a
=
A
=
,
i
(18-1)
0.
system of 2A + 1 equations for 2h + 1 variables. So we expect correspondence between the present variables (xi, yi)
isolated solutions. The
physical
and the
variables is
given by (17.165), (17.166), and (17.167).These
equations involve two parameters, p and AC. Thus, for a given value of of orbits, p, an isolated solution of. (18.1) generates a one-parameter family obtained
18.1.1
by
varying AC.
Properties
1 can never vanish, because of, (18.1b). h 1, Therefore each of them keeps a constant sign along a family. This is an illustration of the principle of the invariance of the side of passage
1. The
quantities
yj for i
=
(Chap. 8).
M. Hénon: LNPm 65, pp. 181 - 197, 2001 © Springer-Verlag Berlin Heidelberg 2001
-
.
.
.
,
1/3
18. The Case
182
<
<
v
1/2
2. For any solution, by applying the fundamental symmetry E of the stricted problem (Sect. 2.7), we obtain another solution: E
:
I
(yi, Xi)
-Xfi+i -i)
(Yfi-i'
-
re-
(18.2)
-
3. For any solution, there exists a symmetrical solution obtained by changing the signs of all variables yj and xi. We call this symmetry E':
V
(yi, Xi)
:
4. For h :2!
(-Yi, -Xi)
-+
(18.3)
-
eliminate the xi and obtain relations 2, successive values of yi: we can
yi-I
2yj
-
1 + yj+j +
=
0,
i
1
=
involving three
(18.4)
.
Yi
5.
(18.4)
be rewritten
can
yi(yi-i
2yj
-
+
yi'+,)
It follows that there
system of A
as a
are
+ 1 =
0,
at most
-
1
quadratic equations:
(18-5)
i
2fi-1 solutionIS.
18.1.2 Number of Solutions
We
0. We start from yo forget temporarily the last equation yf, given value of x, and we compute successively the other variables xjj, yfj, using alternately (18.1a) and (18.1b), rewritten as =
=
1 Yi
=
yi-i
-
Xi
All variables
Proposition dyi dxl
Xi+1
,
are
dxi dxl
Proof. this is
obviously
Yi
There is
(18.7)
i
' 0
true for i
dyi
dxi
1
dxl
dxl
j2 dxl
and from
(18.6a)
-
dyj+i From
variables
dxj+j
dyi =
dxl
-
dxl
(18-6)
Xi +
=
1, since (18.6a) reduces then
Assume that it is true up to i. From.
dxi+l
a
thus functions of xj.
18.1.1.
< 0
=
0 and
yi, X2) Y2)
dxj
> 0
(18.6b)
we
to yj
=
-xj.
have
(18.8)
,
<0.
(18-9)
(18.6) one can easily draw up the table of variations of the successive (Table 18.1).
This table shows that when x, takes all possible values by increasing from -oo to +oo, yj vanishes once, Y2 vanishes twice, and generally yj vanishes
2i-1 times. In particular yf, vanishes 2f'-' times. Reintroducing equation yf, 0, we find that =
now
the last
183
18.1 R-Arc
Table 18.1. Variations of the successive variables when
increases from
x,
-00
to
+00.
+00
XI
-00
Y1
+00
X2
-00
0
0
Y2 +00 -00
X3
0
Y3 + 00
Proposition
+00
-00
00
+ 00
-
0
+00
-00
-00
+00
+00
-00
00
+ 00
-
The system
18.1.2.
_CK)
(18.1)
+00
0
0
has 2F
_00
+00
-00
00
+ 00
-
+00
0
-
00
distinct solutions.
We remark that this is the maximal number found in the previous section
(Property 5). Moreover,
as can
Proposition signs of yi,
be
easily from the table,
There is
18.1.3.
...I
-seen
ing
can
symbolically
It will be convenient to write also
if A
5, the solution corresponding
=
represented by
solution
one
for
each choice
a
each solution
(17.169)
letter R for each basic
to yj >
0,7
Y2 >
0)
arc.
Y3 <
0,
signs of the R-arc,originates at v
shows that the a
S-arcs, while ai' solutions correspond
18.1.3 From is
instance,
(18.10)
S-arcs. Rom Table 17.1 and
(see
For
y4 > 0 will be
the sequence
two
S-arcs
the
by the correspond-
R+R+R-R+R.
We recall that
of
YA-1.
therefore represent sequence of signs.
We
exactly
=
to the
below Sect.
Stability
(17.3),
-1 for
an
yj
are
=
1/3
the relative sides of passage U from the fusion of consecutive
we find that o i antinode inside
=
+1 for
a
a
S-arc.
Thus, the 2f,.-'
node between
2fI-1 possible decompositions of the R-arc into
19.3.3.1).
and Jacobian
Proposition 18.1.1 we know that dyf,/dxl never vanishes, i.e. the R-arc critical. Reasoning as in Sect. 13.1.4, we find that the Jacobian never
never
vanishes.
18. The Case
184
1/3
<
1/2
<
v
18.1.4 Small Values of ii
(18.4)
The system solution xi
can
0. For ft
Y1
be solved for h =
2, the
=
1 to 4. For h
two solutions
1,
we
have the
(18-11)
v/2-
For ft
3,
we
have the two solutions
(18.12)
Y2
Y1
single
are
and the two solutions 1 Y1
-Y2
-v/3-
4, there
For h
(18.13)
=
are
the two solutions
F4
17 Y3
Y1
Y2
where either
theupper sign
or
the lower
(18.14)
sign should be taken
in both equa-
F1'2-1-3
(18.15)
tions ; the two solutions
jr13
2:2
V
YJ
Y2
3
:F
Y3
and four other solutions obtained
by changing all signs. can only be solved numerically. Table 18.2 gives the value of yj for A = 2 to 6 and for all sign sequences beginning with a'+. Thus, only solutions with yj > 0 are listed; the others are obtained by changing the signs. Values of yj are listed with 8 digits because the R-arcs are strongly unstable: errors are amplified during the computation of the successive yi. For h >
4,
it
seems
that
(18.4)
18.2 R-Orbit In that case, we have a total bifurcation, and the R-region coincides with the whole bifurcation orbit. The number of basic arcs is A = n. The
equations
Yi
Yi-1 + Xi
-
Yi(Xi+i
-
Xi)
are
=
-
1
0, =
(18.16)
0
where i is to be taken modulo
n
and takes all values from 1 to
n.
This is
system of 2n equations for 2n variables. So here again we expect isolated solutions. As explained in Sect. 18.1, each such solution generates a oneparameter family of orbits. a
18.2 R-Orbit
Table 18.2. R-arcs for A
h
sequence
=
2 to 6.
sequence
n
Y1
6
Y1
1.40472828
R+R+R+R+R+R
2
R+R
0.70710678
R+R+R+R+R-R
1.28123317
3
R+R+R
1.00000000
R+R+R+R-R+R
1.14898685
R+R-R
0.57735027
R+R+R+R-R-R
1.12992127
R+R+R-R+R+R
0.95398640 0.94868391 0.92357558
4
R+R+R+R
1.17914724
R+R+R-R
0.94010422 0.59967641 0.53185593
R+R-R+R
5
R+R+R-R+R-R R+R+R-R-R+R
R+R+R+R+R
1.30656296
R+R-R+R+R-R
0.90676764 0.61196435 0.60587822
R+R+R+R-R
1.14267956
R+R-R+R-R+R
0.59651693
R+R-R-R+R
0.95043145 0.91921106 0.60746124 0.59586158 0.54119610
R+R-R+R-R-R
0.59452140 0.54444346 0.53958978 0.51070145 0.48686874
R+R-R-R-R
0.505.26000
R+R-R-R
R+R+R-R+R R+R+R-R-R R+R-R+R+R
R+R-R+R-R
18.2.1
R+R+R-R-R-R R+R-R+R+R+R
R+R-R-R+R+R
R+R-R-R+R-R R+R-R-R-R+R
R+R-R-R-R-R
Properties
properties are essentially the same In regard to properties 4 and,5 we
The
as
yi-,
-
2yj
i
0
=
we
-
2yj
+
1,...,n
yi+,)
+ 1
=
0
i
,
=
Yi/Z
(18-17)
.
Yn- It
yl,
1,...,n
introduce homogeneous coordinates Y1,
Yi
(Sect. 18.1.1).
R-arc
Yi
a
yi(yi-i
we
=
a
now
system of n equations f6r n variables system of n quadratic equations:
as a
If
1 + yj+j +
for
have
,
This is
185
...,
can
beyewritten
(18-18)
.
Yn, Z, defined by
(18-19)
,
obtain
Yi(Yi-i This is
Yj
a
set
=
1,
-
2Yj
+
Yi+,) +Z2 =0,
i
1,...,n
of n quadratic homogeneous equations. i
=
1,...,n
Z
=
There is
a
double solution
(18.21)
0
This double solution is not valid since it
corresponds
to infinite values of the
inhomOgeneous coordinates yi. It -follows that the system 2 solutions. 2n -
(18.20)
.
(18.18)
has at most
1/3
18. The Case
186
We
proceed
ables
as
as
1/2
in Sect. 14.1.2. To compute the
Jacobian,
we
order the vari-
follows:
YOi XI; Y1 7 X27 Y27
A
=Y1
Yn-1, Xn
....
f2i-1
=---
f2i
Yi(Xi+l
Yi
=
f2n
YO(X1
a
be written
YO
-A =0
Yi-I + Xi
-
f2n-1 `
can
-YO+X1 =0
f2 =YI(X2 -X1)
=
(18.22)
-
equations (18.1)
The fundamental
For
<
v
and Jacobian
Stability
18-2.2
<
-
-
Xi)
Xn)
,
0
Yn-1 + Xn
-
0
==
1
-
0
=
0
=
given value of I W 1, this
(18.23)
-
is
a
system of 2n equations for 2n variables.
The Jacobian is
OYI,..., f2n) Nyo, X1, Y1, Xn) -
We
can
-
(18.24)
-,
(18.23) by adding
also "unroll" the system of equations
two vari-
ables Yn and Xn+1, considered as distinct from yo and xi, modifying the last two equations of (18.23) and adding two equations, so that the end of the new
system reads:
f2n-I
=
f2n
Yn(Xn+l
=
f2n+1
Yn
=
YO
f2n+2.=
X1
-
-
-
0
Yn-1 + Xn
Xn)
-
Yn
=
Xn+1
-
1
0,
0 =
0
(18.25)
-
equations (18.25), with the last two equations system of 2n equations for 2n + 2 variables. Starting from given values of yo and x, and applying the equations one by one, we obtain successively yi, X2, Xn, Yn, Xn+1 In the same way, starting from variations dyO and dxj, we can compute successively dyl, dX2, dYn, We consider the system of
deleted. We have then
a
-
-
-
-,
...,
dxn+l.
For the whole
dYn dXn+1
orbit,
6Yn OYO 19Xn+l
5Y-0 The
stability index
we
have then
(9Yn OX, (9xn+l ex,
(dxl) dyO
is the trace of the matrix divided
(181.26) by
2:
187
18.2 R-Orbit
Oy. Z
Proceeding
as
(18.27)
axi
in Sect.
14.1.2,
we can
show that
of a R-orbit vanishes if 1). critical orbit of the first kind (the stability index is z
Proposition a
+
ayo
2
In
The Jacobian
18. 2.1.
and
only if
it is
=
Actually any pair of variables can be used to compute the stability index. practice it will be convenient to use yo and yj rather than yo and xj. We
can
then
(18.4)
use
dYij dyi+
to obtain
0
1
-1
Ui
dyi-1 dyi
(18.28)
with 1
=2+
ui
(18.29)
2
y, We
then compute the matrix
can
aYn
aYn eyo (9Yn+1 19YO and the
It
-
0
1
0
1
0
1
-1
Un
-1
Un-1
-1
U,
)(18-30)
-
TY ,
stability index is 1
z
Oyj (9Yn+l
=
2 can
(19Yn O yO
'
(18.31)
O yj
be shown that
Proposition Proof:
+
we
18.2.2. R-orbits
will
always unstable.
that, from (18.29):
the fact
use
are
(18-32)
ui > 2.
We write
dyi-1 dyi
bi di
ai
Ci
dy-1 dyo
(18.33)
There is ai
b
ci
di
(0 -
1
1
ai-1
ui-1
ci-1
bi-, di-1
(18.34)
or
bi
di-1
ai
=
ci-1
ci
=
-ai-1 + ci-jui-i
,
We show first that
=
,
,
di
=
-bi-1
+
di-jui-I
.
(18.35)
1/3
18. The Case
188
ci +
di
> 1
ci +
,
This is true for i
di
=
>
2(ci-i
di
(ai
-
+
di-1)
+
a,
+ =
bi)
di-1)
-
(ai-1
-
0, b,
(ai-1 +
(18-36)
> 0
1. There is ui-1 >
-
(ci-1
ui-1
1/2
<
v
1, because
=
that it is true for i ci +
<
1, cl -1, di 2, therefore =
=
uo > 2.
Assume
bi-,)
+
bi-,)
=
(18-37)
> 1
and ci +
+
bi)
(ci-1
+
di-1)
-
>
Next cj
(ai
di
we
< -i
=
(ui-1
-
(ai-1
-
1)(ci-i +
bi-,)
+
di-1)
(ai-1
+
bi-,)
'(18-38)
> 0
show that ai
,
This is true for i ci
=
=
-
ci
> 1
(18.39)
1. Assume that it is true for i
-ai-I + ci-jui-i < -ai-1 +
2ci-1
=
1. There is
-
(ci-1
-
ai-1)
+ ci-1
<
(18.40)
and ai
-
ci
=
ai-1)
-
ci-1
(ui-1
Finally, adding (18.36a) ai
+di
Taking
i
-
and
1)
>
ai-1)
(18.39b),
we
-
ci-1
>
(18.41)
obtain
(18.42)
> 2.
n,
=
we
have > 1
Z
2
(18.43)
.
It follows that the Jacobian
18.2.3 Small Values of
never
vanishes.
n
We solve the system of n equations (18.17). For n = 1, there are no solutions. 18.2.3.1
2. There
n
are
two solutions. The first is
I YO
Y1
=
2
(18.44)
2
and the second is obtained 18.2.3.2
3. After
n
a
by
a
little
shift of the origin of
algebra
we
one
basic
arc.
obtain the 2 solutions
2 YO
=
T6
Y1
=
Y2
=
and 4 other solutions obtained
(18.45)
T-T6 by
shifts of the
origin.
This
gives
a
total of 6
solutions. We remark that for
number 2n
-
2.
1
n
=
1 to
3, the number of solutions equals the maximal
189
18.2 R-Orbit
18.2.3.3 YO
=
n
4. After
=
Y1
-
some more
Y2
v _2
computation,
Y3
find 1'solution
we
(18.46)
,r2-
the '2 solutions + YO
Y3=17
Y1
2
1
-
Y2
-03
(18.47)
2
by shifts of the origin. This gives a total of Adding the two spurious solutions with sub-period 2, we have 2. total of 14, which equals the maximal number 24 Numerical values for the solutions (18.44), (18.45), (18.46), (18.47) are
and 9 other solutions obtained 12 solutions. a
-
listed in Table 18.3.
Table 18.3. R-orbits for
n
2 to 6.
=
YO
Y1
2
0.500000
-0-500000
0.408248
-0-816497 0.816497
-0.816497
-0.408248 0.707107
0.707107
-0.707107
-0.707107
1.366025
1.000000
-0.366025
-1.366025
-1.000000
0.366025
1.000000 -1.000000
3
4
5
6
Y5
Y4
Y3
Y2
n
0.81649.7
1.712938
1.712938
1.129146
-0-340271
1.129146
-1-712938
-1-712938
-1.129146
0.340271
-1.129146
-0.450837
0.799673
0.799673
-0.450837
0.516750
-0.799673
-0.799673
0.450837
-0.516750
0.450837
0.652966
0.652966
-0.878507
-1.271686
-0.878507
-0.652966
-0.652966
0.878507
1.271686
0.878507
1.224745
0.816497,
-0.816497
-1.224745
'-0.816497
1.345162
0.973460
-0.425507
0.525667
-0.425507
-1.345162
-0.973460
0.425507 1.228446
-0.973460
1.965259
0.425507 1.228446
-0.525667
2.193233 -2-193233
-1.965259
-1.228446
-1.228446
-1-965259
1.618034
1.618034
1.000000
-0.322404 0.322404 -0-618034
-1.618034
-1.618034
-1.000000
0.618034
0.816497 0.973460 1.965259 1.000000
-0.618034
0.618034-
-1.000000
-0.720707
0.720707
0.774597
-0.462509
0.462509
-0-774597
0.720707
-0.720707
-0-774597
0.462509
-0.462509
0.774597
18.2.3.4
n
=
5. A frontal attack
on
the system
(18.17)
seems
difficult. We
remember, however, that a R-orbit originates at v = 1/3 from the fusion of consecutive S-arcs. Using the notation of Sect. 6.2.1.3, in which each S-arc is
represented by
its
length
5,41, 32,311,221,
we
m,
2111
,
find that there
are
30
possible combinations:
(18.48)
18. The Case
190
1/3
<
v
<
1/2
and 24 other combinations deduced
by shifts of the origin. We remark then correspond to symmetric orbits. This suggests
that all of these combinations
that
look for symmetric solutions of the system
we
(18.17).
One of the crossings of the symmetry axis lies in the midpoint of arc. We choose the origin in such a way that this is the first basic arc take h Y2
=
1,
=
Y4. After
5y61
elimination,
20Y41
-
7.3.2). There following equation
in the notation introduced in Sect.
+ 17
Y21
-
4
we
0
=
obtain the
a
basic
(i.e.
is then yo for y1i
=
we
yi,
(18.49)
,
which has 6 real
solutions, listed in Table 18.3. 24 other solutions are obtained by origin., We have thus a total of 30 symmetric solutions. This the maximal number of solutions 25 equals 2; therefore we have in fact shifts of the
-
found all solutions. 6. We follow the same strategy as for n 18.2.3.5 n 5. The possible decompositions into S-arcs fall into three groups with respect to the symmetry E: (i) symmetric orbits with the two crossings of the symmetry axis in =
=
collisions:
(18-50)
6,42,411,2211,21111
(ii) Symmetric 51,3111
orbits with the
crossings
in
midpoints:
(18.51)
;
(iii) asymmetric
orbits:
(18.52)
321,312. We consider each
case
(i) Symmetry (h 0). Then yj
axis
=
Yo
1
=
in turn.
through
Y,5; Y2
=
collisions. We take the
Y4. After
Y'=Y5=
42
some
Y2
algebra,
=
origin
we
Vj2
Y4
in
one
crossing
obtain the solution
Y3
43 (18-53)
and the 4 solutions
(V7- V/7 4V2W2 [( 'F W2) N/7 (N5 W2) V 7 4V 2w2 [(v2--W2)v7-(*/2+W2)v7, 4NF2W2] V2- (v -7--V7-4V2W2)
wl
yo
=
Y1
=
Y5
=
Y2
=
Y4
=
2
+
r2i
W1
-
2
-
-
+
+
-
W
wl
Y3=
where w,
2
=
117
W2
=
1, independently of each other.
(18.54)
191
18.2 R-Orbit
(ii) Symmetry
through midpoints. We
axis
point of the first basic
V5_ + Yo
=
Yi
(h
are
1).
=
take
one
crossing 1
1 Y2
2
=
in the mid-
We find the 2 solutions
Y3
Y5
=
-
Y4
V5
2
(18.55) (iii) Asymmetric
orbits. It turns out that
explicit solutions
can
also be
+--+-+.
The sequence of signs for the asymmetric orbit 321 is: This sequence is nos symmetric under E. However, it is it is invariant
under
change of the signs
obtained in this
a
followed by form Yo
=
a
case.
(as
of the yi
allowed
E symmetry. This suggests that
_Y1
Y2
=
-Y5
by Property 3, Sect. 18.1.1), we
look for solutions of the
(18.56)
-Y4
Y3
We find indeed 2 such solutions:
:F l (47+ V-15 )
e
1
yo
=
Y3
=
-Y1
=
-Y4
=
2 1
T
The 9 solutions With the shifted two
2
(Vi7
1 _
v/1-5
)
)Y2
=
-Y5
5
(18.57)
.
(18.53), (18.54), (18-55), (18'.57) solutions,
we
have
a
are
listed in
total of 54 solutions. If
Table we
18.3.
add the
spurious solutions with sub-period 2 and the 6 spurious solutions with 6 we have a total of 62, which equals the maximal number 2 -2.
sub-poriod 3,
Therefore all solutions have been found. n = 7, there appears for the first time sequences of of devoid signs any symmetry, even when Property 3 is used. One example is the sequence +++-+--. It seems that solutions for n > 6 can only be obtained
18.2.3.6'n > 6. For
numerically. For instance the solution corresponding to the above sequence was computed as: yo 0.956199, Y3 0.880142094, yj 1.302150709, Y2 -0-678047. -0.435560, Y4 -0.761411, Y6 0.468576, Y5 =
==
=
=
=
=
1&.2.4 Sign Sequences
example of the R-arcs (see Proposition 18.1.3) suggests that we look signs of the yi. A detailed examination of Table 18.3, completed by the addition of the shifted solutions, reveals indeed a remarkable property: there is exactly one solution for each choice of the signs of yo, and for y,,-,, with the exception of the two sign sequences +++. which there are no solutions. This property holds true up to n 6, and it is tempting to conjecture that it is true for all n. This conjecture will indeed The
at the sequence of the
.
.
.
---.
.
=
be proven in Sect. 18.3.
.
.
.
.
,
192
18. The Case
We
ing
can
1/3
<
v
<
1/2
therefore represent symbolically each solution by the correspondsigns. As in the case of R-arcs (Sect. 18.1.2), we write also
sequence of
letter R for each basic
arc. As was done earlier (Sect. 8.5), for the sake of repeat the first sign at the end of the sequence. For instance, the solution quoted in the previous Sect. 18.2.3.6 for n 7 is represented by a
symmetry
we
=
the sequence -R+R+R+R-R+R-R-.
Study
18.3
(18.58)
of the
Mapping
We will show
now that the equations (18.1) or (18.16) of a R region have remarkable property- they are equivalent in a sense to the classical baker transformation. This equivalence accounts for the properties observed in the
a
previous sections.
Eliminating
the xi from
(18.1)
(18.16),
or
we
obtain
a
relation b,etween
three consecutive values of the yi: yi-1
-
2yj
0
+ yj+j +
(18-59)
.
Yi
Consider that
ai we
a
plane. yj
=
plane with coordinates (yi, yi+,). (18.59) defines a mapping This becomes obvious if we rename the coordinates )3i
,
=
F in
(18-60)
yj+j
have then F
Oi
ai+i
This is
a
Oj+j
,
=
-ai +
2#j
(18.611)
-
one-to-one, area-preserving mapping; its inverse
can
be written
explicitly: F-1
:
ai
=
2ai+l
-
1
Oj+j
,
aj+j
Oi
=
aj+j
(18.62)
.
Consider a point Po (yo, yi)- of the plane. By applying the mapping F repeatedly, we obtain a sequence of points P, Pi (Y1) Y2)7 (Yi, Yi+1), Equivalently, we may consider that we generate a sequence of values =
=
...
-
....
Y1) Y2
Two
cases are
possible:
1. All yj in the sequence sequence
(18.63)
)Yii---
(18-63)
are
different
from
zero.
We obtain
an
infinite
(18-63).
The results of the previous sections suggest that the sequence of the signs of the yj plays a fundamental role, and acts as a "signature" of the orbit or arc. We associate therefore to the sequence (18-63) an infinite sequence of
binary digits
18.3
wi
4017 W2
0
if Yj <
1
if Yi > 0.
0,
(18. 65)
(18.66)
;
*
2
2. We reach
wi is related to the relative side of pas-
+ 1
I
=
value yi
a
193
by
Incidentally, (17.169) shows that sage ai by Wi
-
(18.64)
...
where wi is defined Wi
Study of the Mapping
=
0 for some'i > 0. Then
(18.59)
becomes undefined
and the sequence (18.63) cannot be continued beyond yi. (As an extreme case, this may happen'for i == 1, i.e. the initially given value yi -equals zero.
The sequence (18.63) is then limited to its first term.) The rule fails to define wj for j 2! i. In that case, we still. define an infinite
(18.65)
binary digits (18.64), by writing
sequence of Wi
=
1
wj=O
for
j>i.
(18.67)
for
j>i;
(18-68)
We could also write Wi
I
as
=
0;
wj=1
will be shown
in the limit of Yi+1
-
a
these two definitions
below,
These definitions
can
are equivalent. intuitively justified by considering what happens small but non-zero yi. Then (18.59) gives
be
very
-2yi
Yi+2
-Yi
1
-jyi
yi+j
(18-69), If yi > 0, applying the rule (18.65), we obtain the sequence Similarly, if yi < 0, we obtain the sequence (18.68). Thus in all
cases we
PO. We consider
point) V
and is
of =
a
O-W W2
the
now
number
v.
Wi
...
associate wi as
In other
(18.64) to the point successive.binary digits (after the binary
an
the
words,
infinite sequence v
is written in
binary
as
(18.70)
...
formally defined
(18.67).
as
00
2-'wi
V
(18.71)
.
We remark that in the
same
V
=
value for
O.W1
...
case
2
above, both
sequences
(18.67)
and
(18.68) give
v:
Wi-10111
...
=
O.W,
...
Wi-1 1000
2-jwj
...
+
2-'
.
j=1
(18.72)
1/3
18. The Case
194
Thus,
is
v
v(yo, yi). 0 < We
<
v
<
1/2
uniquely defined for
any
(18.70)
Rom the definition
point PO, i.e.
it is
(18.71),we
have
or
study
(18.73) this function
by keeping
yo constant and
varying yi. The table of back,to the variables xi,
(Table 18.4) is easily established by going and using (18-6). (This is a generalization of Table 18.1.)
from yj
definite function
<
V
variations yj
a
When yj increases to +oo, x, decreases from +00 to -00; X2 becomes infinite for
-oo
0, and decreases from
=
+oo to
in each of the intervals
-oo
-oo
< yj <
0,
0 < Y1 < +00; Y2 increases from -oo to +oo in each of these two intervals; X3 decreases 4 times from +oo to -oo; and so on. The last line but one shows the beginning of the sequence WlW2W3 in the successive intervals; the last ...
line shows values of
v
when
one
yj vanishes.
Table 18.4. Variations of the successive variables when yo is constant and yj increases from
YJ
-00
xi
+00
X2
+00
Y2
-00
X3
+00
Y3
-
-oo
to +oo.
0
+00 -00
+oO
-00
0 + 00
00
0
00
+ Cx)
+ 00
-
00
-
0
oo
+00
-00
0
-00
+ 00
-
+00
-00
0
00
+00
+ 00
-
-00
0
00
+ oo
X4
600... V
]
001...
8
010...
I
Oil...
1
1
3
1
4
8
2
100...
101.-..
110...
8
4
8
Fig. 18.1 shows V as a function of yi, for the arbitrarily chosen value 1; other values of yo give similar curves. This figure, and Table 18.4, strongly suggest that v is a continuous and increasing function of yi, taking yo
=
all values in the interval 0 <
V
< I
(18-74)
-
v(yi) is a highly irregular function, with a "devil's staircase" appearance. (The steps which seem horizontal on the figure are actually slightly inclined.) Similarly, starting from the point Po (yo, yi) and applying. F- 1 repeatedly, we obtain a sequence of points P-1 P-i (y-,, yo), (y-i,, y_j+j), =
=
....
Equivalently, YO) Y-1) Y-2)
we
...
I
may consider that
Y-i7---
we
.
generate
.
a
.
,
=
sequence of values
(18.75)
18.3
Study
of the
Mapping
195
V
0.75
0.5
0.25
0
Fig.
18.1.
function of yi, for yo
v as a
2
0
1
Y1
0.
Again two cases are possible: the sequence (18.75) may be infinite, or finite 0. (As an extreme case, this may happen for i 0, ending in some y-i i.e. the initially given value yo equals zero.) Proceeding as above, we define and
=
in all
cases an
COO)
W- I
i
infinite sequence
W-2
We consider
number
=
W-il
... )
now
...
(18.76)
1
the successive terms of
(18.76)
as
the
binary digits of
a
u: 00
U
O.WOW-1
=
...
W-i
1: 2-'-lw-i
...
(18.77)
.
i=O
uniquely defined for any point P0. We can study the function u(yo, yj) by keeping yj constant and varying yo. We obtain a table similar to Table 18.4, and figures similar to Fig. 18.1, which strongly suggest that u is a continuous and increasing function of yo, taking all values in the interval u
is
0 <
U
< 1
(18.78)
.
We have thus defined 0 <
U
p
< :
1, 0
<
(yo, yj)
V
-+
a
map W from
the plane (yo, yi)
to the square
< 1:
(p, q)
(18.79)
Fig. 18.2 illustrates this map by showing some lines with constant u or v (yo, yi) plane. These lines are easily computed. v 1/2 corresponds to
in the
=
18. The Case
196
1/3
<
v
<
1/2
the horizontal axis y, = 0. v = 1/4 and v = 3/4 correspond to Y2 = 0 (and y, < 0, y, > 0 respectively); therefore the corresponding lines are obtained
by applying the mapping F-1 to the horizontal axis. Similarly, v 1/8, 3/8, the lines obtained and 0 to are corresponding by correspond 5/8, 7/8 Y3 In F` the axis. the horizontal to same u corresponds applying 1/2 way, Const. are obtained by to the vertical axis yo 0; and the other lines u applying F or F2 to the horizontal axis. =
=
=
=
CO
CN
N
3
40
00
IV-.
t)
YJ 2
v=1/2
0
-3
-3
-2
1
1
0
3
2
YO Fig.
18.2. The map W:
curves u
Const. and
v
Const. in the
(yo, yi) plane.
figure, together with the previous results (Table 18.4 and Fig. 18.1) suggests the following conjecture: the map V is continuous and one-to-one (it is a homeomorphism). This conjecture was indeed proved by Devaney (1981). (Devaney uses a slightly different f6rm of the mapping, with xi and yi rather This
than yi and yi+l as coordinates.) As a result, we can consider the passage from (yo, yl.) to (u, v) through the map V as a simple change of coordinates, and instead of studying the original
18.3
mapping F mapping B
B p
in the in the
pF
=
But the
Study
of the
Mapping
197
(yo, yi) coordinates, we can consider the corresponding (u, v) coordinates. B is formally defined by (18,80)
.
mapping B corresponds
to
a
simple Bernoulli Shift
on
doubly
the
infinite sequence ...
so
W-1WO-W1W2
that B is
(18.81)
...
simply
the well-known baker transformation
Ui
Ui+1
2 ui +
Ui+1
2vi
Vi+J
=
2
We have thus
1
if
2vi
=
-
Vi <
2
if
1
Vi >
(Devaney, 1981)
Proposition 18.3.1. transformation.
The
mapping F
topologically conjugate
is
All observed properties of the R-arcs and R-orbits corresponds to yo = 0, yi 54 0'for i
derived. A R-arc Therefore the u
=
(18-82)
2
binary representations
0.1000...
V
=
O.W,
...
of
and
u
Wfi-11000
to the baker
can now
be
1,
simply yf,
=
0.
v are
(18.83)
...
Conversely, values of u and v of that form, with arbitrary values W1, correspond to a R-arc of order A. We obtain thus 2f1-1 distinct R-arcs. Each R-arc corresponds to one sequence of signs. (This was already proved in Sect. 18.1.2 by a different method.) In the (u, v) plane, the representative wij-1,
points lie bn u 1/2 and have ordinates v 1/01, 3/2F1, 5/2f',... (2i!- 1)/2f'. For a R-orbit, the sequence of the wi must be periodic. Conversely, any periodic sequence with a period n corresponds to a R-orbit, with two excep=
=
tions: the sequences
correspond
...
000
...
0 and
and
...
111
.
.
..
This is because these sequences
respectively, and these end values are excluded. As shown by Table 18.4, they correspond to infinite values of all yi. Another way to show this is- to sum (18-59) over all i, obtaining to
u
=
v
=
u
=
v
=
1
n-1
E Yi
-
=
0,
(18.84)
i=O
which shows that the yi cannot all have the same sign. We obtain thus 2n distinct R-orbits. Each R-orbit corresponds to one sequence of signs.
Finally, deviation is R-orbits
all orbits in the baker transformation
multiplied by 2 at are always unstable.
each iteration. This
-
2
highly unstable: any explains why R-arcs and
are
19. Partial Transition 2.1
equations obtained for transition 2.1 in Sect. 17.5. bifurcating arc is then formed of T-arcs and R-regions. The values of the variables for a T-arc are given explicitly by (17.157) and (17.158). For a R-region, on the other hand, we have only the implicit system We
study
in detail the
now
A bifurcation orbit
or a
equations (17.154), which we consider now. We consider first, in the present chapter, the case of a partial transition, and a particular R-arc. To simplify the notations, we renumber the encounters and the basic arcs starting from the origin of the R-arc. The equations are of
then
-XIi Z 71 Yi
-
+W
=0,
W(X Z)
Yi-i +
yi(x Z+1
-
2
YO =0
Yfj
+
?1
-
1 =
4)
=
=
2
i
0' i
0,
=
1'...'A
-
1;
(19.1)
0
These equations form
a
system of 3A + 1 equations for the M + 2 variables
to A. Therefore we expect one-parameter families w, yo to yfj, x'1 to x A, x" 1 A solutions. For of a given value of p, there is a one-to-one correspondence, given by (17.138) to (17.141), between the present variables w, yi, x , x ' and
the
physical variables AC, hi, uj', u'j/. Thus, the one-parameter families of (19.1) correspond simply to the ordinary one-parameter families
solutions of
of orbits
(see
Sect.
2.3).
Properties
19.1
1. The
equations (19.1) do
not contain any
parameter: they
are
the
same
for
all transitions 2.1, independently of the values 1, J, E' which characterize the bifurcation. This differs from type 1 where the equations contained a
parameter K
(see (13.1)).
h 1 can never vanish, because of (19.1c). keeps a constant sign along a family., This is illustration of the principle of the invariance of the side of passage
2. The
quantities
yj for i
=
Therefore each of them an
(Chap. 8).
M. Hénon: LNPm 65, pp. 199 - 224, 2001 © Springer-Verlag Berlin Heidelberg 2001
-
19. Partial Transition 2.1
200
3. Assume that
w
=
it follows that all yi
are
vanishes, and has a constant sign along a family. For any solution, by applying the fundamental symmetry E of the restricted problem (Sect. 2.7), we obtain another solution: Thus: for h e ! 2,
4.
shows then that all yi are equal. Rom (19.1d)> zero. If A 2, this contradicts the previous point.
(19.1c)
0.
F,
w never
(Yi, Xi , X D
:
(Yfi-i7
-+
-X,'h+l-i)
1
-Xn+1-il
(19.2)
-
by (17.140) and (17.141h), has the sign of -c'AC. Thus: the symbolic sign of the branches, as defined in Sect. 6.2.1.3, is the opposite of the sign of w. For A 2, we can eliminate the x and x ' by extracting them from and (19.1b) and substituting into (19.1c). We obtain relations (19.1a) three successive values of yi: involving
5. w, defined
6.
yi-1
2yi
-
Taking into given w, we
2w + yi+1
+ 2w
2
1
i
0, and for a boundary conditions yo yf, variables I h for 1 A of equations yl, system system has been solved, the x and xY can be computed
account the
have
yf,-,. After this
(19.3)
.
Yi =
a
=
-
-
.
.
.
I
by X/ 7.
(19.3)
+W
yi-i-yi =
2
2w can
be rewritten
yi(yi-i
-
2yi
as a
yi+,)
+
Yi-1
X11
,
system of A
2W2Y, + 2w
-
W
Yi
-
(19.4)
2
2w -
1
quadratic equations: i
0
=
1'...'fi
-
1
.
(19.5) It follows that for
a
given
w, there
are
at most
2f'-' solutions.
8. We define A
Rom
pi
=
Yi
(19-3) =
qi we
qi_1
=
yi+1
(19.6)
.
have then qi
=
-pi-1 +
2qi-l
2w -
+2w
2
(19-7)
qi_1
Thus, 'the problem is formally equivalent to the study of a plane mapping, which is easily shown to be area-preserving. A R-arc corresponds to an initial point on the q axis (po 0), such that after fi I iterations obtain of the mapping wea point on the p axis (qf,-, 0). Numerof the ical explorations (H6non, unpublished) mapping (19.3) exhibit the characteristic mixture of regular and chaotic orbits found in nonintegrable systems. This supports the conjecture that the system (19.1) is not explicitly solvable in general. =
-
=
19.1
Branches for
Asymptotic
19.1.1
w -+
Properties
201
oo
oo, we enter the region JACI > 111/3, or v' < 1/3. The region are known (Sect. 17.4). For a sequence of consecutive S-arcs, they are given by the equations (17.123), (17.124), (17.127) of Sect. 17.4. If we substitute these expressions into the change of variables (17.138) to (17.141) for a R-region in the case v 1/3, we obtain In the,limit
-+
w
solutions in that
=
_j(M
Y,+j Yi
X
Ma + Mb
Z+j
j)WI[l + O(W-3)]
_
2
W-1 [1
inside
O(W-3
+
at
a
a
S-arc,
SS node,
_m-2j+2W[j+0(W-3)]' 2 m
Xi+j
23
-
2
W[j
+
O(W-3)]
(19-8)
be verified that these asymptotic values satisfy the equations (19.1). Conversely, we show now that for w -+ oo, the one-parameter families of solutions of (19.1) have no other asymptotic branches than those which correspond to a decomposition of the R-arc into S-arcs, and which are described by the equations (19.8). Consider one particular branch. We assume that the yj behave asymptotically as powers of jwj: It
can
yj
-
bilwiPi
IWI
for
-+ 00
The bi
are non-zero
constants. The pi
of the
R-arc,
yfj
PO
=
Pfi
=
yo
=
0,
=
are
arbitrary
constants. The two ends
be included in this formulation
by taking
-0<).
Assume that there exists
totically
can
(19.9)
.
a
pi > 2. In
(19-3),
the five terms behave asymp-
as
1W1` 11, M" 1W1Pi+1' M", IWI,
(19.10)
Terms 4 and 5 are negligible in comparison with term 2. Therefore the latter must be balanced by terms I and/or 3, and at least one of the following
inequalities A- i
is true:
(19.11)
A
Pi+1
A
is true. It follows in
particular that the distinguish several cases. (i) pj+j > pi. Consider (19.3) with i increased by 1. The terms 1, 4, 5 are negligible in comparison with term 2, which must therefore be balanced by Assume.that the second
inequality
encounter i + 1 cannot be the end of the R-arc. We
term 3. It follows that
Pi+2
=
Pi+1
7
bi+2
=
2bi+l
So encounter i + 2 is not the end of the R-arc. we
obtain in the A+3
=
Pi+1
same
(19.12)
.
Increasing
i
again
in
(19.3),
way
bi+3
=
3bi+l
(19.13)
19. Partial Transition 2.1
202
equal and the b constantly increase. This is impossible some point we must reach the end of the R-arc, (ii) pj+j pi, bj+j 0 bi, and bj+j bi has the same sign as bi+,. Increasing
and
The p
so on.
are
because at
=
i,
we
-
obtain
Pi+2
=
Pi+1
bi+2
=
bj+j
+
(bj+i
Pi+3
=
Pi+1
bi+3
=
bj+j
+
2(bi+l
-
bi)
-
bi)
(19.14)
,
again leads to an impossibility. bi has a'sign opposite to that of pi, bj+i 54 bi, and bj+j bi+,. Then bi bj+i has the same sign as bi, and proceeding in the opposite direction (i decreasing), we obtain the same result as in the previous case. bi. We have then (iv) pj+i pi, bj+i and
on; this
so
(iii)
pj+i
=
-
-
=
=
Pi+2
=
Pi+1
bi+2
=
bj+j
Pi+3
=
Pi+1
bi+3
=
bj+i
(19.15)
so on; this is again impossible. Therefore pi < 2 for all i. Assume now that there exists a pi < -1, with 0 < i < A (i.e. excluding the two ends). In (19.3), the terms 2 and 5 are negligible in comparison with
and
term 4. Therefore the term 4 must be balanced
least
one
following inequalities
of the
impossible
But this is
I
Pi+i
A
Pi-1
since pi is
-
by
terms I
and/or 3,
(19-16)
A
never
and at
is true:
exceeds 2.
conclusion, there
In
-1!
< 2
pi
for
i
=
1'...'A
-
1
(19.17)
.
node if pi = -1, an antinode if pi > -1. We consider a part of the R-arc lying between two consecutive nodes, or between one end of the R-arc and the next node. It consists of m basic arcs and We call
contains i +
m.
m
-
11
numbers
01
i
we
=
-W
Xi+3
2
.
.
.
Yi+j
to encounters i and ,
M
-
Yi+j-i
(19.18)
2w
(19.19)
.
x +-lw 3
0!', Z+3 can
-
=
that the xi also behave asymptotically as powers of w. We call and x" 1w for jwj 00 (note that these the limits of '
be
04i
zero). (19.1b) =
Z
-1
On the other hand, for j
Yi+j-1
have,
correspond 1, j
for
obtain
O(W)
i+j
assume
0!+j
we
2w
2
X
and
-
(19. 1c)
-
Using (19.17),
We
and
Yi+j
W
X ?,+j
a
1 antinodes. The two ends
(19. 1b)
From
X
encounter
an
m
-
1
i+j
gives
(19.20)
.
dividing (19.1c) by
wyi+j and using pi > -1,
we
obtain
19.1
04j+1 Summing
)3i'+j
I
04i
-
over
=
j,
8i'+1
find
we
i
M4,
+
-
(19.22)
i
We call yi+j the limit of y,+j /W2 for I w oo be zero). Dividing (19.1c) by w 2, we obtain
^ti+j
-
7i+j-1
There is also yj we obtain ,yi+j
=
484i
=
=
203
(19.21)
0.
" __
-
Properties
+
0.
-yi+,,,
fli4j)
(note
that these numbers
can
(19.23)
-
Substituting
the values
(19.22)
and
summing,
-2jPj'+j +j2
(19-24)
Rom the condition
0
have then
we
M
(19.25)
and
'8
M =
+j
-
-
2
j
+ i
M
X"+j
,
-
2
-j
7i+j
-j(M
j)
-
We have recovered the asymptotic expressions (19.8a), (19.8c), results from (19.1). This completes the proof
.
(19.26)
(19.8d).
Fi-
nally, (19.8b)
19.1.2 Variational
Equations
for
w -+
oo
We derive here variational equations similar to those which were obtained for type 1 in Sect. 13.1.2. We call N the number of S-arcs which make up the R-arc. We
io) il that the M"'
=
=
arc
ic,
-
U1, U2,...' U ...... UN the successive S-arcs. We call iN the positions of the nodes, with io h7 SO 07 iN extends from io,_1 to ic,. The number of basic arcs in U,, is
name
ia)
=
ic,-,.
From the two initial values yo and xj, using (19.1a), (19.1b). and (19.1c) We in turn, we can compute successively the values of x", x", yl, x', 2 1 2 compute now the corresponding variational equations: we assume arbitrary ....
infinitesimal variations ations of
dZ
x1',
=
dyo and dxl and
We have
yl,
we
compute the corresponding vari-
.
I
dx
dyi
=
dyi-1
-
w(dx
+
dZ)
dx +j
=
dZ
dyi
Y? (19.27) 2
Consider the yj
=
arc
0(w 2),
U,. From (19.8a) i
=
i'_1
+
from the initial variations
(19.27),
find that
i"
have for the antinodes -
1
(19.28)
.
dyi. -1 and generally
Starting relations
we
we
dx .
and
applying
the
19. Paxtial Transition 2.1
204
dZ
dx
dyi
=
O(W-4 )dyi. 2(i
dyj,_
dx . i,-I)wdxi'._,+,
-
+
I,-,
+
il-1
+
(19.29) where
[1 + O(w-3)].
have used the abbreviation
we
In
particular
we
have the final variations
dx'i'
=
dyi,,,
=
Z"'
O(W-4 )dyi,,,-,
dx'j. _I+,[],
+
Z
2m,,,wdx'i
dyi.,
(19-30)
We compute now the initial variations of the next arc dyi,,,. Since i,,, is a node, we have from. (19.8b): yi.
U,,+,. We already
know
(19.27c)
O(w-1).
Rom
obtain
we
dyi,
dx'iz"'+1
Yj!
[]
+
dx'iz'-- + 1 J]
2m,,w,
(19-31)
-
y
i2c
The equations (19.30b) and (19.31) give the initial variations of U,+, as functions of the initial variations of U,,,. We can iterate these equations and we
obtain
dyj,,
=
(2w)'-lM2M3
...
dyO
Ma
Y11 ...Y S-1 2 .
-(2w)',rn,7n2
-(2w)'-' Tn2Tn3
dx'i
...
rn,,
Yj?l
09yi", =
0YO jqx
71.
+1
magnitude
=
OYO The variations
19.1.3
are
=
09X1
()(W3a-1)
...
Yj"'
H
T-n.,
dxl
-2
...
(19-32)
Y?
ax
E)(W3a-2
11"'+1
-
axi
strongly amplified
Asymptotic
i2'
y t"
.2
...
are
19yi,,_
E)(w 3a-3)
dxl
Tn.
Y
...
+(2w)'Ml'rn2 The orders of
...
Branches for
=
O(W 3a)
(19.33)
after each node.
w -+
0
0, we enter the region JACI < p 1/3 or v > 1/3. The region are known"(Sect. 17.6 and Chap. 18). For a R-arc, are they given by (17.191). In these equations, the xj+j and yi+j are numerical constants, given for small values of n in Sect. 18.1.4, and more generally R obtained by solving the system (18.1). We call these constants R xj+j, yi+j to avoid a confusion with the notations of the present chapter. In the limit
w
-+
solutions in that
,
19.1
If
we
(17.141)
substitute these for
a
R-region
205
Properties
expressions into the change of variables (17.138)
in the
case v
=
1/3,
to
obtain
we
R
1+j'X11i+j
Xi
Z+J[j+0(jWj3/2]'
-
V2_ w
Y,+j
=YRjV 2W(j+O(jWj3/2]. i+ (19.34)
the equations that, if the numerical values XR i+ j, YR i+j satisfy the equations (19.1). (18.1), then the asymptotic values (19.34) satisfy Conversely, we show now that for w -+ 0, the one-parameter families of solutions of (19.1) have no other asymptotic branches than those which correspond to a R-arc, and which are described by the equations (19.34). Consider one particular branch. We assume that the yj behave asymptotically It
can
be verified
powers of
as
yj
The bi
-
jwj:
bilwIPi
PO
Ph
=
IWI
for
are non-zero
of the arc, yo =
-
=
(19.35)
.
constants. The pi
yf,
=
0,
are
arbitrary
constants. The two ends
be included in this formulation
can
by taking
+00.
Assume that there exists
.
0
-+
asymptotically
a
pi <
1/2.
In
(19.3),
the five terms behave
as
1WIpi-1, 1WIpi, IWIpj+j, IWII-pi, IW12 Terms 4 and 5
are
must be balanced
inequalities
negligible in comparison with term 2. Therefore the latter by terms 1 and/or 3, and at least one of the following -
is t1rue:
(19-37)
A
Pi+1
A
Pi-1
(19.36)
particular that the distinguish several cases. i increased with Consider by 1. The terms 1, 4, 5 are (19.3) (i) pj+j < pi. which therefore be balanced by with must in term 2, comparison negligible Assume that the second
inequality
is true. It follows in
encounter i + 1 cannot be the end of the R-arc. We
term 3. It follows that
Pi+2
=
Pi+1
,
2bi+l
bi+2
(19.38)
.
So encounter i + 2 is not the end of the R-arc. we
obtain in the Pi+3
and
=`
so on.
because at
(ii) i,
we
Pi+1
same
,
Increasing
i
again
in
(19.3),
way
bi+3
=
3bi+l
(19.39)
,
are equal and the b constantly increase. This is impossible point we must reach the end of the R-arc. bi has the same sign as bi+,. Increasing pi, bj+j 54 bi, and bj+j
The p some
pj+j
=
-
obtain
A+2
=
Pi+1
A+3
=
Pi+1
,
bi+2
=
bi+3
=
bi) bj+j + 2(bi+l bi) bj+j
+
(bj+j
-
,
-
,
(19.40),
19. Partial Transition 2.1
206
and
again leads
on; this
so
to
an
impossibility.
(iii)
bi has pj+j = pi, bj+j 0 bi, and bi+l Then bi bj+j has the same sign as bi, and
bi+,.
-
-
(i decreasing),
direction
(iv)
pj+j
bj+j
pi,
=
we
=
obtain the
=
Pi+1
bi+2
=
bj+j
Pi+3
=
Pi+1
bi+3
=
bj+j
so
result
as
in the
previous
case.
bi. We have then
Pi+2
and
same
a sign opposite to that of proceeding in the opposite
(19.41)
again impossible.
on; this is
Therefore pi 2! 1/2 for all i. Assume that there exists a pi > 1/2 with 0 < i < A. Then (19-3), the terms 1, 2, 3, 5 are negligible in comparison with term 4, which is impossible. We have thus shown that pi
This
1/2
=
E)(IWII/2)
=
W -
2
2w
from which
X 'X ' Here
we
only
have
19.2.1),
Yi-1
=
(19.1b) give
and
-
Yi-1)
1
W ,
-W(Vi
2
(19.44)
Yi-1)
0&1-1/2)
=
(19.45)
and
it can happen that the order of x , x ' happens for instance in the case 2. 1P I (see generally when there exists a symmetry entailing
0() relation;
a
IWI-1/2
is smaller than
Sect.
(Yi
(19.43)
0 < i < A.
have
we
z
71
for
(19.1a)
The equations
X
(19.42)
0 < i < h.
also be written.
can
Y,
for
.
more
This
Yi.
proved
We have
the relations
(17.161),
which
had been guessed from the
examination of small values of fi.
We show
now
that
values. Assume that Yi
/IW11/2
for
w
---
a
for
h >
1,
w
can
tend toward 0
branch exists with
w
0-. There is yo = 0, yfj = 0 for 0 < i < ft. Dividing
(19.3) by JW11/2
above, we have -yj 0 w -+ 0-, we have -yi-1
which
-
can
2,yi
2 + -yi+l
h
i
=
only through positive
0-. We call -yj the limit of 0 from (19.1d). Rom (19.43)
-+
-
-
letting
(19.46)
1
7i
be written 2
741
and
^fi
=
^/i
-
(19.47)
N-1 +
have -y2 -yj > 0; hence and so on. The 7i 72 > 0, -y3 > 0; 0. form a positive increasing sequence. But this is in contradiction with 'Yfj If -yj < 0, an identical reasoning holds. We have thus. shown that for A > 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: <
w
:F -
v
0.
<
1/2
we move
studied in
19.2 Small Values of fi
19.2.2 ft
2! 2, the
For A
x'j'
simplest method of solution of the fundamental equations solving the equations (19.3) for the yi, and then computing the xi'
(19.4).
from
In the 2
Y1 +
and
2
=
consists in and
209
case
W
2
YJ
ft
W
-.-
single equation
a
for yj:
(19.57)
0
/W--4 + 4w
-W2
2
The
X:z
4w
/I
X1
-
,
2
(19-58)
-
W -
4w
_
asymptotic branches for
I
V/W--4 + 4w
2
-+4W V/W-4
-3W2
j,
=
w
x,2
X1
=
-X1i
X1
=
obtain
we
obtain the two explicit solutions
we
YJ
2,
=
-
-W
Y1
-
2
oo
w
are W
1
X2
-
W
-
X2
-
2
-W
(19.59)
-
2
and X
W,
X
I
Y,
2W2
_
_W2
X2
-
W
2
X2
W.
(19.60) The
asymptotic branches for
X1
1
1
I
X1
2 vlw-
X2
-
T
are
Y1
2 V,'w-
-
T
w
1
/1
X2
2 V,'w--
0+
w
T
-
2,V -w
(19.61)
'
represented on Fig. 19. 1. The asymptotic branches help of (19.8). We recall that the easily the is branches of the opposite of the sign of w (Sect. 19.1). symbolic sign The limit point yj does 0 not belong to the characteristic: there w 0, is w 54 0 from Property 3, Sect. 19.1. (We remark also that (19.61.) would 0.) give infinite values of xi, etc. for w The characteristics
for
w
-+,oo
are
identified with the
are
=
=
=
The notation used for the branches is the
same as
the notation introduced
in Sect. 6.2.1.3: each S-arc is
The
represented by a number equal to its m value. the sign of e'AC, i.e. the'sign of -w (Sect. 19.1).
sign of a branch is stability can be determined from (19.54):
The
dY2
-2w(w +4)::F2 W2 V/W-(W3+4). 3
-
dxl
3 only for w forbidden; see, Sect.
This vanishes w
=
0 is
=
-4,
19. 1).
i.e.
w
=
(19.62) -2 2/3
(extremum). (The
value
19. Partial Transition 2.1
210
2
Yi 0
-
-
-
-
-
-
-
-
-
-
-
-
-
--
+
-2
2
0
-2
w
2
+2
Fig.
19.1. Characteristics for 2. IP2 transitions.
19.2.3 ii
= 3
(19.1d)
From
2w
-2yj
+ Y2
Subtracting,
(Y2
(19.3)
and
+ 2w
2
yj
-
2Y2
_2w
=
we
+ 2w
2
(19.63)
Y2
Yi
have
yi) (3y, Y2
-
obtain two equations for yj and Y2:
we
2w)
+
=
(19.64)
0
and the two solutions Y2
Y1
-W
AVFW4
2
w
+ 2w
X2
-X2
-
2
V/W- 4 +2W -X3
X1 as
well
as
=
T
X1
2w
='
-X3
-=
+ 24w VF9-4 W
-3w 2 T
9W2
:F
Fq-W4 + 24w 12w
,Fq-W4 + 24w 6w
I
V/9-W4
+ 24w
12w
The characteristics
It
11
-3
-
excluded.
VF9-4 W
+ 24w
are
represented
'Fq-4 W + 24w 12w
-+24W -3 W2 ,,vF9W4
X2
=
3w 2 :F
-3w 2 T
X1
=
3w 2 +
X3
(19.65)
6
6
X2
2w
the two other solutions -3w 2
X1
4 V/W-jU--+2 W
T
W
6w
-+24W :FV/9W4
-9W2 -
12w on
Fig.
19.2. Here
(19.66)
again the origin
is
19.2 Small Values of fi
211
2
Y1 0
+ 12
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
12
+
-2
-2 +21
Fig.
+3
-3
The
-21
(solid- lines).
19.2. Characteristics for 2. 1P3 transitions
Also shown: 2. 1P2
(dotted lines).
transitions
we
2
0
stability
be determined from
can
(19.55).
For the first solution
(19.65)
obtain
dY3
4w[-(w
dxl
3
+2 )2 ::F
W(W3
This expression vanishes for W3 tremum et
_
dxl
3) vlw;4
+
2vj]
(extremum)
-2
(19.67)
.
and for W3
-8/3 (ex-
intersection).
For the second solution
dY3
=
+
_6w(3W3
This vanishes for
8)
+
W3
(19.66)
we
obtain
(19.68)
-
-8/3 (intersection).
19.2.4 h > 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
<
1/3
,
It
was
shown in
asymptotic branches than' those which
the R-arc into S-arcs.
given large value of IwI, using the condition given value of x1, we can compute the successive
a
from any
0 and
successively Y2 _- 2y,, 0: this is impossible.
oo
w --+
are no
19.3.1.1 Variations. For
vertical asymptote in the
starting from (19.1). We consider the value of x'1 which x", Y2 yi, X12 X117 2 1 to some particular branch of order n (so that the computation corresponds ends in y,, 0). We apply now a small variation dx',, and we compute the We can use the computations made corresponding variations of x1', yi, 0. We obtain in particular in Sect. 19.1.2, setting dyo yo
=
values
...
i
=
....
=
(2w)C'M1M2 Y?T. Y?'11 Y?S2
dyi.
...
dx'1
We will need the order of
dyi. dx'1
Ma
[1
O(W-3)]
+
(19.70)
...
=
()(W3a-2)
-
i
magnitude sign
and the
(y
dx'1
is used. From
-(signw)'
(19.71)
dxl. On the figures showing the char(19.27) we find that these two variables
We have referred the variations to
acteristics, however, y, are simply related by
sign
19.3 Positional Method
dy,
Therefore
dyi. dy 1
(19.72)
-2wdxl
=
have
we
=
213
E)(W&Y-3)
sign
(dyjr)
signw)"+1
(19.73)
19.3.1.2 Relative Positions of Two Branches with Initial Common
Arcs. We consider
two branches for which the
now
arcs
U,
to
U,,,
are
the
same, but the continuation is different: either the arcs U,,,+, are different in the-two branches, or the arc U,,+, does not exist in one of the two branches
(the
i,).
R-arc ends in
i,, given by (19.8) are asymptotic expressions of x , x ', yj up to i the same for the two branches; therefore these quantities differ by O(wbetween the two branches. For jwj large enough, this can be made as small The
as
=
desired. Therefore these differences will be called
dxi, etc.,, and
the above
variations can be- applied. If U,,+, exists, we have from (19.8b)
results
small
on
2 W-
Yi.
1
ma + Ma+1
[1 +IO(w-')]
(19.74)
does not exist, i.e. the R-arc ends in i,,, we have yi,,, = 0. > 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
<
<
(19.73b)
to determine the order of the values of yj for the
0, 'or w 0 and a is odd,
if,
w
>
< 0 and
is even, that order is defined
a
it is defined
by
by (19.76).
19.3.1.3 Relative Positions of Two Branches with Different First
Arcs. The method of the
branches differ to 6
=
0).
already
In that
section does not work when the two
previous
in their first
case we
compare
arc
(this
can
be taken to
directly the values of
correspond
yi. We have from
(19.8a) YJ
(This
=
-(Mi
-
1)w
2
+
is true also if mi
O(W-1) =
order of the values of yj is
1, i.e. yj corresponds to given by the order of the
(19.77) a
node.)
Therefore the
two values of mi in the
sequence
3,2,1.
(19.78)
214
19. Paxtial 'I ansition 2.1
(19.74)
19.3.1.4 Packets. From
dyi. From
=
dy,
have, for
a
0,
>
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
<
0),
there
h
=
1; for
(Sect. 19.1.3).
are
simply ordered. Consider
...,
two branches which have
different continuation: either the signs 0 for one of the branches (the R-arc ends in
yi, but
a
yj+j = Then the order of the values of yj for the two branches is order of the two signs of yj+j in the sequence
i+
Sects. 17.5
2fi-1 branches. Exactly one each choice of the signs of the yi. As shown by Ta-
(e'AC
signs for yl, different for yi'+,, or same
are
0),
solution, therefore
ble 18.1, the branches
the
>
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
<
R+R+R+R R+R+R-R
R+R R+R-R
<
R+R-R+R R+R-R-R
R
R-R+R
<
R-R+R+R R-R+R-R
R-R' R-R
n
n
Fig-
0+,
w
=
3
fi < 4
R-R-R+R R-R-R-R
n
.
(yi
increases
=
4
upwards.)
Results,
use now
the
positional method systematically
19.3-3.1 The Case
w
with
>
w -4 +oo, ordered as in determine the junctions.
for
For fi
=
1, there
are
on
junctions.
Fig.
the left side the
asymptotic branches for
right side the asymptotic branches 19.3. We apply now Proposition 19.3.1 to
and
only the
to determine the
represent schematically the characteristics
> 0. We
(w 0, yl) half-plane, 0+, ordered as in Fig. 19.5,
in the w -+
n
19.5. Order, of the branches for
19.3.3
We
2
=
<
on
the
two branches 'R'
on
the left and
'-
1'
on
the
joined. For A 2, we represent the already known junction as a solid line (Fig4 19.6). It divides the half-plane into two regions. In each of these regions, there is one branch on theleft and one branch on the right; these two branches must be joined (dashed lines in Fig. 19.6). right; these
two branches must be
=
.
R+R R R-R
Fig.
19.6. Junctions for
w
>
0,
-
-
-
-
-
-
-
-
-
-
-
-
fi < 2.
2
19. Paxtial Transition 2.1
218
For A
=
3,
again represent the already known junctions as solid lines divide the half-plane into four regions. Again in each retwo branches, which must therefore be joined (dashed lines in we
(Fig. 19.7). They gion there are Fig. 19.7).
R+R+R R+R R+R-R R R-R+R R-R R-R R
Fig.
19.7. Junctions for
There is
a
>
w
0,
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
111 11 12 1 21 3
R < 3.
strong suggestion that this regular construction will continue
and indeed it is not difficult to prove it. On the left side, it is clear that the ordering described in Sect. 19.3.2 results in exactly one new
indefinitely;
branch in each
considering the
region. On the right side, the
and the construction of the Thus -the
same
relative sides of passage, i.e. the
ordering
junctions of R-arcs
tablished for all A in the
case w
property
signs of the
can
yi
described in Sect. 19.3.1
across
proved by
.
trivially espositional method is not
the transition 2.1
> 0. In fact the
be
(see (17.143)),
are
necessary in that case; it is sufficient to consider the relative sides of passage,
i.e. essentially to apply Broucke's principle (together with the fact that the sign of AC does not change). For A 11 2, 3, we recover the results of Sect. 19.2. =
19.3.3.2 The Case
w
< 0. This
case
simple. All relative sides of principle gives no information
is less
and therefore Broucke's
negative, junctions. We represent schematically the characteristics in the (w < O,yl) halfplane, with on the left side the asymptotic branches for w -+ -oo, ordered as in Fig. 19.4, and on the right side the single,-asymptofic branch for w -+ 0-. We apply now Proposition 19.3.1 to determine the junctions. V on the left and 'R' on the For h 1, there are only the two branches right; these two branches must be joined. For A > 1, there are no right -branches; the left branches, corresponding to w -+ -oo, must be joined between themselves. For h 2, we represent the already known junction as a solid line (Fig. 19.8). There are two other branches, which must be joined. For, A 3, again we represent the already known junction as solid lines (Fig. 19.9). There are 4 new branches, which lie in the same region. They form passage on
are
the
=
=
=
19.3
Positional Method
219
R
+
+ 11
+2
Fig.
a
19.8. Junctions for
<
0,
ii < 2.
(Sect. 13.33.2). In the present case, the R-arc is identical with the arc (there are no T-arcs), and we can use Restriction 7.3.2.
trident
whole
w
bifurcating
symmetric orbits. The asymmetric orbits, but are changed into each other by the symmetry. We can reason as in Chap. 7 and conclude that the two symmetric branches must be joined, and the two asymmetric branches must be joined. Then the two branches
two other
branches,
+
+
111 and
12 and
+
21,
+
3
are
are
made of
made of
R
+ I + 12
+ + 11
+2 +21 +3
Fig.
19.9. Junctions for
w
<
0,
fi < 3.
Fig. 19.9 agrees with Fig. 19.2. 4, using the same method, we find that the 8 new branches form two tridents (Fig. 19.10). All junctions are established. Since Figs. 19.8 to 19.10 are topological in nature, they can be distorted. We notice then that they can be made simpler by "unfolding" them: we Note that For A
=
consider all branches for h 2! 2 we
in
(i.e.
all branches except '+ V and
W)
and
let the lower half of these branches slide around the bottom of the frame a
counter-clockwise direction, until
is thus transformed into
they lie on the right side. Fig. 19.10 which has a simpler appearance. (Note, Fig. 19.11,
however, that this figure is not anymore a sketch of the (w, yl) plane.) This regularity can be explained. By studying the process of construction, 1 to A, the new branches (for A) one can show that when passing from A and the old branches (for 2 to h 1) alternate regularly as follows:'2 new 2 new branches. branches, 2 old branches, 2 new branches, 5 (Fig. 19.12, left). We use this new representation for the next case, A The new branches are separated into 4 groups of 4. Two of these groups -
-
=
220
19. Partial T ansition 2.1
R
+ 1 + 13 + 121
-
-
-
-
-
-
12
+
-
-
-
1
+
+ 1111 + 112 + 11
+2 +22 +211 +21 +3
+31 +4
Fig.
19.10. Junctions for
w
---------
<
0,
ft < 4.
+ + 13
-
-
-
-
-
-
-
R +4 - 31
+ 121
+3 + 12
+ 21
+ + 1111 + 112
+
Fig.
19.11. Junctions for
w
<
-
-
-
-
-
-
-
211 +,22 + 2 -
0, ft: 4: unfolded figure.
19.4 Results for Bifurcations of
correspond
tridents; their junctions
to
established
are
Type
2
221
(dashed lines). is made of
But
1211, the two other groups contain onlv asym'metric orbits; one + 122, + 32, + 311, and the otl er consists of + 11121, + 113, + 23, + 221. Moreover, the symmetry E exchanges these two groups, and therefore no +
information is obtained from the second half of Proposition 19.3.1. Thus the positional method is unable to establish all junctions. We must resort to a numerical computation, which shows that the junctions are as indicated by new
the dotted lines For A
on
Fig,
6, the
=
19.12
new
(Fig. 19.12, right). Using
(left).
branches form 6 groups of 4 and 1 group of 8 the second half of Proposition 19.3.1, we can split
this last group into 2 groups of 4, so that we have 8 groups of 4 branches to solve. 4 of these groups form a trident and are immediately solved (dashed lines). For the 4 other groups, numerical computation must again be used,
produces the junctions shown as dotted lines. Figs. 19.11 and 19.12 exhibit some regularity. Going down, we find that the junctions involve successive groups of 4 branches, except at the top and at the bottom where only two branches are involved. This can be stated in a different way. A given branch is completely defined by specifying whether 1 to h each collision (i 1) is a node or an antinode. Then the following 6: in two branches which are, joined, only rule is found to hold up to A 1 can differ. For instance, in the junction between + the collisions 1 and A 1 changes from node to antinode. In the 1311 and + 132, only the collision h + the collision 1 goes from node to antinode + and 15 between 51, junction and
=
-
=
-
-
1 goes from antinode to node. It would be of interest to prove this rule for all A. On the other hand, even if proved, this 'rule would not be sufficient to establish all junctions.
and the collision A
-
19.4 Results for Bifurcations of
Type
2
chapter we have considered only a single R-arc. partial transition 2. 1, which may contain more than
So far in this now
to, the
19.4.1 The Case
w
We go back one R-arc.
> 0 j
A consequence of the regular structure found in Sect. 19-3.3.1 is that for w > 0, every characteristic of a R-arc runs from w = 0 to w = +00. In other
words,
the T-arcs
a
are
R-arc
simply
traverses the transition 2.1. On the other
not affected either
transition 2.1
(Sect. 11.5),
hand,
It follows
or
w > 0 corresponds to a symbolic obtain, for partial and total bifurcations
and the fact that
sign (Sect. 19.1, Property 5), respectively: -
(Sect. 17.6).
by bifurcating arc passes through the partial being affected. Recalling the definition of a partial
that the whole bifurcation orbit transition 2.1 without
transition 2.1
we
222
19. Partial 'kansition 2.1
+
+14 +131
+13 +121 +1211 +122 +12 + 111
+1112 11111 + 1111 +112
-
-
-
+
+1121 +113 + 11
Fig.
.......
.......
19.12. Junctions for
w
<
0, fi! 5 (left) and fi:! 6 (right).
19.4 Results for Bifurcations of
19.4.1. All branches with
Proposition
the transition 2.1 at
=
1/3
and go
e) higher values of v.
bols i and to
on
a
in Table 8.18 pass
through
a
sign
-
223
2
in Table 8.12 pass
higher values of
to
19.4.2. All branches with
Proposition on
v
Type
through
v.
sign and including T-arcs (sym-
-
the transition 2.1 at
v
1/3
=
and go
Nothing happens to these branches at transition 2.1; their junctions will only be determined at transition 2.2, for v 1/2 (Chaps. 21 and 22). =
19.4.2 The Case
This
< 0
w
branches, as defined in partial transition 2.1 corresponds to a partial bifurcation. We have a particular case of Fig. 17.1c where only S-arcs are present. It follows that the whole bifurcating arc forms a single R-arc, with A n. Thus, the junctions found in Sect. 19.3.3.2 are immediately applicable to the branches with a + sign in the partial bifurcation of type 2, without any case
corresponds
Sect. 6.2.1.3. As
a
to
a
symbolic sign
consequence, -there
+
are no
for the
T-arcs. The
=
change. In the
particular
case
(Sect. 19.2.1). (Sect. 19.4.1). So we
w
=
0
Proposition sition 2.1 at For A
=
themselves
v
n
>
1/3 1,
=
n
=
1, the characteristic same as
runs
for the
from
w
general
=
-00
case w
to
> 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 <4). +and 2
(sign
224
19. Paxtial T!ransition 2.1
Table 19.1. Partial
bifurcation, type 2, n branches, determined by the quantitative study.
2P2-S
2P5
=
----
S
2P6
-----
S
2 to 6:
junctions between
2P6 -----A
*
2
+
131
+
141
+
*
11
+
5
+
6
+51
+
11111
+
1221
2P3--S
+212 +
+
132
+1311
3
2P5
+
2P3--A
+
----
+21 +
12
+42 +411
+
11211
+222
+12111 +
1212
+3111
+312
+32 +311
+22
+
+2112
122
+1211
2P4---S
A
14
+41 +
+33
15
+
123
+321
+
+1112 +2111
+
4
+
121
+
1113
1121
+221 2P4
+
+213
---
+
11121
+2121
A
+113 +211 +
112
+-31 +
13
+23
+
11112
+21111 +
1122
+2211 +1131
I JA
+24
the
+
20. Total Transition 2.1
The the
of
case
is
R-region
made of Xi Yi
Xi +
-
-
R-orbit which
a
basic
n
arcs.
W
=
0
The
equations
=
0
,
(20.1)
,
where i is to be taken modulo
equations form
These
are
a
n
and takes all values from 1 to' n.
system of 3n equations for the 3n + 1 variables
As in the to x". w, yo to yn-1, x'1 to x'n , x" 1 n
ordinary
20.1 The
properties 0, all
=
yj
X Z+1 _X and
mostly
are
In the are
the
same as
in the
case
of property 3, we find Rom (20.1a) and (20.1c)
case
equal.
of
partial
a
again
correspond
we
transition 2.1
(20.1c)
from
that. if
obtain then
1
(20.2)
-
2
summing 0
bifurcations
partial
Properties
(Sect. 19.1). w
of
case
we expect one-parameter families of solutions, which one-parameter families of orbits.
(Chap. 19), to
whole bifurcation orbit,
then
,
W(x'j + X'j') Z) i 0
Yi-1 +
yi(x +,
total bifurcation, where
corresponds to a corresponds to the
total transition 2.1
a
Yi over
the whole
orbit,
n
(20.3)
=
Yi
impossible. Therefore w never vanishes, and family. In regard to properties 6 and 7 we have now
which is
along
has
constant
a
sign
a
,
2w yi-1
This is as a
a
-
2yj
+ yj+j
+ 2w
system of n equations for
system of
yi(yi-i
-
n
2yj
2
i
=
1,...,n
(20.4)
.
Yi a
variables yl,
.
.
.
,
yn. It
can
be rewritten
quadratic equations: +
yi+,)
-
2W2Y,
M. Hénon: LNPm 65, pp. 225 - 238, 2001 © Springer-Verlag Berlin Heidelberg 2001
+ 2w
=
0
i
=
1,...,n
.
(20.5)
20. Total IYansition 2.1
226
It follows that for
duce
homogeneous
Yi we
a
Yi/Z
=
w, there
given
at most 21 solutions. But if
are
coordinates Y1,
...,
we
intro-
Y, Z, defined by
(20-6)
,
obtain
Yj(Yj_j One
2Yj
-
+
Yi+,)
particular solution Yi
=
1,
i
2wZ2
2 W2yjZ + 4
-
=
0,
i
n
=
.
(20.7)
is
(20.8)
Z=O.
n
=
This solution is not'valid since it
corresponds
to infinite values of the yi. It
follows that the system (20.5) has at most 2n I solutions. In regard to property 8, the bifurcation orbit corresponds to -
period
n
Asymptotic
20.1.1
a
cycle
of
mapping (19.3).
of the
Branches
w -+ oo, the equations (19.8) are still applicable. It can be shown that the one-parameter families of solutions of (20.1) have no other asymptotic
For
branches than those which
correspond to a decomposition of the R-orbit into S-arcs; the proof is the same as in Sect. 19.1.1, with slight modifications. For w --+ 0, the equations (19.34) are still applicable. It can be shown that the one-parameter families of solutions of (20.1) have no other asymptotic branches than those which correspond to a R-arc; the proof is the same as in Sect. 19.1.3, with some modifications. It can be shown also that for n > 1, there exists
proceed
ables
as
as
0-.
w -4
and Jacobian
in Sect. 18.2.2. To compute the
Jacobian,
we
order the vari-
follows: I
YO i
branch, with
Stability
20.1.2
We
no
X 1,'X1
I
-
Y1 i X2) X2; Y2,
(20.1)
The fundamental equations
X1,
-
X1 +
W
Yn-l,xnxn-
....
0
=
f2=Yl-YO+W(Xl+ XI) 1 f3,
=
Y1
(XI2
-
f3i-2
=
4 11
f3i-1
=
Yi
f3i
Yi(Xz +j
=
X111 )
XZ
-
+
-
Z) z
I
W
Yi-1 +
be written
can
=
=
0
=
0,
,
0
W(X Z
+
0
X% I)
=
0
(20.9)
20.1
f3n-2
=
X11n
f3n-1
=
YO
f3n
YO(Xi
For
=
given
a
IIn
-
+W
J
-
X11) n
,
W(X'n
Yn-1 + -
0
=
227
Properties
+
X11)
=
n
0,
(20.10)
i =0
value of w, this is
a
system of 3n equations for 3n variables.
The Jacobian is
I il
f3n)
09(fl, a(YO, X,1 X", 1 yi,
=
-
,
-
xl
-I
n1
-
(20.11)
x1l) n
also "unroll" the system of equations (20.10) by adding two variconsidered as distinct from yo and x'j, modifying the last ables Yn and x,,+,, n of (20.10) and adding two equations, so that the end of the two We
can
equations
new
system reads:
f3n-I
=
f3n
Yn(X'n+l
=
Yn
f3n+l
=
YO
f3n+2
=
X1
Yn-1 +
-
=
X1n+1
-
W(X'n -
n
Yn
-
X11)
-
1
0,
+ Xn
=
0
,
0 =
(20.12)
0
system of equations (20.12), with the last two equations a system of 3.n equations for 3n + 2 variables. Starting
We consider the
deleted. We have then
given values of yo and xi and applying the equations one by one, we X' 4, Yn, X' +1 In the same way, starting successively x", yi., 1 dYn, from variations dyO and dxl, we can,compute successively dxl', dyl, dx 1+1. For the whole orbit, we have then
from
obtain
n ,
n
n
-
...,
n
(9Yn
dYnj) 1
dxn+ The
stability
(9X1
dyO
n
09X,+ 1
dx'
09YO
09X1
n
ax'n+l
jy-O
Proceeding
0YO
09x,+,
index is the trace of the matrix divided
( 19Yn
1 z
OYn
,
as
by
2:
(20.14)
axi in Sect.
(20.13)
14.1.2,
we can
show that
Proposition 20.1.1. In a total transition 2.1, the Jacobian vanishes if and only if the R-orbit is a critical orbit of the flrst kind (the stability index is z
=
In
1).
Actually any pair of variables can be used to compute the stability index. practice it will be convenient to use yo and yj rather than yo and xi. We
can
then
use
dyi dyi+
(19.3)
to obtain
0 -1
Ui
dyi-1 dyi
(20.15)
228
20. Total Transition 2.1
with ui
We
2w
2 +
=
can
then
Yn
2Yn Oyl
I9Yn+1
aYn+1
OYO
Oyl
stability
index is
(19Yn
1
It
easily compute
ayo
and the z
(20.16)
Y?
=
ayo
2
+
the matrix
0
1
0
1
Un
0
Un-1
1
(20.17)
U,
(20.18)
i9yi
be shown that..
can
Proposition Proof.
we
20.1.2. R-orbits
will
the fact
use
always
are
that,
from
unstable
for
w
> 0.
(20.16),
ui > 2.
We
(20.19)
then
can
use
the
proof
same
follows that the Jacobian
20.2 Small Values of
as
in Sect. 18.2.2 to show that
vanishes
never
for
w
>
z
> 1. It
o.
n
We will represent the characteristics in the (w, x1) plane. A point of this plane is not sufficient to define an orbit: the three values w, yo, x, are needed. As a consequence, the intersection of two characteristics on the figures does not
necessarily correspond to a true intersection of the two families; it can be merely a projection effect. Here again,, the simplest method of solution consists in solving the equations (20.4) for the yi, and then computing the x'i and x'i' from (19.4). 20.2.1
This
n
=
1
corresponds
case
to
a
bifurcation 2TI and will be treated separately in context of
Sect. 23.1. However, it is of interest to consider it also here, in the transition 2.1.
The
equations
are
1
1
.
Yo
The
W
stability index 2
+
W
=
-
X1
solved into
W
11
X1
=
UO
immediately
(20.20)
-
-
-
2
is 3
The Jacobian vanishes
(20.21) only for
w
0.
20.2 Small Values of
20.2.2
This
n
2
=
case was
From
already studied by Guillaume (1971,
(19-3)
obtain the two equations
we W
subtracting, =
-W
The solution yo twice. Therefore
(yo
,
yj is
=
(20.22)
obtain
we
2
+W2.
YO
Y1
and
YO'+Yi
Yi -YO
139-145).
pp.
.
U)
W2
+
YO-Yi
Adding
yi) (2yoy,
-
w)
0
=
corresponds
it
spurious:
+
(20.23)
.
to
a
2T1 orbit described
have y0y, = -w/2. Solving for yo, yj and and x ' from (19.4), we obtain the. two solutions
x
-W
we
V W--4 + 2w
2
YO
W
-
W
2
1
2
VW- 4 +2W
2
X2
-w2
I
X2
X1
2w
computing the
V-W-4 + 2w
T-
Y1
1
2
X1
229
n
V/W--4 + 2w 2w
(20.24) and :F symbols, either the upper signs or the lower ones should everywhere. asymptotic branches for w -+ oo are for the upper signs
In the
selected The
Yi--W
YO-T, W X/1
-
W
X11 1
,
-
and for the lower YO--W
2
X/2
W2
TW-2
X
It
1
-
-
W,
X2
asymptotic branches for
Y0
Y1
/I
1
-
X/I 2
- W2
(20.25)
-W
yi-T-, W
The
X
w -+
-
I
0+
X2
T
v -2-w
(20.26)
'jW2
are
X1
T-
X2
v "2-w,
W
-
(2-w X2
-
:F
2w 72=w
(20.27)
*
are represented on Fig. 20.1. The asymptotic branches easily identified with the help of (19-8). This is essentially Fig. V-2 in Guillaume (1971, p. 140+). (N in Guillaume proportional to -w here.) (See also Guillaume 1973b.) The stability index is obtainedfrom (20.17) and (20.18):
The
for is
2
signs
I
X1
be
characteristics
w -+
oo
are
UOU1
z=
There is
-2-1=8w z
=
1 for
3
(20.28)
+17.
w3= -2. This corresponds
to the'extremum
in w.
20. Total Transition 2.1
230
-2
2
0
+-11-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
/
-
-
-2
2
0
2
W
+2 for 2. IT2 transitions.
Fig. 20.1-Characteristics
20.2.3
This
n
=
3
case was
also studied
(19.3) gives Y2
yo
-
-
by Guillaume (1971,
pp.
158-164).
the three relations 2w
2yo
+ yj
2y,
+ Y2
+ 2w
2
+ 2w
2
+ 2w
2
Yo
_2w Y1
yj
-
2Y2
Subtracting
2w + YO one
relation from
(Y2
-
yi) (3y, Y2
+
2w)
=
0
(YO
-
Y2) (3Y2YO
+
2w)
=
0
(yi
-
yo) (3yoy,
+
2w)
=
0
One solution is yo
corresponding equal. On the and
(20.30c) 3Y2YO
(20.29)
Y2
to
a
=
+ 2w
=
Y2
we
obtain
(20.30)
.
11w. However,
this is
a
spurious solution
2T1 orbit described three times. Thus not all yj
other
we
yl,
another,
hand,
assume
that
no
two yj
are
equal.
Rom
can
be
(20.30b)
have then =
0
3y0yj
+ 2w
,
(20.31)
20.2 Small Values of
231
n
from which yj ` Y2 follows: contradiction. Therefore two yi must be equal, while the third is different. The equations are then easily'solved. In the case YO
:A
Y1
:---
Y27
find the two solutions
we
+ 12w V/'9--4 W
-3w 2 YO=
1
6
1
X1
4w
-2
9W4 + 12w
11-W
I
X2
X2
4w
The four other solutions
The
3
=
X3
1
V9--4 W + 12w
, Fq-+12W W4
3w2
if
-X3
=
-W X
3W 2 TY2
-
2
obtained by rotating the indices.
are
(20.32)
branches of the two solutions
asymptotic signs
(20.32)
-
for
00
w
are
for the upper
Y2
Y1
YO
-
-2w
2
W
3w X
X3
1
and for the lower YO
X/1
-W
-
W X
2
X3
1
2
Y2
Y1
-X3
r!3
X1
X3
2-
(20-33)
2
3w
X1
-,!YW2
YO
-X
-
I
-X3
=
asymptotic branches for
The
=
signs
if
=
W
X12
2
-
0+
w -+
L FL3
-
W
I
-X2
X2
W
(20.34)
-
2
are
W
Yl=Y2-:F
x"1
iw W W
=
'
-3 3 T TW w
_X,3
X,2
-
=
-x" 2
W -
2
(20.35) The characteristics for
oo
w -+
are
The.stability computation:
represented on Fig. 20.2. help of (19.8).
index is obtained from
UOU1U2 Z
are
The
asymptotic branches
identified with the
-
UO
-
U1
-
(20.17)
and
(20.18),
after
U2
2
351W3
+
+ 332 T-
(27W4
+
-+12w 27w)V9-w4
8
equation
(3W3 The
little
=
81W6 The
a
+
z
4) (2W3
case
=
+
W3
-3/2, (20.36) gives
(20.36)
1 becomes
3)2
(20.37)
0
-4/3 corresponds
to
an
extremum. For the
case
W3
20. Total Transition 2.1
232
-3
-21
X1 2 12 -21-
+-12-
0
+ -111-
+ 12 +-21-
-
-
-
-
-
--
-
-
2
0
-2 +21
2
+3
20.2. Characteristics for 2. 1T3 transitions.
Fig.
-49 T- 81 Z
(20-38)
32
which shows that z
=
11
1. This
we
must take the lower
corresponds
case
to
sign
in
(20.32)
intersection with
an
an
in order to have
orbit 2T1 described
three times.
20.2.4
n
=
4
4 seems difficult. We will on the equations (20.1) for n simplify the task by noting that the asymptotic branches for w -4 00 must correspond to a decomposition into S-arcs. There are 24 possible branches of this kind (Table 6.5): 211, and the other branches obtained by a 4, 31, shift of the origin. All of these branches correspond to symmetric orbits (see
A frontal attack
Table
7.5).
=
Since
values of w, orbits.
large
we are
interested in families of solutions which
we can
limit
our
search for solutions of
(20.1)
to
come
from
symmetric
crossings of the symmetry axis can either lie both in collisions, junctions of two basic arcs, or both in the midpoints of basic arcs. We
The two at the
consider these two 20.2.4.1 First was
*studied
origin
is at
by one
cases
in turn.
Case:'Symmetry Guillaume
(1971,
crossing of the
Axis
146-157). symmetry axis (h pp.
Collisions. This
Through We =
assume
0).
case
first that the
We have then from
20.2 Small Values of
the symmetry: Y3
Y1, X13
=:
x" -x", 2 3
=
=
x14 -xl, 2
=
x" -x", 4 1
233
n
The -xl. 1
=
equations (20.1) can be solved with a little algebra. After elimination of the spurious solutions corresponding to a 2T1 orbit described four times and to a M orbit described twice, one finds the solutions q + w, Yo
Vlq2
+ 4w Y1
2 -q + w,
X1
X4
X1
X4
X2
X3
Vlq2
Y3
=
=
q +
w
q
2
-w,Vq2
+4w
2
+ 4w
4w -q
-
4w 2 + w,
Vq2
+ 4w
4w
4W2
q +
+ W,
Vq2
+ 4w
q + wi
X3
X2
4w
A,/q2
+ 4w
4w
(20.39) with -5W 2 + W2 V9--4 W + 8w q
(20.40)
=
2
and w,
=
provide
1,
W2
1, independently of each other. Thus (20.39) and (20.40)
=
4 solutions.
The characteristics of the 4 solutions The
asymptotic
branches for
w -+
oo
(20.39)
represented on Fig. 20.3. help of (19.8).
are
identified with the
are
(Here there is a disagreement between our results and those of Guillaume (1971, p. 150++). Guillaume's results indicate that the among the 4 branches 0 subset (Table 8.18), + 4 is joined to + 121 and + -1111- is of the 2T4 ----
joined to
to
+
,-22-. On the other hand,
-22- and
+
+
realize that the
the interval
joined quantity p2+ 4w -1111- is
-9/4
<
w
3
<
-
to
+
our
results indicate that
121. It
becomes
negative
+
4 is
joined
that Guillaume did not
seems
in the
case
W2
=
+1,
in
1.)
are obtained by shifting the origin by one basic arc, 2. The characteristics are rotating the indices. This corresponds,to h represented on Fig. 20.4. The. stability index is obtained from (20.17) and (20.18):
4 other solutions
i.e.
=
2 Z
-
(UO
+
U2)(Ul
+
Z
z =
+ U0U1U2U3
(20.41)
2
For the solutions h
W2
U3)
+
=
-
[p2
1 is the
+17
W 3=
+
=
0 and h
4w] [(p +
=
2, this gives
4W2) 2+ 4w] [(p + 4W2)2 2W3
+
2w]
(20.412)
product of three factors. The first factor, p2+ 4w, vanishes for 3 -1 and for W2 +1; W -9/4. The second factor vanishes =
=
3 1. The third factor vanishes for w = -8/9 (for both for W2 = + 1, W 3 = signs Of W2). All these cases correspond to extremums, which are visible on -
Figs.
20.3 and 20.4.
234
20. Total Transition 2.1
-4
2 121
+-22+ -1111-
0
-
-
-
-
-
-
-
--22-
-
+121
-2
0
2
+4
Fi
20.3. Characteristics for h
0 in the 2. 1T4 transitions.
+-13-
2 112
+ 112
2
-2
2 w
+-211+-31-
Fig.
20.4. Characteristics for h
2 in the 2. 1T4 transitions.
20.2 Small Values of
Symmetry Axis Through Midpoints. We ascrossing of the symmetry axis in. the midpoint of
20.2.4.2 Second Case:
first that there is
sume
the first basic YO
=
=
_X1
IfI
We obtain then the two solutions
V/W4
+
Y2
W
=
X 2 =-X 4
2
_X4=
+
W
X3
W
it
(20.43)
X3
by rotating the indices; this corresponds
to
3.
=
The characteristics of these 4 solutions on
N/W4
2w
2., FW_4+ W obtained
_T
_4+ 2 V/WW
-
2w are
_W2
Y3
W2
-W
2 other solutions
h
1).
-Y,
I
=
=
a
W -
1
X2
(h
_W2
Y1 it
I
X1
arc
235
n
Fig.
(h
1 and
h3)
are
represented
20.5.
-31 Xf
13
0
+ 13
--112-
+ -211-
-2
0
+,31
Fig.
20.5. Characteristics for h1 and h3 in the 2. 1T4 transitions.
The
stability index 3
z=1+96(w +1) There is
z
=
1
is
computed from (20.41);
we
obtain
2
(20.44)
only for
1, which corresponds
w
to
an
extremum
(Fig. 20.5). We remark that in all are
included,
the maximal
cases
n1, 2, 3, 4, when the spurious solutions
number
of solutions 2n
-
1(Sect.'20.1)
is reached.
236
20. Total 1 ansition 2.1
(In the case n 4, this proves that there are symmetric solutions which we have computed.) =
20.2.5
Here of
other solutions than the
> 4
n
2 to 4 suggest that for larger values again the solutions found for n might exist also solutions of the form (19.69). It is found, however, =
there
n
that n
no
such solutions exist for
no
> 4
20.2.5.1
n
5. It
=
seems
that the system
(20.1)
for
only be solved numerically.
can
> 0: Relative Sides of
Passage. The positional method used partial transition. However, it was noted in Sect. 19-3.3 that in the case w > 0,Ahe positional method is not necessary: the consideration of the relative sides of passage, i.e. Broucke's principle, is sufficient to establish all junctions. The same approach can be used for the w
in Sect. 19.3 works
only
for
a
total transition. in
We consider first the asymptotic branches for w -+ +oo. The orbit consists sequence of S-arcs. yi is positive in a node, negative in an antinode
a
(Sect. 18.1.2). Therefore the sequence of signs defines completely the orbit. But if the sequence of signs has a sub-period (for instance +R-R-R+R-R-R+), the corresponding orbit also has a subperiod, and therefore must be excluded. Thus, for a given n, there signs having no subperiod. We consider
now
is
one
the branches for
n > 1, the 23.1). (see that there exists one and only
Sect.
and
For
subperiod. Therefore, all junctions
study one
only w
-+
one
branch
0+. The
of the R-orbits
for
each sequence
case n
=
I is excluded
(Sects. 18.2, 18.3)
orbit for each sequence of
of
shows
signs having
no
indeed be
trivially established for total transicase w a sequence of S-arcs is joined to a R-orbit, which is immediately determined by considering the signs. As in the case of, a partial transition (Sect. 19.3.3.1), we have the result that all branches with a sign in Table 8.18 pass through the transition 2.1 and at v to on higher values of v. Their junctions will only be 1/3 go can
> 0. Each orbit made of
tions 2.1 in the
-
=
determined at transition 2.2, for 20.2.5.2
of passage on
< 0: Numerical
w
negative, junctions.
the
The For
are
case n n
>
branches with The For
n
>
a
cases n
4,
a
=
1/2 (Chap. 22).
Computation. In that case, all relative sides Broucke'sprinciple gives no information
and therefore
1 is excluded
here; it will be treated in Sect. 23.1. R-orbits, and therefore no branches w -+ 0-. It branches w -+ -oo must be joined between themselves: all + sign in Table 8.18 are joined in the transition 2.1. 2 to 4 have been solved analytically in the previous sections.
=
1, there
follows that the
v
are no
=
numerical
computation
seems
necessary. A program similar to
described in Sect. 14.2.2 for the total bifurcation of type 1 was used. The computation of a given branch is begun at a large value of JwJ. An the
one
20.3 Results for Bifurcations of
method, based
iterative relaxation
orbit into S
arcs
and
on
on
2
Type
237
asymptotic decomposition of the equations (19-8), works well in that
the
the asymptotic
case.
The branch is followed towards
decreasing JwJ. Usually
at some
point the
iteration does not converge anymore, because the decomposition into S arcs ceases to be a good approximation. We shift then to a trial-and-error shooting
method, which works well for moderate values of JwJ where the branches well separated.
20.3 Results for Bifurcations of 20.3.1 The Case
It
was
w
0 to
=
w
Type,2
> 0
shown in Sect. 20.2.5.1 that for
from
runs
w
are
=
> 0, every characteristic of a R-orbit words, a R-orbit simply traverses the
w
+oo. In other
total transition 2.1. The R-orbit coincides with the whole bifurcation orbit. Thus
we
obtain 20.3.1. All branches with
Proposition
(symbols and go
i and
on
to
e)
in Table 8.18 pass
higher
This result
values
w
-
sign and containing
through
one
sign
Sect.20.2,5.2
v
T-arcs =
1/3
found in Sect. 19.4.
< 0
Since the R-orbit coincides with the whole bifurcation found in
no
the transition 2.1 at
of v.
complements the
20.3.2 The Case
a
are
immediately applicable
in the total bifurcation of
Table 20.1 lists the
orbit, the junctions
to the branches with
a +
type 2.
junctions
up to
n
=
6,
in the
same
format
as
Ta-
ble 8.18, which was obtained using only qualitative methods. It can be verified that there is agreement for all branches which appear in both tables (sign +
and 2:!
n
<
4).
238
20. Total Transition 2.1
Table 20.1. Total
determined
+2 -112T3
----
0
by
the
bifurcation, type 2, n quantitative study.
-111-
2T3
----
1
2T3
+141
+-1113-
+-123-
+212
+-2112-
+411
+-222-
+42
-11111-
+5
+
+-212-
+2112
+2211
+'-111111-
+12111
+1221
+-11121-
+6
+-24-
+-33-
+-51-
2T5
2
+-12-
+21 2T4
-----
------
1
+
122
+
1121
-1111-
-
+1131
+-11112-
+.-21111-
+1311
+15
+-213-
+-312-
+51
------
2
+23
2T6
+-1211-
+114
+321
+2111
+24
+312
+-14-
+-3111-
+132
+-311-
+-321-
+-1311-
------
3
2
+311 +-31-
+32 +-1121-
+211
+1112
112 -----
+31
2
2T6
-------
+11121
+-1131-
+-12111-
+231
+21111
+213
+
+-13-
2T4
-------
-
2T5
+
5
113
13 -----
-------
1
+-211-
2T4
2T6
+14
+
+121
+
1
0
+-22-
-----
-------
+11112
+-32-
2T5
2T4
11211
2T6
+4
+
branches,
+-141-
+-221-
----
+
+-131-
+-21-
+12
junctions between the
+131
1 +3
2 to 6:
3
------
+-1212-
+-42-
+-1122-
-------
+-1112+-122-
3
+-2211-
+-21214
+1113 +3111
+-112-
+123
+-15-
2T6
+-1132T5
1122
+-114+-411-
A
21. Partial Transition 12
study
We
,
in detail the
now
corresponding
transition 2.2,
equations, (17.205) obtained to
v
chapter the
We consider first in the present equations are then
in Sect. 17.7 for
1/2.
=
case
of
a
partial
transition.
The
Xi2
Yi
_
Yi-1
-
Yj(Xj+i' YO
ziXi
-
-
Xj)
-
Yn
0,
=
i
Zi _W=O, =
1
=
0
=
0
=
1,...,n
i
,
i
0
,
n
=
,
1'...'n
=
-
1
(21.1)
-
equations form a system of 3n + 1 equations for the 3n + 2 variables W, Yo to Yn, X, to Xn, Zi to Z,,,. Therefore we expect one-parameter families of solutions. For a given value of M, there is a one-to-one correspondence, given by (17.198) to (17.199), between the present variables, W, Yj, Xj, Zi and the physical variables AC, hi, ui, Aaj. Thus, the one-parameter families of solutions of (21.1) correspond simply to the ordinary one-parameter families These
of orbits
21.1
(see
Sect.
2.3).
Properties
1. The
quantities Yj for
i
Therefore each of them 2. For
n
zi
n
keeps
-
1
vanish, because of (21.1c). sign along a family.
can never
constant
a
1, there is < 0
,
i
=
(21.2)
1,...,n
Proof- Assume that there exists
Xi 2: 0, Yj > 0. Rom (21.1c) Zj+j >. Zi ! 0; from (21.1b),
a
we
Zi
> 0. We consider first the
have then
case
where
0; from (21.1a), Xj+j 0; from (21.1c), Xi+2 > Xi+l; >
Xi
>
Yj+j > Yj > increasing sequence.. This contradicts (21.1e). We consider next the case Xi 0, Yi < 0. Then from (21.1b) we have Yi < 0; from (21.1c), Xj_j > Xi > 0; from (21.1a), Zj_j > Zi Yi-1 0; from (21.1b), Yi-2 < Yi-1; and so on. The Yj towards lower i form a decreasing sequence. This contradicts (21.1d). If Xi < 0, the same proof holds with all signs inverted for the Yj and Xi. and
so on.
The Yj form
an
M. Hénon: LNPm 65, pp. 239 - 269, 2001 © Springer-Verlag Berlin Heidelberg 2001
21. Partial Transition 2.2
240
0. We reason in a similar Way,' Finally, in the case i n, there is Yi of instead Yi. using Yi-1 It follows immediately from (21.1a) and the previous Property 2 that, for n > 1, =
3.
W > 0
=
(21.3)
.
Since W has the sign of -c'AC from (17.198) and 17.199), it follows that only branches with a sign are involved in transition 2.2. This agrees with -
what
joined is the
found in Sects. 19-3.3 and 20.2.5. Branches with
was
at transition 2.1 and case n
can
be refined. From
IXiI< WI/2, Rom
(21.1b) (for
ly,I
<
IZil 1)
i
(21.1a)
and
Property
(21.1d),
we
obtain
(21.5)
hand, (21.1c) gives
_W-1/2.
(21.6)
2
Combining
we
derive then
1
>
2
(21.4)
< W.
and
a + sign are only exception
in Sect. 23.2.
w3/2.
On the other
ly,I
reach transition 2.2. The
1, which will be treated separately
=
(21.3)
The result
never
these
results,
we
obtain
W > 2 -1/2
(21-7)
4. For any solution, by applying the fundamental symmetry E of the stricted problem (Sect. 2.7), we obtain another solution:
(Yi, Xi, Zi)
E
(yn-i, -Xn+l-i, Zn+l-i)
1-4
re-
(21-8)
-
5. For any solution, there exists a symmetrical solution obtained by changing the signs of all variables Yi and Xi. We call this symmetry V:
(Yi, Xi, Zi) By combining VE 6.
For
V,
=
the
Xi
Zi, 1
+
Y
(-Yi, -Xi, Zi)
-+
we can
(21.9)
obtain
we
(Yi, Xi, Zi)
Eliminating Xi+i
E and
-+
(-Yn-i, Xn+1 -i, Zn+i -0
write the
Yi+1
Yi
(21.10)
equations (21.1) as'
+
Xi+i (X i+1
-
W)
.
(21.11)
given W, this is an area-preserving mapping ofthe (X, Y) plane. explorations (H6non, unpublished) exhibit the mixture of and chaotic orbits characteristic of non-integrable systems. This regular the supports conjecture that the system of equations (21.1) is not explicsolvable in itly general. a
Numerical
21.1
Asymptotic
21.1.1
In the limit W
Branches for W
+oo,
-4
The solutions in that
we
enter the
--+
+oo
region JACI
>
M1/2
(Sect. 17.6); they
are
known
equations (17.187). to changes of variables (17.198) and (17.199) for the find:,
9
2W
[1
X, with
<
1/2.
-
1/2,
we
+
a
X, with
sign for
a
1)Wl/2[l + O(W-2)],
T'
are
and
a
-
sign for
a
(21-13) T'
With the
arc.
new
sign for
a
Tf
notations
this becomes
Wl/2[l + O(W-1)],
=
+
a
-
(Sect. 17.1),
T9
(21.12)
by (17.195) and (17.196); and
sign(a
=
Tf and
are
and
a
(21.14) -
sign for
a
T9
arc.
R-arc:
Z,
=
yAi
=
_W[l + O(W-2)], W1/2 Yj [1 + Off-1)]
W-112Xj[1 + O(W-1)],
Xi=
where
j
Chap.
(21.16)
i"-1
-
is the relative
in
(21..15)
have defined
we
i
18
position inside the R-arc. yj, xj
(see
Tables 18.2 and,
are
the numbers
computed
18-3).
TT node:
'W-1/2[j+O(W-2)],
Yj with -
case v
O(W-1)]
+
where g is defined
-
v
are
T-arc:
Z1
-
or
,
given by the (17.194) of Sect. 17.6. If we substitute these expressions region
into the
-
241
Properties
TR
a
or
+
a
sign for
a
T9Tf node and
a
-
sign for
a
Tf T9 node.
RT node:
W-1/2[l + O(W-1)],
Y
with
(21.17)
2
+
sign for
a
TgR
or
RV node, and
(21-18) a
-
sign for
a
Tf R or'RT9
node.
asymptotic values satisfy the equations (2 1. 1). expression for a TR node, distinguishing between different R-arcs. This expression is easily obtained by substituting (21.14) for Xi and (21.15c) for Xj+j into (21.1c): It
can
be verified that these
We will need
a
finer
21. Paxtial 'Ransition 2.2
242
Yi
=
Xi+1
1- Xi
=::FW-'1/2
1 -
W-112X,+, W-3/2 Xi+1
_
9-
+
W1/2
+
O(W-3/2)
Off -5/2)
(21.19)
sign for a Tf R nodesign for a TgR node and a that for show W -4 +oo, the one-parameter families we now Conversely, of solutions of (21.1) have no other asymptotic branches than those which correspond to a sequence of T-arcs and R-arcs, and which are described by (21.12) to 21.19). Consider one particular branch. We assume that the Yj and the Xi behave asymptotically as powers of W: with
a
Yj
+
-
-
aiWPi
Xi
-
bi Wqj
for
W
-4
(21.20)
+00.
The ai and bi are non-zero constants. The pi and qj are arbitrary constants. The two ends of the bifurcating- arc, YO = Y,, = 0, tan be included in this
formulation
-00by taking po Pn equations (21.1) as =
=
We rewrite the
Xi+1 Yi
-
Xi
-
Yi_1
1
(21.21)
=
Yi
=
Xil
-
(21.22)
XiW
1. Assume that there exists
a
1/2.
qj >
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 <
to term 4. Therefore at least
1/2.
one
of
(21.32) is true. We
distinguish again several
cases.
-bi
(21.33)
.
=
bi
to term
2; therefore
(21.34)
.
ai+j
,
=
-(j
+
1)bi
(21.35)
,
impossible. 1 + qj. This case is treated as above. pi-1 > 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 <
-
2
Therefore still true,
W-1/2'
(22.2)
from which
-1/2 > W
lyi-Y On the other
Combining, W > 1 In
period
hand, (22.1b) gives
y_11
ly
we
<
W3/2
(22.4)
obtain
J22-5)
.
regard n
(22.3)
to
of the
property 6, the bifurcation orbit corresponds
mapping (21.11).
M. Hénon: LNPm 65, pp. 271 - 281, 2001 © Springer-Verlag Berlin Heidelberg 2001
to
a
cycle of
22. Total Transition 2.2
272
Stability and Jacobian
22.1.1
We proceed as usual (see Sects. 20.1.2, 18.2.2, 14.1.2). To compute the Jaco bian, we order the variables as follows:
(22.6)
YO XI; Z1 Y1) X2 Z2) Y2)... Yn-1) Xni Zn i
)
i
7
We obtain
Proposition 22.1.1. In a total transition 2.2, the Jacobian vanishes if and only if the bifurcation orbit is a critical orbit of the first kind (the stability index is
z
=
1).
Any pair
of variables
be convenient to
use
dXj dYj_j
can
0
-Yi-,
dXjj
dXi+
It will
dXjj
0
1
3Xj2
dYj
stability index.
dXi-1 dXj
12
Yi!-,
(dXj)
be used to compute the (22.1) we find
X0 and X1. Rom
-
W
dYj_
1
1
0
1
_1/y2 i
dXj dYj
(22.7)
(dXj_j
(22.8)
from which 0
dXjj
yi
dXi+
1 Ui
1
y2 i
y2 i
3Xj2
W
dXj
with
Yj j
ui
We
can
Yj2
+
09XI
j X_O
OX,
I9Xn+1
09xn+l
Xo 1 2
=
ax,
stability
=
(22.9)
.
then compute the matrix
OX,
and the
Z
-
(IM OXo
(
1
0
Yn
y2 _n
1
Un
y2 0
Ul
2
j 1_2
y2 1
Yn
index is +
(22.11)
ax,
22.2 -Small Values of
n
We will represent the characteristics in the (W, YO) plane. Note that of this plane is not sufficient to define an orbit: the three values W, are
needed.
(22.10)
point YO, X,
a
22.2 Small Values of
22.2.1
There
n
=
I
solutions in that
are no
term vanishes. This n
=
and that
cannot be satisfied since the first
(22.1c)
case:
corresponds
(Sect. 18.2-3),
1
273
n
to the fact that there
cannot either have
we
a
R-orbits for
are no
T-arc, because
(Proposition 4.3.2). (see Chap. 23). However, it does
we
would then have successive identical T-arcs The total bifurcation 271 exists
not visit
the transition 2.2. It will be studied in Sect. 23.1.
22.2.2
The
n
=
2
equations
are
X21 _Z1 _W=0,
Y1(X2 2
'X2
-
X1)
Z2
-
YO(X1
-
-
1
-
W
X2)
=
=
=
-
YO
-
ziXi
=
0,
YO
-
Y1
-
Z2X2
=
0,
0.,
0;
-1
Y1
(22.12)
0
Combining these equations,
obtain
we
(22.13)
YO
Yi and
(X1
+
2 X2) (Xi -'X1 X2
+
X22_W)=O.
.(22.14)
Because of the symmetry E' (Sect. 21.1, Property 5),we can consider-only solutions with YO > 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). (
For this
replaced by
f and
respectively; 17.1.) Table 22.2, together with Table 20.1, gives all junctions for total bifurcations of type 2, from n 2 to n 5. The cases left undecided in Table 8.18 are now solved; in addition all junctions are established for n 5. Nothing prevents in principle the solution of higher values of n, using numerical com6 are listed in Table 20.1 for branches with putation. (The junctions for n a + sign.) However, the amount of work grows exponentially. g
see
Sect.
=
=
=
=
22.3 Results for Bifurcations of
Table 22.2. Total determined
2T2+-+O -
2
-fg 2T2+-+A -
-
fI
Ig
2T2-+-O
by
the
bifurcation, type 2, n quantitative study.
2T3-++-O -
-
-Ill-
glf
2T3-++-A -
-
lif
gIl
2T3-+--2
=
2T4++--+l -
-
13
Iflg
2T4++--+A -
-
If2
12g
2T4+-+++2 211
2T2-+-A -
-
if
gl
2T3++-+i -
-
-
-
2T3-+--A -
-
-if I-
g2
2T3--+-l
12
Ifl
Ilg
lgf 2T3--+-A -
-
2f
-lgl-
2T3+-++2
2T4+++-+2
-21
-112
-
fgl
2T3+-++A -
fil
2T4+-+++A -
-
-
-
-
-
-
Igl
2T3+--+O
-
2T4++-++O
-
2fl
-
f12
-
-
-
-
-
3
fig
2T3+--+A
-
-
121
lfgl
2T4++-++A
fg2 2fg
21g Ig2
lifi
lllg
Igfl fIlg
2T4+-+-+A
-
-
Igll
2T4+-+-+2
Ilfg
2T4+++-+A
f ill
2T4+-+-+O -
Ifg
2T3++-+A
fgIl
gfl
gf
lgfg fgIg fgf I flfg
-
-
-
-
-
-
-
ifli
-
llgl
-
-
2gl
gf2
-1111-
2T4-+---A -
-
2T4--++-l --211-
lglf
glif
Ilif
gill
-If 2-
g3
2T4--++-A -
-
21f
-igli-
--22-
-
glf 1
2T4-++--A -
-
-ilf I-
g12
2T4-+-+-O -
-
-
2T4-+-+-A
-
-
-
-Igf I-
2T4--+--A -
-
-2f I-
-lg2-
2T4---+-2
-If gl-
-gfll llgf
-
-
g2f
M-- -+-2
f Igl
f2l'
--13-
--112-
31
2T4+--++A
2T4-+---2
2T4--+--O
-
2g
3g
-
2T4-++--3
-
-
f3
2T4-+++-A -
between the
f2g
2T4-+++-O
-
-
4
2T4+---+A
2T4+--++3 -
junctions
2T4+---+O
-
;.-
f2
2 to 5:
Type
-Ilglg2l -if 1112f
gfgl glgf gf If if gf
2gf 2T4---+-A -
-
3f
-2gl-
2
279
branches,
28,0
22. Total, Transition 2.2
Table 22.2.
2T5 ...... 3
(continuation)
-
-
-
-
1112
2T5+-++-+A
-
2T5 ...... A
-
IfIfl
-
12fg IfIfg
Illig
-
-
2T5 ...... I -
-
-
tlfgl
-
.5...... A
-
-
131 If Igl
-
-
-
-
-
IIIgl
-
-
2T5 ...... 2
llf lg
-
-
-
If2g 14
2T5 ...... A
IIf2
112g
21fg Iglf g f lif I
f glf 1 f llfg
fgllg
-221
-
-
f lIgI
lgf 11 fgfgl
2T5+-+-++A
-
2T5 ...... 4
If3
13g
-
-
Ig2l lglgl
2T5+--+-+A
-
-
-
2T5 ...... 2
1211
-
-
If gll
-2111 -
2T5 ...... A
f gill
-
-
-
IlgIl
-
-
2T5++-+-+l
f lill:
lglII
-
2T5+-++-+O
122
If lig -
-
-
llgfl lfgfg
-212 -
-
-
-
2T5++-+-+A
-
-
-
llglg IM
121g
lglfl f llIg
fglfg
3fi f 2f I
-
f22
-
M
flf2
-
-
M
-
-
22g
-
lgf2 Igf Ig
-
-
-
-
-
fgf2 fg2g
-
f 121
f 12g
fg2l fgIgI
-311
2fll f if 11
f2fg 2gfg 2gfl f Igf I
f2gl
-41 2T5 ------ A
2T5 ...... 3
-
f2lg f IgIg
f If Ig
21gl
-
2g2
2T5 ------ 4 -
-
31g 2glg
-
-
-
f lgll
-
f3l
3gl
2T5 ------ 0 2T5 ...... A 5 -
-
-
-
lg3 lg2g
-
-
2T5 ...... A
If ill
fg3 fgfIg
-
-
-
-
2fIg
2T5+-+--+A
-
-
3fg flgfg
-23 -
f 112
f Ig2
-
-
-
-32 -
2T5+-+--+2
2T5+-+-++4
-
2T5 ...... A -
12gl
2T5 ...... 1
-113 -
If 21
lg12
-'f g12
2T5 ...... A
lif It
fgf It Igfgl
2T5+--+-+3
-
-
-
21fl
2fgl f IfgI
-
-
-
2T5 ...... 0
1121
2T
lfgfl llgfg
-
-
-
-
2lIg IgIIg -
-
-
illfl
12fl
-
-
-
Illfg
-
-
Ifg2 lfgIg
f2ll
2gI1
f3g 2T5 ------ A f4
4g
22.3 Results for Bifurcations of
Table 22.2.
2T5 ...... 0
2T5 ...... 3
2T5-+-+--A
-
1
-
-Ilg2g1g2
gif 2
-11111-
--113-
g1lIf
-
-
-
-
2T5 ...... A
-If 12-
2T5 ...... A
-if if I-
-
111if
-
glill
-
-11f2-
2T5-+-++-2
--1112-
--1211-
glif I
11glf
2T6 ...... A
-
-
-
-
-
2T5-++-+-3
-
111gf gifgf
2T5-++-+-A
-
-
-111glgllgl
-
-
-
-
-
-lif 11-
g121
-
-
-
-
-11fg1gif gl
-
112f
-
lif if
-
-
gf If I
-
-
-
Ilfgf gIIgf
-if 3-
281
g4
g2lf gIg1f
-If Igi-
-
-Ig1II2T5 ------ 0
2T5-+--+-A
-1g1f I-
g3l g2gl
If lif
If 2f
gf I If gfg1I
-
-
-1f21-
-
-
-
-
-21f I-
-1g12-
-12gl-
2T5--+-+-l
gf21 gf IgI
--221-
g2f I -If g2-
1g2f
-2fgilgf gf
gfgfl 2T5--+-+-A
-
-
-21gl-1glgi-
-Igf 2-
-2f 2-
-1g3-
2gif
31f
-2gll-
2T5 ------ I --32-
-2gf I2T5 ------ A
-
-3f I-
-2g2-
2T5 ------ 3
--41-
-
-lgf 11lgf If
2T5 ------ A
-
-
-lgfgl-
--311-
-
Of
If g1f
2f g f
2T15------ 2
--2122T5 ------ A
-If g11-
-
-
g3f gf Igf
21gf Ig1gf
2T5 ------ A
211f
-
-if III-
--122-
lgIlf
-lg2l-
--23-
M -+++-A
121f
-11gil-
-
22f
2T5 ------ 4
--2111-
-
gl2f g1fif
-
-
2T5 ...... 1
g211 g1gil
2T5-+-+--4
-
-
-
-
gf 12
-
-
-
gf 111 gf gIf
gf3
2T5 ------ A
-If gf Igfg2
-
-
-
-
-
-
-
-
-
M-i---+-O
--1121-
gif 11
12gf if lgf
-
-Ilgf Ig1gf I
2T5-+-++-A
-
-
-
--14-
g 22
_11if I-
g112
-
2T5 ------ 2
-12f 1-
g13
2TS ...... 4
gf2f g2gf
-
-
-
2
(continuation)
-
-
Type
3gf
2T5 ------ A
-
-2f 11-
-
-
2f1f
-
4f
-3gl-
23. Bifurcations 2T1 and 2P1
The two
cases
with
n
1
=
are
very different from the
in the
general
case 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 <
v
<
1/2
0, AC :A 0, corresponding to v > 0. We will 0. This previous. section for v corresponds to the asymptotic branches of the families of orbits for p 54 0, which are close to the branches for M =. 0 (see Fig. 1.1). We consider
try
the
now
case
p >
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)
<
2
(23-8)
.
5. We want all terms in the
This is the 0 <
we
VY -
(23.6)
ACz*
second
a
-
2(a
=
0(t,'AC-2), z
1
Step
AC,
suggest the change of variables
in the fundamental
dominant terms in the left-hand
YXI
0
Aal
Y
Step
=
3. It will be convenient to make
Step X
the
1 and 2. The results for
Steps
=
right-hand members
to vanish for p -+ 0.
if
(23-9)
23.1 Total Bifurcation of
Type 2,
n
=
1
(2TI)
285
general case n > 1, where the limits of the validity were 0 < v <.1/3 (Equ. (17.103)). asymptotic equations are then obtained by equating the right-hand
Note the difference with the
interval of The
members to
yx'
zero:
2x'
0
=
+
6. We solve
Step
0
z
y
these
now
-
1
-
Z
=
0
(23-10),
.
asymptotic equations. There
are
two
so.lu-
tions. The first is 0
x
Z
=
0
y
1
=
It is the continuation of the first
(23-11)
.
species solution (23.4). The second solution
is y
=
0
-1
Z=
,
x'
,
=
1/2.
(23.12)
-
It is the continuation of the second species solution (23.5). Step 7. We compute the Jacobian of the system (23.10) for the three
x',
variables y,
X
.
IJI
=
z:
I
0
y
0
2
1
1
0
-1
-2x'+
(23-13)
y
1 for the first solution, IJI -1 for the second solution. gives IJI Thus, for 0 < v < 1/2, there exist for p 0 0 solutions of the fundamental equations (23.8) close to the solutions (23.11) and'(23.12) of the asymptotic equations. The error is of the order of the largest term in the right-hand
This
=
members of
=
(23.8),
i.e.
O(AC' jAAC-2)..
8 and 9. We go back to the physical variables, using (23.7) and and we refine -the error estimates for some variables. For the first
Steps
(23-6),
solution
ho
=
-
2
I
ul
(23.11),
=
Aal
I
a
=
+
O(AC' IAAC-2)],
-
O(AC2, IIAC-1) O(AC2' tZAC-I)
_U11I =
=_
Aal
obtain
(aVy 1) AC[l =
(23-14)
.
Using (23.2a) cipal term for Ul
we
an
'd (23.2b), ul', Aal:
we can
refine the estimates and obtain the prin-
ul,
I/=Ul
aVyv
4a sign(vy)
37rIVy
For the second
AAC -1[1 +0 (AC' /IAC-2)),
,AC- 1 [1 + 0 (AC' /'jAC-2)]'.
solution,
we
have
(23.15)
286
23. Bifurcations 2T1 and 2P1
O(AC2, IZAC-1)
ho U
3-7rl(a
ul
-
1)Vy
-
2V2
AC[j
+
O(AC tjAC-2)]
Y
Aal
a2VY =
AC[l
-
VY
Using (23.2a), ho
2v
O(AC, tIAC-2)].
refine the
we can
estimate
(23.16)
for ho:
_/_,AC-ffl + O(AC /.jAC-2)j
=
31rl(a
+
(23.17)
,
1)Vy
-
These equations are particular cases of (17-122), (17.123), (17.127), for'rn For p -+ 0, the equations (23.14), (23-15), (23.16), (23.17) reduce to the equations (23.4) and (23.5) of the previous section.
23.1.3 The Case
We
study
to AC
=
'Step
now
v
what
=
1/2
happens when
we
reach the value
v
=
E) (fil/2). 1. The results of the
1/2, corresponding
previous section show that when the value
reached, all quantities ho, ul, ul', Aal become of the order of 1/2 1/2 for both solutions. IL , Step 2. This suggests the change of variables v
is
=
ho
=
IL1 /2 y*
1
Ul
,
=
A
1/2 x 1*
,
Aal
=
p 1/2Z*
AC
=
p
1/2 W*
.
(23.18) 'Step
3. It will be convenient to make the second
V:3 7r:I Y
I
av
Y*
Yx,
+ 1
2 W
IrI
VY 4.
(a
_v
( aIrIZ
a z
Step
X/*
Substituting =
=__
VY
1)
2x'
+
Z
=
of variables
/-3-7rlx
V V3
V::v _ IrIa 3,7rIa
in the fundamental
O(y 1/2),
change
(23.19)
W.
equations (23-2),
O(P1/2)
2y
+ 2w
-
we
z
obtain
=
O(p 1/2)
(23.20) Step equations
Yx,
5. The are
+ 1
=
right-hand members tend to zero for by equating them to zero:
p
0. The
2x+
0,
z
=
0,
2Y +
2w
-
z
=
and
-
1
=
0
(23.21)
0.
Step 6. The system of asymptotic equations (23.21) Eliminating x' and z, we obtain
Y2+,Wy
asymptotic
obtained
is
easily solved.
(23.22)
23.1 Total Bifurcation of
-
-2 V/W+4
W
Y=
Z
2
AVFW2
W
-
Type 2,
n
=
-
+
4,
1
(M)
WT
x
287
-2 +4 V/W-
2
(23.23) The characteristics in the
(w, y) plane
are
shown in
Fig.
23. 1.
+ E
4 1
1
-77
0
2
Y 2
+ 1
0
-2
-4 -4
-2
4 W -
Fig.
E
23.1. Characteristics of the 2T1 bifurcation.
Step
7. We compute the Jacobian of the system
variables y, x, I
Y 2
0
0 2
0
-1
x I
IJI
(23.21)
for the three
z:
1
-2x' + 2y
=
2v rO
(23.24)
+ 4.
0 solutions Therefore, for v 1/2, there exist for of the fundamental equations (23.20) close to the solutions (23.23) of the asymptotic equations. The error is of the order of p 1/2 Step 9. We can go back to the physical variables, using (23.18) and
It
vanishes.
never
(23.19). ho
=
We obtain
Fy2AC2 7r-Ia P) 4(a 1) (VYAC Fy2AC2 TirIa O(A) 2vy (-VYAC 37rI(a 1) (VYAC::F FV2AC2 Y7-rIa'o) O(A) 16v
a
=
+
Y
+ 0
-
Aal
a2
16v
=
+
+
Y
16v
-
Ul
4V2
Y
+
+
(23.25)
23. Bifurcations 2TI and 2P1
288
If
decreased, so that p becomes negligible compared with AC', these expressions reduce to those of the case 0 < v < 1/2: either (23.14a) and v
(23.15),
is
or
(23.16b), (23.16c),.and (23.17), depending
allows the identification of the branches shown in
Fig.
on
the
23.1. We
This
sign.
recover
the
junctions deduced from Broucke's principle (Table 8.18). 23.1.4 The Case
v
>
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 <
Steps
-
-
-
I and 2. The results for
Aal
ACZ*,
=
U1
v
<
v
2/3
0 suggest the
change
(_61Aq1/2x/*
=
U
I
1
=
of variables
(_r1AC)112X11* (23.32)
3. It will be convenient to make
Step Z*.=
a
2VY
second
IVYx'' VE'E'V Y
X'*
Z'
a
VY
change
of variables
VYXII. "' ev Y
X11*
(23.33) In the last two
equations,
that way, X' and X"
Step
4.
are
take the
we
sign
+ for
a
T' arc,
-
for
a
T'
Substituting in,the fundamental equations (23.26),
we
obtain
O(AC1/2), O(AC' tIAC-1) 0(,AC, j1AC-31 2). Z(X + X11)
X11
X1
X12
1
=
_
Z
I
Step 0 <
The
v
5. All terms in the
X11
right-hand members
vanish for it 4 0 if
2/3.-
(23-35)
asymptotic equations
bers to
(23-34)
=
<
arc; in
always positive.
are
then obtained
by equating the right-hand mem-
zero:
-
Step
x/
=
X12-_l_Z=o
0,
6. The
I
asymptotic equations (23.36)
Z(X' are
+
X")
=
0
-
easily solved.
(23.36) We obtain 3
solutions:
Z=O,
xl=x"=i;
Z=O,
X,
Z=-1'
=
x/
x" =
=
x/1
-1; =
The relevant solution is determined p
=
0: it is
(23.37a). (This are positive.) can
(23-37)
0.
by continuity
with the previous case as mentioned
also be shown from the fact that,
above, X' and X"
Step variables
7. We compute the Z:
X', X ',
Jacobian of the system (23.36) for the three
292
23. Bifurcations 2TI and 2P1
JJJ
=
Thus, for
-1
1
2X'
0
-1
Z
Z
X, + X"
0 <
2/3,
<
v
0
-
equations (23.34) close The
error
(23.34),
Steps
for
p
54
0
a
=
(23.38)
-4.
solution
of the fundamental
(23-37a) of the asymptotic equations.
largest term,
in the
O(ACI/2' pAC-3/2).
8 and 9. We go back to the
right-hand members of
physical variables, using (23.32) and
We obtain
,/---VYAC [1 + O(A'C1/2'jZAC-3/2)]' Vf VYAC [1 + O(AC1/2' ttAC -3/2)], O(AC3/2' MAC-1/2).
U, u
Aal The
there exists
to the solution
is of the order of the
i.e.
(23.33).
-2X(X'+.X")
(23-39)
=
error
Aal
estimates
be refined. From
can
(23.26c)
and
O(tAC-1/2).
=
(23.39)
we
obtain
(23.40)
Note the difference with the 2P I is in fact the
only
(23.26b)
Rom
we
case
general case (17.127). The present bifurcation where g' and g" both vanish.
have then
V/-vY AC [1 + O(AC' IZAC-3/2)],
Ul
I
VY
Ul It is not
depends
(23.41)
(23.26a)
and from
on
Ac [1 + O(AC' MAC-312)].
(23.42)
possible here to compute the principal term of Aal, because it the last term O(M) in (23.26c), which is not known in the present
order of approximation.
23.2.3 T-Arcs: The Case
v
2 : 2/3
We consider
now
In the
limit,
the first term of the
O(IL),
i.e. of the
terminate. The
the limit
same
v -+
order
2/3,
as
still using the change of variables (23.32). right-hand side of (23.26c) becomes of order
the
error
0(tt) approximation,
a
it would be necessary to go quantitative description of the
23.2.4 S-Arcs: The Case 0 <
v
< I
For p = 0, the two branches + 1 and tion arcs. We assume as usual that the
-
tion
7.3.2) U1
=
1 are made of
symmetric bifurcasymmetric (Restricsymmetric. Then there is
complement
and that for M > 0 the orbits remain
-Ul
equation is then inde-
Sect.11.3.2, is no longer to a higher-order approximation to T-arcs in the range v > 2/3.
sufficient; obtain
term. The
introduced in
is
(23.43)
23.2 Partial Bifurcation of
(23.26c)
identically satisfied,
is
37r-T(a
I
Ul
-
1)
-
2a 2 vy
Steps I and. 2. change of variables Aa,
The
+
a
2
VY -
x1*
,
2x' +
z
The
V
112
2 ,
U1
,AA
(23.44)
-
v
=
0
a
second-change
O(AC, y)
suggest the
of variables
x/
(23.46)
VY
equations (23.26)'become Z
5. All terms in the
Step
to
=
1) Vy 2
4. The fundamental
293
(23.45)
31rI(a
VY
Step
(2PI)
1
ACx'*
=
=
=
reduces to
O(AC, Aal, ul
3. It will be convenient to make
=
0 <
(23.26a)
n
equations (23.30) and (23.31) for
U/1
ACz*
=
Step Z*
Aal [1
and
Type 2,
+ 1
=
O(AC' ILAC-1
right-hand sides
(23.47)
vanish for p
0 if
(23.48)
< 1
asymptotic equations
are
then obtained
by equating
the
right-hand
sides
zero:
2t-I
+
=
=
0,
6. The
Step Z
Z
-1
Z
=
2
1
0
1
0
(23.49)
.
['
-
are
immediately
solved:
(23-50)
1/2.
Step 7. The Jacobian
111
=
asymptotic equations (23.49) x'
,
+ 1
is
(23.51)
2.
Thus, for 0 < v < 1, there exists for p 54 0 a solution of the fundamental equations (23.47) close to the solution (23.50) of the asymptotic equations. The error is of the order of the largest term in the right-hand sides of (23.47), i.e.
O(AC, pAC-1). 8 and 9. We go back to the
Steps
(23.46). Aal
Ul
physical variables, using (23.45) and
We obtain
=
=
_a 2V A C[1
VYY
37rI(a
+
O(AC' 11AC-1)]
Vy 21) AC[1 2v,
+
,
O(AC' /_IAC-1)]
.
(23.52)
23. Bifurcations 2T1 and 2PI
294
23.2.5 S-Arcs: The Case
We consider In the error
now
limit, the
term
the limit
v -+
1, still using the change of variables (23.45).
(23.26b) become of the same order as the again, our O(p) approximation fails; it would be
main terms in
O(p). Thus,
here
necessary to go to a higher of the S-arcs in the range
23.2.6
> I
v
approximation to obtain v
a
quantitative description
>
Recapitulation
quantitative method, as developed in the present volume, does not entirely describe the 4 branches of the bifurcation 2P1, and is thus unable to establish their junctions. These junctions, however, have been determined in Chap. 7 on the basis of symmetry considerations (Table 7.2). We reproduce them here in Table 23.2 The
(using
the
new
notations f and g;
see
Sect.
17.1),
so as
to make the
present
volume self-contained.
Table 23.2. Partial
bifurcation, type 2,
n
=
1:
junctions determined by symmetry
from Table 7.2.
2PIS +
I
2P1A -
f
-9
The Tables 19.1, 21.3, and 23.2, taken together, give all junctions for 1 to 5. partial bifurcations of type 2, for'n =
23.3 Conclusions for
Type
2
study of bifurcations of type 2, begun in Chapter 17, is now complete. 1 are given earlier in the present chapter, in Tables 23.1 n and 23.2. For n > 1, branches with a + sign are joined at transition 2*.1 (see Sects. 19.3-3.2 and 20.2.5.2); results are given by Table 19.1 for partial bifurcations and by Table 20.1 for total bifurcations. Branches with a sign are joined at transition 2.2 (see Sects. 21.4 and 22.3); results are given by Table 21.3 for partial bifurcations and by Table 22.2 for total bifurcations. The
Results for
=
-
23.3.1 The Newton
Approach
The Newton approach, which equally well be applied to type
was
developed
2. In the
case
of
in a
Chap. 15 for type 1, could partial bifurcation of type 2,
23.4
have
we
system with
a
Aaj, uj', uj" (i
1,
=
.
.
np
.
,
4n + 1 variables p, AC, hi = 1 nonlinear 4n
=
n),
and MB
(17.66).
Thus here again, MB dimensional manifolds.
to
MB
nB
-
(i
=
295
3
n
-
1),
equations (17.63)
2, and the solutions lie
on
two-
4n + 2 variables, of a total bifurcation of type 2, there are nB 2. nB equations, and again MB Here again, the Newton polyhedra are simple: the fundamental equations
In the
I
=
-
Type
case
=
4n
=
=
-
significant terms, while (17.65) and (17.66) have 4 polyhedra are simplexes, either triangles or tetrahedra. Thus, a detailed analysis of the polyhedra, faces, truncated equations, and normal cones can easily be made as in Sect. 15.3, and used to find the valid combinations in some simple cases, as in Sects. 15.5.4 and 15.8.1. We will not make this analysis here, as it would take up too much space, and also because the Newton approach has the same fundamental shor'tcomings as in the case of type 1 (Sect. 15.9): (i) it can only be applied for a specific value of n; (ii) the amount of work grows exponentially with n (even faster than
(17.63)
and
significant
have 3
1).
for type
23.3.2
(17.64)
terms. The
Proving General
Results
possible to develop for type 2 an approach similar to that preChap. 16, i.e. to prove general properties of the solutions directly from the fundamental equations. We will not attempt this here. Such an approach would be likely to be more involved than for type 1, since the system of fundamental equations is more complicated. It
might
be
sented in
23.4
Type
3
Having completed the quantitative analysis of bifurcations of types 1 and 2, should now logically tackle the last type, i.e. type 3. However, the analysis of type 3 proves to be much more complex than that of types 1 and 2. In fact, this analysis is still unfinished at the time of this writing. Transitions have been identified for the following values of v: we
1
2
-
5
1
-
7
9
n-1
1
1
n-1
2
(23-53)
-
7
4 '4n
-
5
7
3 '2n
-
1 '2 '3
and it is quite possible that other transitions remain to be discovered. This should be contrasted with the single transition at v = 1/2 for type 1, and the two transitions at
the value of In the introduce
of
arcs
v
v
=
1/3 and
depends
case
on
of type 2,
one new
kind of
v
=
1/2
for type 2. Note also that in two cases,
n, the number of basic arcs in the bifurcation. in addition to the S-arcs and T-arcs, we had to arc:
have to be introduced.
the R-arc. For type
3,
at least 12 new kinds
23. Bifurcations 2TI and 2PI
296
In any case, the study of type 3 could not fit into the present volume II. this study should be completed and then published in a third volume.
Ideally But it
seems now
unlikely
that I will be able to fulfill this program. 400 pages of manuscript notes, in french,
I have accumulated about
type 3; any interested colleague is welcome
they
are
worth.
to
a
on
copy ofthese notes, for what
Index of Definitions
This index refers to the section in the present Volume II where a word or expression is first encountered and defined. (See also the Index of Definitions in Volume I for terms
already defined
in that
volume.)
antinode
12.4.2
antinode*
16.6
arc*
16.6
arc
12.1.1
relation
asymptotic equations boundary subset coherent aggregate of boundary subsets cone of the problem cone
arc
critical
bifurcating point
critical
15.2 15.5
15.2 15.5
of truncation
critical
11.4
13.1.4 arc
13.1.4,
21.1.4
15.7
critical R-arc
19.1.5
dimension of the truncated system
15.6
dominant term
15.2, 16.2
encounter
11.3.1
encounter relation
12.1.3
face
15.2
fundamental system of'solutions intermediate arc
15.5.1
intermediate orbit
11.3.2
keplerian keplerian orbit matching relations maximal linear subspace
11-3.2
minimal dominant subset
15.3.1
arc
Newton
Newton
approach polyhedron
11.3.2 11.3.5 15.5.1
15 15.2 12.4.2
node
16.6
node* normal
11.3.2
cone
15.2
O-expression
11.2.1
O-relation
11.2.1
Index of Definitions
298
order
(R-region)
17.5
packets partial transition 2.1 partial transition 2.2. partial T-sequence polyhedral cone polyhedral forward cone power transfor,mation
13.3.2.5, 19.3.1.4, 21.3-1.4
relative side of passage R-antinode
17.1
R-arc
17.5
R-Jacobian
17.5
R-node
17.5.
R-orbit
17.5
R-region
17.5
S-arc
12.4.2
S-arc*
16.6.2.2
set of basic
arcs
set of basic
arcs
17.5 17.7
12.2, 16.8.1 15-5.1 15-5.1 15.7 17.5
11.3.1
(R-region)
set of internal collisions
17.5 11-3.1
set of internal collisions
(R-region)
17.5
simplex
15.3.1
support
15.2
supporting hyperplane
15.2
T-arc
12.4.2
T-arc*
16-6.2.2
Tf
arc
17.1
T9
arc
17.1 16.8
T-sequence total
T-sequence
12.2, 16.8.2
total transition 2.1
17.5
total transition 2.2
17.7
transition
17.5
transition 2.1
17.5
transition 2.2
17.5, 17.7
trident
13.3.3.2
truncated equation truncated function
15.2
truncated system of truncation
valid combination
15.2
equations
15.6 15.2
15.5
Index of Notations
This index refers to the section in the present Volume II where a notation is (See also the Index of Notations in Volume I for
first introduced and defined. notations
already defined
in that
volume.)
Notation
Section
Definition
ai
12.1.1
value for intermediate orbit
Ai
12.1.1
value for intermediate orbit
A
11.3.1
set of basic
bi
15.2
coefficient of
arcs
asymptotic form
for Bruno's variables
B
15
subscript
B, Bj Ci
15.5
maximal linear
12.1.1
value of C for intermediate orbit
C
15.5.1
polyhedral
C
11.3 *1
set of internal, collisions
subspace
cone
d
15.6
dimension of the truncated system
Ej
15.3.1
unit vector
fi 9 1,9It,
15.1
fundamental equations in Newton switching variable
G1, G2, G3
12.4.21 17.4 12.1.4
i"
13.1.2
node
111
11.4
Jacobian
K
12.1.4
K
15.2
L, L2 L3
17.5 17.7
quantity characterizing of the problem auxiliary constant auxiliary constant auxiliary constant
Ma, Mb
12.4.2
values of
MB
15.1
number of
equations
M,
13.1.2
value of
for
nB
15.1
number of variables
A
12.2
number of T-arcs in
ft
17.5
number of basic
N
13.1.2
number of S
N, N'
15.5.1
O(X)
11.2.1
fundamental system of solutions of the order of x
A
15.2
exponent of asymptotic form
17.6
approach
constants
position a
bifurcation
cone
m
m
before and after collision
arc a
or
arcs
T
a
T-sequence a R-region
in
arcs
Index of Notations
300
Q
(ql,.
=
I
qnB 15.2
exponents in
R-region,
a
monomial
formed
by fusion
of S-arcs
R
17.5
Si
12.1.1,
S
15.2
set of
ti
11.3.1
t0i
11.3.2
time of passage near collision i time far from collisions on basic
12.1.1
time of intersection of unit circle
12.1.4 indicates whether collision i is in P in Newton
points
or
Q
approach arc
i
by
intermediate orbit
Tf,
T9
notations for T-arcs
17.1
new
U ' UY
17.2
radial
U.
13.1.2
velocity unspecified arc
U ikd)
15.3
normal
V" VY
17.2.2.1
VX, VY
17.2.2.1
velocity velocity
Wi
15.7
new
12.4 Xil Yi 12.5 W, Xi, Yj X ' X 1' Xil Xil 1, Yi,
Xj" Xi", Xil Xil I
Yi, zi,
yi, zi,
7
=
(Xi'
in
rotating
in fixed
axes
axes
variables in Newton approach
scaled variables for type 1, 0 < nu < scaled variables for type 1, nu = 1/2
1/2
scaled variables for type 2, 0 <
1/3
scaled variables for type 2,
nu
<
nu
1/3
=
scaled variables for type
2, 1/3
<
nu
<
1/2
Yi, Zi scaled variables for type 2,
17.7 X
collision, collision,
Zi
17.6
W, xi, Xi
at
Zi
17.5
X ' X " Xi" Xi"
at
Zi
17.4 W,
cone
nu
=
1/2
XnB) 15.1
variables in Newton
zi
12.1.1
value for intermediate orbit
a
13.1.2
are
r
15.2
Newton
AXj Ai
11.3.2,
12.1.1
approach
number
polyhedron a quantity
variation of
x
for intermediate orbit
15.5.1
coefficient
V
11.4
exponent defining the relative orders of
H
15.5
cone
P
11.3.1
vector from
A
11.3.2
011
17.1
p for intermediate orbit relative side of passage
El
13.1
symmetry in reduced variables Y, X, W
T
15.2
independent
0
11.3.1
second species bifurcation orbit
magnitude
of AC and it
of truncation
M2
to
M3
variable in Newton
approach
References
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(1981):
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L.
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Perko,
an
0(y'),
1 3
<
v
<
1,
near-
Moon passage. Celest. Mech. 16, 275-290. Perko, L. M. (1981a): Periodic orbits in the restricted
neighbourhood
of
a
1st
problem: an analysis in the species-2nd species bifurcation. SIAM J. Appl. Math. 41,
181-202.
Perko,
L. M.
(1981b):
Second
species solutions with
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an
0(y'),
0 <
v
<
1,
near-
