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D-6912i
Michel Henon
Uenerating Families a
J*IIN
in the Restricted
Three-Body Problem
(11 Springer
.1
-
1811,
Author
Michel H6non
CNRS, Observatoire de la C6te d'Azur
B.P.4229 F-o6304 Nice Cedex
CIP data
4, France
applied for. Die Deutsche Bibliothek
-
CIF-Einheitsaufnahme
Hinon, Mchel:
Generating
families in the restricted three-body problem / Michel Berlin ; Heidelberg; New York ; Barcelona ; Budapest Kong ; London ; Milan ; Paris ; Santa Clara ; Singapore
H6non.
-
Hong Tokyo : Springer, 1997 (Lecture notes in physics
:
N.s. M,
ISBN 3-540-63802-4
ISSN 0940-7677
(Lecture
Monographs; 52)
Physics. New Series m: Monographs) Springer-Verlag Berlin Heidelberg New York
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Preface
The work described in this
monograph has grown, somewhat erratically, over period of more than thirty years. My interest in the subject was first aroused by the beautiful computations and drawings in Broucke.'s thesis (1963; see also Broucke 1968), where families of periodic orbits in the restricted three body problem were investigated for the Earth-Moon mass ratio (/.I 0.012155). These drawings suggested that a natural explanation for the existence of the observed families and for the shapes of the orbits could perhaps be found by a
=
a recourse
to the limit y
---+
0.
first step, it appeared necessary to catalog as completely as possible the generaiing orbits obtained in this limit. Generating orbits of the first As
a
specZes had been studied
by
Poincar6
(1892)
and other authors.
Surprisingly,
however, the two other species had apparently been neglected. Orbits of the
collisions, present a comparatively simple problem, using only the equations of the two-body problem; yet no systematic study had ever been done. An inventory of the constituent arcs was presented in H6non (1968). second species,
or
orbits with consecutive
Also very little work had been done on farmlies of orbits of the third to Hill's problem. A numerical investigation was pub-
species, corresponding lished in R6non
My
(1969).
encounter with Pierre Guillaume in 1966 marked the
beginning
of
a
fruitful collaboration. At my suggestion, he started work on the "quantitative" side of the problem, using analytical methods to describe to first order the behaviour of the families of
periodic orbits for small I.L. For my part, I "qualitative" side, using invariants given by symmetries and by Broucke's principle to determine which branches of families were joined in bifurcations. We could then compare notes and verify that our results were in agreement. Pierre obtained his Ph.D. from Li6ge University in June 1971. We continued to correspond until his untimely death in December 1973. In 1975 1 received from Donald Hitzl a preprint describing computations of the criiical arcs, which correspond to an extrernum. in the Jacobi constant C and play an important role in families of second species orbits. I told him that I had myself done, but not published, similar computations. Dori then generously proposed to replace his paper by a joint publication. This led to a very interesting exchange of approaches and results, and resulted in two papers (Hitzl and H6non 1977a, 1977b). worked
on
the
V1
Since 1972 1 have also been in contact with Alexander Bruno, who independently studied the same subject (1972-1981). 1 was so impressed by his work
that,
in
spite of
my meager
knowledge
Russian, I translated two of read them and use them comfortably of
English, to be able to (Bruno 1972, 1973). His papers have now been collected into a book (Bruno 1990), which has been translated into English (Bruno 1994). A continuation of this work has appeared in a series of more recent papers (Bruno 1993a1996). Another high point was the visit of Larry Perko to Nice Observatory from February to May 1977. In his Ph.D. work (1965), Larry had developed powerful mathematical tools for matching different parts of an orbit, and these tools proved to be just what was needed to put the study of second species orbits on a rigorous basis. His visit and our numerous discussions, which continued after his departure, rekindled my interest in the subject and led me to begin the composition of this monograph. During these thirty years there were long interruptions and also periods of discouragement, when the subject seemed too complex and too academic to be worth pursuing. I found that the difficulty of the problem resides not in the use of any deep concepts or sophisticated techniques, but simply in the number and the variety of details which one must keep in mind simultaneously to make any progress. For this reason, I have spent much effort in trying to find, in each place, the simplest method of presentation, as well as the most appropriate terms and notations. 'This is not a mathematical work. Concepts are sometimes advanced on the basis of intuition, or numerical evidence, rather than rigorous proofs. The his papers into
reasoning is sometimes of
Among
a more or
all the
colleagues with would like to thank in particular Roger Broucke, Alexander Bruno, pecially Lawrence Perko, who read
less heuristic nature. whom I have had
helpful discussions,
I
those who have been mentioned above:
Guillaume, Donald Hitzl, arid esearly version of the present work and
Pierre an
made many constructive comments and criticisms. I thank also Uriel Frisch for many stimulating conversations (on this and on all other conceivable subjects), and for encouraging me to finish and publish this work. I seize also this occasion to thank
people whose names should appear colleagues who have unselfishly devoted the development of wonderful software tools, and then put them in the public domain. In particular I wish to thank Donald Knuth, for T X; Leslie Lamport, for LATEX; Tim Pearson, for PGPLOT; and Richard Stallman, for GIWU Emacs. These tools have been of invaluable help to the scientific community at large. In my own modest case, I would never have managed to write this monograph without them. more
often in
some
acknowledgments: those a large fraction of their time to
Nice, September 1997
of
our
Michel H6non
Contents
1.
Introduction
2.
Definitions and
1
.........................
5
.............................................
5
Properties
Notations
2.2
The Restricted Problem:
2.3
Periodic Solutions and Periodic Orbits
2.4
The Perio d-in- Family
2.5
Structure of Families
2.6
Family Segments Symmetry Stability Generating Orbits and Keplerian Orbits Species
2.8 2.9 2.10
Generating
Equations
6
...................................
8
....................................
9
.......................................
12
...............................................
13
...................
14
................................................
1-6
Orbits of the First
Species
....................
21
.................................................
21
Kinds
3.2
First Kind
.............................................
22
3.2.1
The Case
n
I
...........
......................
22
3.2.2
The Case
n
1
..................................
23
...........................................
24
3.3
Second Kind
3.4
Symmetric Asymmetric Orbits Summary Orbits
3.3.2
Generating
................................
26
...............................
31
..............................................
33
Orbits of the Second
4.1
Arcs
4.2
Supporting Ellipses Type I
4.4
5
....................
.............................................
3.1
4.3
.......................
12
3.3.1
4.
......
2.1
2.7
3.
...................
..........................
Species
..................
35
......................................
35
and
...........................
38
................................................
41-
............
Types
4.3.1
S-Arcs
4.3.2
The
4.3.3
S-Arc Families
4.3.4
T-Arcs and T-Arc Families
4.3.5
Overview
Types 2, 3,
..........................................
(A, Z) Plane
4
.................................
...................................
42
48
52
........................
55
........................................
59
...........................................
59
VIII
Contents
4.5
Ends of S- and T-Arc Families
4.6
Extremums of C: Arc
4.7
Orbits
4.8
Second
4.9 5.
68
................................................
73
Species Families Ends of Family Segments
Generating Orbits of the v-Generating Orbits Continuation from 5.2.1 5.2.2
5.3
5.3.2 5.3.3 5.4 5.5
> 0
........................
82 82
...................................
82
1/3
......................
83
.....................................
84
v
v
<
...................................
85
....................................
88
<
Other Families
<
v
--
1/3
1/3
to
v
..........................
=
1/3
....................
88 91 91
........................................
91
........................................
92
........................................
92
........................................
92
...................................
92
...................................
93
5.7
Continuation from
5.8
Conclusions
Orbits for
v
>
1/3
1/3
..........................
1/3
93
....................
94
............................................
94
Bifurcation Orbits
v
-
to
>
v
........................................
Species and First Species
...........................
95 96
6.1.1
First Kind and First Kind
6.1.2
First Kind and Second
............
96
6.1.3
First Kind and Second
...........
97
............
97
...........
97
..........
97
.........................
Kind, Symmetric Kind, Asymmetric 6.1.4 Second Kind, Symmetric and Symmetric 6.1.5 Second Kind, Symmetric and Asymmetric 6.1.6 Second Kind, Asymmetric and Asymmetric Second Species and First or Second Species
98
.................................
99
Total Bifurcation
6.2.2
Partial Bifurcation
...............................
Species Species and Third Species Recapitulation
Third
6.4
Third
and First
96
................
6.2.1 6.3 6.5
v
........................................
v-Generating
6.2
0 to
Orbits for
5.6
First
Species
.....................................
5.5.7
5.5.5
Third
79
5.5.6
5.5.3
76
....................................
=
Family f Family a Family c Family g Family g Recapitulation
5.5.4
................................
Orbits for 0
Continuation from 5.5.2
75
79
Species Second Species Third Species First
v-Generating 5.5.1
6.1
v
.................................
...................
Species Second Species First
v-Generating 5.3.1
6.
Family Segments
62
....................
5.1
5.2
...........................
Second
Species
115
.................
123
..........................
123
.........................................
123
or
Contents
7.
Junctions: 7.1 7.2
7.3
8.
Synimetry Species Bifurcations Third Species Bifurcations Second Species Bifurcations
......................................
125
................................
125
First
7.3.1
Partial Bifurcation
7.3.2
Total Bifurcation
7.3.3
Conclusions
Junctions: Broucke's
...............................
126
..............................
126
...............................
.................................
128
......................................
134
Principle
137
.............................................
137
Definition
8.2
Side of Passage for
8.3
Type Type 2 8.2.3 Type 3 Side of Passage for 8.3.1 Type 1 8.3.2 Type 2 8.3.3 Type 3 8.2.1
I
8.2.2
Node
...............................
139
..........................................
140
..........................................
143
a
..........................................
Type Type 2 Type 3 1
8.4.2 8.4.3
8.5
9.
Type I 8.5.2 Type 2 8.5.3 Type 3 Recapitulation
Fragments 9.1
9. 1.1 9.1.2 9.2
9.3
First
147
..........................................
150
..........................................
151
..............................
152
..........................................
152
..........................................
160
..........................................
162
...............................
162
..........................................
167
..........................................
167
.........................................
167
...........................................
Accidents
.......................................
Explanation Fragment Species Family Segments of the
Tables
9.2.2
First Kind: Direct Orbits
9.2.3
Second Kind
Retrograde
Orbits
9.3.3
......................
..........................
.....................................
Species Family Segments
.........................
Detection of Bifurcation Orbits
Family Segments Fragments Third Species Family Segments 9.3.2
................
...........................
First Kind:
Second
Data
on
162
..........................................
9.2.1
9.3.1
9.4
..........................................
................................................
Introduction
145
147
Results: Total Bifurcation 8.5.1
8.6
Antinode
..........................
an
Results: Partial Bifurcation 8.4.1
127
............................
8.1
8.4
ix
Arc
171
171 172
173 174
174 174 175 179
....................
179
.....................
183
.......................................
190
...........................
196
X
10.
Contents
Generating Families 10.1 Algorithm
......................................
203
.............................................
203
Explanation
205
.......................................
206
10-2.1
........................................
207
10.2.2
........................................
208
10.2 Natural Families
Family a Family b 10.2.3 Family c 10-2.4 Family f 10.2.5 Family g 10-2.6 Family h 10.2.7 Family i 10-2.8 Family I 10-2.9 Family m 10.2.10 Summary 10.3 Other Families 10.4
of the Tables
.........................
10-1.1
........................................
209
........................................
209
........................................
210
........................................
211
.........................................
213
.........................................
214
........................................
214
.......................................
214
.........................................
214
Comparison with Computed Families 1.0-4.1 Family a 10.4.2 Family b 10-4.3 Family c 10.4.4 Family f 10-4.5 Family g 10-4.6 Family h 10.4.7 Family i
.....................
221
........................................
223
........................................
224
........................................
225
........................................
225
........................................
225
.........................................
10-4.8 Families I and 10.5 Final Comments A.
B.
C.
229
........................................
232
........
235
..................................................
235
A. I
Arcs
A.2
Arc Families
A.3
Critical Arcs
...........................................
237
...........................................
240
The Domain D2 B. I
The Curved
B.2
S-Arc Families
..........................................
Boundary in
Type Type 2 C.1.3 Type 3 I
C.1.2 C.2
Type
C.2.2
Type 2 Type 3
C.2.3 C.3
Conclusions
I
............................
245 251
......................................
251
..........................................
251
..........................................
253
..........................................
254
Total Bifurcation
C.2.1
................................
243 243
......................................
Partial Bifurcation
C.I.I.
r
Domain D2
Number of Branches C. I
225
.................................
m
Between Old and New Notations
Correspondence
220
........................................
.......................................
255
..........................................
256
..........................................
258
..........................................
260
............................................
264
Contents
Index of Definitions
...........................................
265
............................................
269
....................................................
275
Index of Notations References
XI
1. Introduction
problem
The restricted
of three bodies has attracted the attention of many was first considered by Euler (1772)
mathematicians and astronomers since it and Jacobi
(1836).
The most obvious
the model of the restricted
problem
reason
number of real situations in astronomy A
deeper
motivation
has not been it
seems
now
found, highly
comes
probably
for this continued interest is that
can serve
(and
from the fact that the
non-integrable dynamical systems
such systems
general
solution
spite of the apparent simplicity of the problem. In fact, likely that such a general solution will never be found:
in
numerical studies indicate that the restricted class of
first approximation in a recently in astronautics).
as a
more
are
known to have
an
problem belongs to the general degrees of freedom, and
with two
inexhaustible richness of detail in the
simplicity, the restricted problem problem for the study of non-integrable
behaviour of the solutions. In view of its can
then
serve
as
a
good
model
systems.
already recognized by Poincar6 (1892), periodic orbits play a fundamental role in the problem: they are "the only opening through which we can try to penetrate the stronghold". Also, periodic orbits constitute the "skeleton" (Deprit and Henrard 1969) around which orbits in general are organized. Accordingly, most studies have concentrated on periodic orbits, and this work is no exception. In particular, since the advent of high-speed computers, many periodic orbits have been computed numerically. (A review can be found in Szebehely, 1967.) So far, however, this numerical exploration has been largely descriptive. The only partial attempts at a synthesis seem to have been those of Str6mgren (1935) for the case of equal masses, and Deprit and Henrard (1970) for the families emanating from the Lagrange point L4There is a need for a more systematic approach, which should ideally: (i) account for the existence of all known orbits; (ii) explain their properties; (iii) classify these orbits according to a methodical scheme; and (iv) predict the existence and properties of as yet undiscovered orbits. One way to develop such a systematic theory is to start from a solvable case of the restricted problem, namely the case where the mass tt of the second body vanishes. In that case, one obtains the problem of two bodies, and a complete description of periodic orbits is possible. One can then try to extend the results to non-zero values of y, using perturbation methods. As
was
M. Hénon: LNPm 52, pp. 1 - 4, 1997 © Springer-Verlag Berlin Heidelberg 1997
1.
Introduction
This is the program of the present work. The idea is of was already the basis of Poincar6's "M6thodes nouvelles"
However, Poincar6 restricted his attention
to
first
species
those for which the minimal distance from the second
course
not new; it
(1892, 1893, 1899). periodic orbits, i.e. body to the third
remains finite in the limit p --+ 0. He was aware of the existence of second species orbits, i.e. orbits such that the minimal distance tends to zero in the limit y -+ 0: they work (ibid., Chap.
are
mentioned and
32).
briefly discussed near study them in any
But he did not
the end of his detail because
"ces solutions s'6cartent trop des orbites r6ellement parcourues par les corps c6lestes" (these solutions are too different from the orbits actually described
bodies) (ibid., p.371; see also Poincar6 1890, p. 269.) We shall however, that second species orbits play a major role in the problem; indeed, they tend to dominate the picture. Besides, second species orbits are of practical interest in space navigation, where close approaches to planets and other bodies are frequently used to change the velocity of a probe without expending energy (this technique is known as "flyby", "swingby", or "gravity assist"). Second species orbits will therefore constitute the main object of the present investigation. Since we shall use a perturbative approach around y 0, the results will in principle be applicable only to small values of y; they will not cover, for inby
celestial
see,
=
stance, the beautiful results of the Copenhagen school for P
1935).
be
=
1/2 (Str6mgren
observed, however, practical applications of the restricted problem correspond to small values of y. Besides, numerical investigations have shown that families of periodic orbits preserve sometimes their main features when IL increases from small to large values, so that we can hope to explain at least some of the discovered families for large P. It will be necessary first to enumerate systematically all generaiing orbits, defined as the possible limits of periodic orbits of the restricted problem for 0. This is done in Chaps. 3, 4 and 5. Much is already known about these p orbits, but in the form of partial results scattered throughout the litterature; the aim of Chaps. 3 to 5 will be (i) to assemble these results into an organized and (as far as possible) exhaustive classification of generating orbits; (ii) to It
can
that most
-- -
present them
so as
to pave the way for the considerations which will follow.
particular, the second species arcs, which are the building blocks of'second species orbits, have been studied in an earlier paper (116non 1968); but they will be presented here in an entirely different way, which offers a better view In
of the whole structure and which is also
more
appropriate
for the
subsequent
work. For
general
a
given
this
value of /-t, periodic orbits form one-parameter families; in remains true in the limit /.i --+ 0, and generating orbits also form
one-parameter families. It is useful point represents
a
periodic orbit;
a
to visualize
family
in orbit space, called a characieristic. Now it more characteristics intersect at a common
or
an
is then
orbit space, where each
represented by a curve happens frequently that two point; in other words, two or
1.
Introduction
families of generating orbits share a common orbit. This will be called bifurcation, and the common orbit will be called a bifurcation orbit. These orbits play a fundamental role in the problem, as will be explained shortly; and our next task is to identify them. This will be done in Chap. 6. In following chapters, we shall consider the vicinity of a generating orbit for y > 0. In the general case of an ordinary generating orbit, i.e. one which is not a bifurcation orbit and therefore belongs to only one family, the situation is comparatively simple; it is represented in orbit space by Fig. 1.1a, where the point 0 represents the generating orbit, and the solid line FO represents the family to which it belongs. For y small but not zero, the family is simply displaced by an amount of the order of p; the new characteristic F,, is represented as a dashed line in Fig. 1.1a. This case has been much studied in the more
a
past and is
now
well understood.
FO F
Y
a
FO
FO
Go
Go
b
Fig.
1.1.
C
Neighbourhood
of
a
FO
11
Go d
0. generating orbit Q. Solid lines: families for /'t a: ordinary generating orbit. b, c, d: bifurcation
Dashed lines: families for A > 0. orbit.
If,
hand, the generating orbit Q is a bifurcation orbit, the complex. Consider the typical case where two families FO and Go intersect at Q (solid lines in Figs. 1.1b to 1.1d). Q separates each family into two branches, so that we have a total of four branches. on
the other
situation becomes
more
4
1.
Introduction
The
problem is then to determine the Junctions between the branches for slightly different from zero. One finds occasionally the situation of Fig. 1.1b: each family is simply displaced (dashed lines). But in most cases the picture is found to be as in Fig. 1.1c or 1.1d: the four branches are joined in a new way, and the characteristics exhibit sharp, hyperbolic-like turns in the vicinity of Q. The continuation of a family through the bifurcation is thus radically different in the cases /.t 0 and /,t > 0. Moreover, the distance from Q is no longer simply of the order of p. When more than two families intersect in Q, the complexity of the problem increases rapidly: for j families, there are 2j branches, which can be joined in pairs in (2j X 1) x (2j 3) x =
-
3
x
I different ways.
Thus, the bulk of this work will have
-
...
to be devoted to
bifurcation orbits and to their
neighbourhood for small p. I have classified the bifurcation orbits into three main iypes, to be defined below; each type must be further subdivided into many separate cases, depending on the nature of the bifurcation orbit; and each case requires a separate study. Thus the whole problem
incredibly rich in its details. objective consists in trying to find simply which branch joins which for p > 0. If this could be achieved for all bifurcating orbits, then the course of families of periodic orbits could be determined for small y, at least qualOne method consists in i.e. itatively. finding invariants, properties which do not change when we follow a family in the vicinity of a bifurcation; then only branches having the same invariants can be joined. A first category of invariants is given by the symmetry properties of the orbits; a second category is provided by what I call Broucke's principle, because it appears to have been explicitly stated for the first time by Broucke (1963, 1968). Roughly speaking, this principle states that the side of passage of the orbit with respect to the second body cannot change. It is shown in Chaps. 7 and 8 that many simple bifurcations can be solved, in the qualitative sense indicated above, with the help of these invariants. However, when the number of families passing through the bifurcation orbit is
A first
increases, the method
soon
fails.
Finally, in Chapters 9 and 10, all the pieces of the puzzle are brought together to determine the course of the generahng famihes, defined as the limits of families of periodic orbits for p 0. These generating families account satisfactorily for the observed families of periodic orbits for small --+
values of y. A planned attack of the the
vicinity
sequel to the present volume will describe a more ambitious problem, consisting in a quantitative analysis of the families in
of
a
bifurcation orbit.
2. Definitions and
Properties
2.1 Notations We
following
the
use
more or
less standard notations for
special
some
func-
tions: -
[xj,
the
integer -
[x],
floor
n
of x, is the greatest < x < n + I n
such that
-
less than
or
to x, i.e. the
equal
ceiling of x, is the smallest integer greater than or equal integer n such that n I < x :! n (ibid.). sign(x) is defined as x1jxI for x :A 0, and is undefined for x 0. H(x), the step function is defined as the
the
-
integer
(Knuth 1973). to x, i.e.
-
--
,
H(x)
for
x
> 0
for
x
< 0
(2.1)
,
.
by sign(x) 2H(x) 1. 0, "f is of order g", means that lf(x)lg(x)l has f(x) 0[g(x)] for x 0. More generally, the 0( an upper bound for x ) notation can be used inside an expression, meaning "a quantity of order ..."; a formal definition can be given in that case by using set theory (Graham et al. 1989, Sect. 9.2). 0 for x 0. This can 0 means that f (x)lg(x) f (x) o[g(x)] for x also be used inside an expression. 0, "f is of exact order g", means that If (x)lg(x) I f (x) 0[g(x)] for x 0 (Graham et al. 1989, has positive upper and lower bounds for x Sect. 9.2). It is related to the notation 0 as follows: f O(g) is true iff and f Ilf O(11g). O(g) For
-
I
0
0, the last
x
two functions
are
related
--
--+
-,
-
-
-
=
...
-*
,
=
--+
-+
,
--
=
=
2.2 The Restricted Problem: We consider the restricted form. The three bodies
are
Equations
(plane circular) three-body problem called
M1, M2, M3. Their
masses
in its usual
are
I
-
/-', Y)
origin in M, rotating axes (x, y) (not in the center of mass) and the positive x axis passing through M2. The distance M, M2 is normalized to unity, so that M2 has coordinates (1, 0). 0, respectively.
We
use a
system of
M. Hénon: LNPm 52, pp. 5 - 19, 1997 © Springer-Verlag Berlin Heidelberg 1997
with the
6
Definitions and
2.
Properties
The coordinates of the third no
confusion
equal
since
simply x, y; this creates are fixed. The gravitational constant is taken angular velocity of the rotating axes is 1.
1; therefore the The equations of the to
2
;i
+
x
-,u
-2i + y
body M3
called
are
M, and M2
-
motion of
lt)xr-
-
(I
_
-3
p)yr
-
M3
3
are
1),0-3
1_,(X
PYP_
-
3
(2.2)
with
VX_-+y 2
r
There exists C
=
(X
2
one
_
fl)2
p
integral +
Vl(--1)2 + y X
-
+
LY
r
of
2
(2.3)
of motion: the Jacobi constant
Y2+
(x, y) system
The
=
axes
-
i
2 -
2
(2.4)
.
P
is the fundamental system of reference for the
three-body problem; its great advantage is positions, so that the equations of motion (2.2)
restricted fixed
that M, and M2 have
are time-independent. study: as will be seen, generating orbits are built in many cases from pieces of keplerian orbits; and these have a more complicated shape in rotating axes than in fixed axes (see for instance Figs. IIIA to 111.7 in Bruno (1972; 1994, Chap. III)). It is therefore convenient to introduce an auxiliary system of axes (X, Y), with the origin in M, and with fixed directions; these directions are defined by
However, this system has
the condition that the
(X, Y)
The
slight
one
(x, y)
defect for the present
and the
axes
system will be referred
abuse of
the center of
language
to
since for Ii
as
(X, Y)
axes
the system
5k 0, M,
has
a
coincide at time t
of fixed
axes.
(This
one
system
(X) (cost -Sint) (X)
circular motion around
to the other is
given by
(2.5)
-
Note in
-
0.
is a
mass.)
The conversion from
Y
--
sin t
particular
Cos
t
.
Y
that the coordinates of M2 in the
(X, Y)
system
are:
(cos t, sin t).
2.3 Periodic Solutions and Periodic Orbits In what
sidered
follows, unless otherwise specified, the parameter p is always conas a given constant, and not as a free parameter: the masses of the
three bodies have fixed values.
particular solution of the restricted problem is then completely deat a given time (we assume here the by the values of x, y, ;i, and existence and unicity of solutions). It is therefore convenient to consider a four-dimensional phase space (x, y,,i, ). A solution corresponds to four given A
fined
2.3
functions
x(t), y(t), i(t), (t),
phase space. instance, define the corresponding point R(O); a
or
Periodic Solutions and Periodic Orbits
equivalently
to
a
function
R(t)
where R is
vector in
Let us, for
solution. A solution will be
0, i.e. by by the values at time t phase space will be called the orZgz*n of the called a periodic solution if there exists a time a
solution
--
in
T > 0 such that
R(T)
R(O)
--
(2.6)
.
periodicity holds only in the rotating axes (x, y). In fixed axes (X, Y), the solution is not periodic in general.) T is called a period of the solution. A periodic solution always has an jT, with j a positive integer, is also a infinity of periods since any T' be Two can cases distinguished: period. 0.) For 1) Equilibrium solution: R is a constant. (It follows that five the this kind: five solutions of a given y, there are only Lagrange points. Any positive number is a period T. 2) General case: non-equilibrium solution. There exists then a Minimal jT0, with j a positive period To, such that all periods are of the form T should be borne in mind that the
(It
--
=
integer.
(2.6) represents five unknowns
are
four
equations
to be
x(O), y(O), i(O), (O),
satisfied;
on
T. The four
the other
equations
hand, are
there
not in-
(2.4),
if
dependent,
however: because of the existence of the Jacobi constant
three of the
Thus, periodic equations are satisfied, really given by a system of three equations for five unknowns; expect that periodic solutions form two-parameter families. This
solutions and is
the fourth is also satisfied.
are
we can
actually
the
case.
of the two parameters corresponds to a trivial transformation of the solution, namely a simple shift in time. Consider a periodic solution But
one
R(to + t), where to period T: then the motion defined by k(t) of solution also is an arbitrary quantity, is a periodic period T. By giving all possible values to to, we obtain an, infinite, one-parameter set of periodic solutions. This set is generally considered as a single object, which we call a periodic orbii. (This definition of periodic orbits is implicit in most studies.) All periodic solutions corresponding to a given periodic orbit are represented by the same curve in phase space; thus, one can visualize a periodic orbit as a closed curve in the four-dimensional phase space (which projects as a closed curve on the physical (x, y) plane). A periodic solution is defined by a periodic orbit, plus an origin. Conversely, a periodic orbit is completely defined by any member of the corresponding set of periodic solutions. In practice, periodic orbits will actually be defined in this way; we shall usually select the solution which has the simplest representation. (Note: in the fixed axes (X, Y), the two solutions R and R' differ not only by a shift in time, R(t),
of
but also
=
by
a
rotation of
angle to.)
One parameter has been thus which we can infer that
eliminated,
and
only
one
remains; from
Definitions and
2.
Proposition
Properties
2.3.1. Periodic orbits
(see
This intuitive result
form one-parameter families.
1974) has been proved by Wintner by the numerical computation of hundreds of families. The remaining parameter, which we call A, is non-trivial. A family of periodic orbiis is described by a function R(t, A). (This definition implies that an origin has been chosen for each orbit of the family. This choice is arbitrary: the same family is described by the function R[to (A) + t, A], where to(A) is an arbitrary function of A.) (1931);
also 116non
it has also been confirmed
2.4 The
Period-in-Family
As has been said
above, a periodic orbit should be considered as having an infinity of periods. For families of periodic orbits, it is useful to change this point of view and to attribute a well-defined period to a periodic orbit considered as a member of a given family. It will be called the penod-in-famZly and represented by T*. The reason is that it is possible to choose T* for each orbit in such a way that T*(A) is an analytic function (Wintner 1931). The fact that T* varies continuously along a family will be useful later. The first idea which comes to mind is to define T* as To, the minimal period. This does not work, however, because of a frequently encountered phenomenon: orbits of a family form two or more loops, which evolve as the family is followed and come to coincide when a particular orbit is reached. The minimal period jumps suddenly, and for that orbit only, to a value j times smaller, with i 2! 2 (Fig. 2.1a). However, for A A0, To tends toward a well-defined limit. This is known as a removable singularity. Thus, the minimal period is not always a continuous function of A. We define instead T* as the new function which is obtained when the singularities are removed (curve F in Fig. 2.1b): --*
Definition 2.4.1. The the limit
of the
period-in-family
minimal
period To
as
T*
one
of
an
orbit in
a
given family
tends towards that orbit
on
is
the
family. We have met here
a first application of a general principle, which we principle of positive definiiion: a definition relating to orbits in a family should not be based on a negative property, such as an inequality ("a is not equal to V). This is because, as one moves along the family, the two sides a and b of the inequality vary, and can become equal at particular points. The inequality property does not hold at these particular points; this introduces undesirable discontinuities. In the present instance, the definition of To is based on a negative property, namely: R(t) is different from R(O) in the
call
whole interval 0
< t
<
To. Therefore To is
not
appropriate
family. Only positive properties, such as an equality, should principle will be used on several occasions.
as
a
period-in-
be used. This
2.5 Structure of Families
T*
TO
F
G
X0
X0
/X
b
a
Fig. a
X
2.1. Variation of the minimal
period To
and of the
period-in-family
T*
along
family.
T*, thus defined, varies continuously along a family F. To is in general T*lj, equal to T*, but can jump to a submultiple at isolated orbits: To with j an integer 2! 2. In such a case, it is invariably found that the orbit belongs also to another family G, in which its period-in-family is To. The situation is represented by Fig. 2.1b, where the period-in-family is shown =
function of the parameter A for both families F and G (it has been assumed for simplicity that A has the same value at the common orbit in both
as
a
families).
that, although really intersect, because
It is essential to realize here
orbit, they do not Therefore, there
a common
not the
change
(Such
a
bifurcation would involve
in the minimal
the
period-in-family is one family to
cannot be any bifurcation from
same.
the other.
the two families share
period To.) The
a
discontinuous and permanent
common
orbit is not
a
bifurcation
period-in-family therefore stipulate that
orbit. This shows that it is essential to include the definition of
a
family
Definition 2.4.2. A
R(t, A) family
periodic
orbits. We
family of periodic T* (A)
and
.
orbiis is
defined by
the two
in the
functions
(2.7)
represented on Fig. 2.1b, where a periodic orbit from a periodic orbit from another family described of interest in their own right. However, they are not relevant present work, and we will generally ignore them.
Occurrences such one
of
as
is identical with
several times,
are
to the aim of the
2.5 Structure of Families
Suppose that we have obtained by some means a finite portion of a family of periodic orbits (i.e. we know the functions (2.7) for a range of values A Al A2), and we try to extend it. What can happen? The principle of
10
Definitions and
2.
natural
termination, obtained empirically by Str6mgren (1934, 1935)
basis of numerical
(1936, -
48),
p.
either the
(which famZly; -
or
Properties
explorations
proved by
Wintner
(1931)
on
the
and Birkhoff
states that
family closes
upon
itself; the characteristic is a closed curve a periodic orbit!). We call this a closed
should not be confused with
family has,
the
which
and
in
each of the two
of the
one or more
following
the dimension D of the orbit
from the
origin),
directions, a natural termination, in quantities grows without limits:
three
(defined
the Jacobi constant
for instance
as
C, the period
the maximal distance
T. We call this
an
open
fa m ily. In the
the
family
towards
original reaches
formulation of the a
principle,
it
was
also considered that
natural termination when the orbit
of the five
as a
whole shrinks
Lagrangian points. However, analysis shows that continued the family analytically beyond Lagrangian point; one recovers then the same periodic solutions, but with a time shift of one halfperiod. It seems therefore more natural to consider that the family of periodic orbits does not end at the Lagranglan point, but continues by "coming back over itself". Following Wintner (1931), we call this a reflectzon of the family. Other examples of reflections are described below. Wintner (1931, pp.334-335) has also shown that a family can be continued in only one way, because of the analytic properties of the equations of motion; a "forking" characteristic, such as represented by Fig. 2.2a, is not possible. A family can intersect another family (Fig. 2.2b); but in that case, each family preserves its individuality through the intersection, and can be continued beyond the intersection in an unique way. the
one
can
linear
be
b
Cl
Fig.
2.2. Continuation of
The
essence
of the
a
family: (a) forbidden; (b) allowed.
principle, then,
is this:
Proposition 2.5.1. Starting from any given orbii along Me family in iwo and only iwo directions.
in
a
family,
one
can move
2.5 Structure of Families
11
now a phenomenon which will be frequently encountered. followed, two loops develop, approach each other, and come to coincidence for a particular orbit Q (upper arrows in Fig. 2.3). (This is an instance of the phenomenon already discussed in Sect. 2.4.) The natural continuation consists then in exchanging the two loops, and coming back along the same sequence of orbits (lower arrows in Fig. 2.3). This will again be called a reflection. (For an example, see Str6mgren 1935, p. 98.)
We describe
As
a
family
is
00
00 <
Fig.
2.3. Reflection of
a
family.
family of periodic orbits comes to an end, new periodic orbits. The answer beyond to this apparent paradox is that the principle of natural termination applies really to periodic soluizons, not periodic orbits (see Wintner 1931). Consider again the sequence indicated by the arrows of Fig. 2.3, this time marking an origin on the periodic orbit so as to transform it into a periodic solution. After the passage through 0, we do not recover the same periodic solutions; on the contrary, since the two loops have been exchanged, we obtain periodic solutions with the origin shifted by one half-period. Thus, there is no problem about continuing periodic solutions. Incidentally, Q consists of a loop described twice, and therefore there exists a periodic orbit consisting of one loop only, with half the period. This orbit generally belongs to another family G of periodic orbits (Sect. 2.4). This analysis indicates that we should really think of the family of periodic orbits as passing through Q and then coming back over itself, rather than as ending in Q. The apparent ending is a simple projection effect, due to the fact that we have projected from the space of periodic solutions to the lowerdimensional space of periodic orbits. (A useful analogy is the transformation of a circular motion into an alternating motion by projection.) As is easily seen, there are four possible cases for a family: (i) a closed family with no reflections. The characteristic is a closed curve. An example appears in Sect. 3.3. (ii) a closed family with two reflections. The characteristic is a finite curve segment, on which one moves back and forth as the family is followed. Ex0.5 (1935). amples are families a and n of Str6mgren for M It would
seem
thus that the
which it cannot be continued towards
=
12
2.
Definitions and
Properties
(iii) an open family with no reflections. The characteristic is a curve ex tending to infinity in both directions. Examples are families 1, m, k, f, r, o of Str6mgren. (iv) an open family with one reflection. The characteristic is a curve extending to infinity in one direction, ending in a point in the other direction. An example is family c of Str6mgren.
Family Segments
2.6
Each family will be identified by periodic orbit, it should then be to which it
only
if
belongs,
we can
find
In order to
a name.
sufficient to
give
the
and the value of the parameter A.
identify
a
particular family
of the
name
However, this works
parameter which varies monotonically along the family. such parameter is known in general. Two or more orbits a
Unfortunately, no family can then correspond
of the
to the
same
value of the parameter, and
additional information is necessary to distinguish between them. The most natural and convenient parameter appears to be the Jacobi constant C, because it is invariant along the orbit; it is thus a common
property of all points of the orbit, and it can be computed at any of these points by a simple application of (2.4). In particular, as will be seen, the value of C is the second
only common property of the species orbit.
In order to be able to
C,
divide each
we
family
distinguish into
Thus,
which make up
between orbits with the
a
composite
same
value of
family segments:
Definition 2.6.1. A which the variation
arcs
family segment is of C is monotonic.
maximal interval
a
in
a
family
in
family segment terminates at each end either in an extremum of C, family. Each family segment will receive a different name. A periodic orbit is then uniquely identified by the name of the family segment to which it belongs and the value of C. in
or
2.7
a
a
natural termination of the
Symmetry
The restricted
problem has
a
fundamental symmetry, defined
by the
trans-
formation E
(I X, y,;i, Y, t)
:
-*
(X,
-Y'
-;i' ' t)
.
(2-8)
E has the property that it transforms any solution into another (or possibly the same) solution. It can be interpreted as a symmetry with respect to the x
axis,
plus
a
reversal of time.
2.8
Other
E",
-
will
-
..
always
mean
a
appear;
the fundamental symmetry E.
"symmetric" unfortunately
The word
13
they will be designated by E', symmetry without specifying which, it
symmetries will occasionally
When reference is made to
Stability
has two different
meanings.
"Q is
symmetric" means that Q is invariant under some given symmetry E; but "the symmetric of Q" means the object EQ, a quite different notion. Both meanings will be needed, and this could result in some awkward sentences such as: "if Q is not a symmetric orbit, then the symmetric orbit is not a symmetric orbit"! I propose to solve this difficulty by reserving symmetric for the first meaning, and using symmetrical for the second meaning. Thus we have: "Q is symmetric", and "Q' is symmetrical of Q". The word "asymmetric", on the other hand, can refer only to the first meaning: "Q is asymmetric" means that Q is not invariant under E. A periodic orbit Q is either symmetric or asymmetric under E defined by (2.8). As is well known and easily proved, a symmetric periodic orbit intersects the x axis twice perpendicularly, at times separated by one half-period. So far, most studies of periodic orbits have been limited to symmetric orbits, mostly because their treatment was easier, but also because many simple orbits of interest fall into this category. As one moves towards more complex orbits, however, an increasing proportion falls into the asymmetric case. This work treats all orbits, whether symmetric or asymmetric, on an equal footing. (This is one of the points where it differs fundamentally from the work of Guillaume
(1969-1975b) and Perko (1965-1983), who restricted their atorbits.) This will be possible because the techniques to
tention to symmetric
be used
apply equally
well to both cases; in
fact, they make
no
distinction
between them.
family of periodic orbits can also be symmetric or asymmetric under possible for the orbits to be asymmetric while the family, taken as a whole, is symmetric; thus, three cases are possible: A
E. It is
-
-
-
symmetric family of symmetric orbits; symmetric family of asymmetric orbits; an asymmetric family of asymmetric orbits. a
a
Instances of all three
2.8
cases
exist.
Stability
keep this work within reasonable bounds, the linear stability of the principle not be considered. In some cases, however, a discussion of the stability introduces itself naturally during the study of a bifurcation, and helps in understanding the relations between the families in the vicinity of the bifurcation. This happens because changes in stability are intimately In order to
orbits will
in
related to intersections of families.
Definitions and
2.
14
Properties
some basic definitions and properties (see 116non 1965b, 1969; Guyot 1970; Szebehely 1967). The stability can be conveniently studied by considering a surface of section. For instance, we consider the successive intersections of the orbit with the axis y 0, and for each intersection we plot a point in the (x, i) plane. For a given value of C, the restricted problem is equivalent to a mapping of this two-dimensional surface of section into itself, and a periodic orbit reduces to a fixed point (or a cycle) of the mapping. The periodic orbit is stable if and only if the -fixed point is stable in the iterated mapping. We consider an infinitesimal displacement (dxo, d;io) with respect to the fixed point. After one mapping, the displacement has become (dxl, d ,), which is related to the initial displacement by
We recall
H6non and
=
(c d) (d o)
dxj dil
b
a
dxo
(2.9)
-
The coefficients a, b, c, d of the matrix can be computed by integrating the variational equations. The mapping is area-preserving, therefore ad
-
We call
bc
I
(2.10)
.
stabilRy a
z
=
index the
quantity
+ d
(2.11)
=
2
The value of the
stability index is an intrinsic property of the orbit: it does not depend on the surface of section which is used to evaluate it. For a symmetric periodic orbit, it can be shown that a d, and therefore the stability index is simply z a. The fixed point is stable if Jzj < 1, unstable if Jzj > L When a family of periodic orbits is followed, the quantity z varies continuously. We call critical orbit of the first kind an orbit for which z +1, and crZiical orbit of the =
=
--
second kind
an
stability
instability along
and
Proposition extremum
in
orbit for which
z
the
=
family.
2.8.1. A critical orbit
C
or
to
an
-1. Critical orbits
It
can
of the first
separate intervals of
be shown
(H6non 1965b)
kind
intersection with another
corresponds eiiher family.
that to
an
Proposition 2.8.2. A cribcal orbit of the second kind corresponds to a pseudo-intersecizon with another family wiih twice the period. R is also an extremum in C for this other family.
2.9
Generating
Orbits and
Keplerian
What has been said up to now (Sects. 2.2 to we concentrate on y = 0 and its vicinity. As stated
2.8)
Orbits was
valid for any p. Now
in the Introduction, we define a generating orbit as a possible 0. More precisely, let us represent by QA an periodic orbits for y orbit of the restricted arbitrary periodic problem for a given p > 0; then
limit of
---*
2.9
Definition 2.9.1. Q i's
orbits
Q., defined for
Similarly, for y
-4
Orbits and
generating orbit if there
0 < p :! po, such that
define
we
a
Generating
a
generating solution
QjL
--+
as a
Q
Keplerian Orbits
exists
for
a
y
limit of
--+
set
15
of periodic
0.
periodic
solutions
origin of specified (compare Sect. 2.3). other hand, we define keplerian orbits as orbits of the restricted the particular case p 0. The equations (2.2) reduce then to
0. It
can
also be defined
as a
generating
orbit
on
which
an
time has been
On the
problem
in
=
i--2 +xwhich
are
Y
x
Y
r3
(2.12)
r3
simply the equations of the two-body problem
seen
in
rotating
axes.
Generating orbits should not be confused with keplerian orbits: generating 0; and correspond to p 0, while keplerian orbits correspond to p these two cases are very different. This is a typical property of singular periurbation problems (Bender and Orszag 1978). In fact, there exist keplerian orbits which are not generating orbits; generating orbits which are not keplerian orbits; and generating orbits which are also keplerian orbits. The point should be emphasized, because it has sometimes been a source of confusion in the past. People have intuitively assumed that the limiting form of a solution of the restricted problem for [t 0 must be a solution of the two-body but this fact the not is in problem; case, because of the singular perturbation nature of the problem. (Poincar6 had already perceived this very clearly, witness his description of second species orbits in Chap. 32 of the M6thodes orbits
=
---,
--
Nouvelles.) expression "generating orbit" is somewhat confusing, since it seems imply that this is an "orbit", in the sense of an orbit of the restricted problem; and this is not necessarily the case. However, this designation is traditional, and we shall retain it. By analogy with Proposition 2.3.1, we have The
to
Proposition
2.9.1.
Generating orbits form one-parameier families.
intuitively clear, and will be confirmed when we will make a deinventory of generating orbits (Chaps. 3 to 5). However, these families defined in two different ways. The resulting families are in general be can very different (see Chaps. 9 and 10), and should be carefully distinguished. Therefore we adopt different names for them: This is
tailed
generating orbits are obtained from the peri0, thus obtaining generating orbits, for p > 0 by first letting p and then grouping these generating orbits in families.
Definition 2.9.2. Families of odic orbits
-*
Definition 2.9.3. riodic orbits
for
a
Generating families are obtained by first grouping given value p > 0 into families, and then letting p
the pe--)-
0.
16
2.
Definitions and
Properties
Families of generating orbits will be somewhat orbits will be
loosely defined; generating
a natural way. This does grouped according convenience, a problem because families of generating orbits will be only an intermediate step (see Chap. 9). As a consequence, these families will not necessarily obey the properties established in Sect. 2.5 for families of periodic orbits of the restricted problem: they may end in other ways than a natural
in
to
not constitute
termination
reflection.
or a
Generating families are more strictly defined, since they are limits of families of periodic orbits which are themselves well-defined. They follow the rules of Sect. 2.5 (see Chap. 10). As in Sect. 2.6, we will need to divide families of generating orbits into segments separated by extremums in C, so that the variation of C is monotonic in a segment. We obtain thus family segments of generating orbits, by application of Definition 2.6.1.
2.10
Species
We divide
now
pens when y
1)
-*
generating 0
(R6non
orbits into three and
all points of the orbit remain at
effect of M2 vanishes in the limit M orbit.
2)
species, depending
on
what
hap-
Guyot 1970)-+
0,
a
finite distance from M2. Then the
and the
generating orbit is
a
keplerian
points of the orbit tend towards M2 while others remain at a finite generating orbit is then made of a succession of arcs, an arc being defined as a part of the generating orbit which begins and ends in M2. Each passage through M2 will be called a collision. The effect of M2 vanishes in the limit y 0 for all points of the orbit except the collisions; therefore each arc is a keplerian arc, satisfying the equations (2.12) of the two-body problem. As we shall see, the arcs join generally with non-zero angles at the collisions; the generating orbit as a whole is then non-analytic. This case is some
distance. The
-*
often called
3)
an
orbit with consecutive collisions.
all
points of the orbit tend towards M2. Then the generating orbit reduces to the point M2. Beautiful examples of these three cases can be seen, for instance, in the numerical explorations of Broucke (1963, 1968). Poincar6 (1892) used the expression solutions de seconde esp ce (usually translated as second species) for case 2. He apparently never defined explicitly a first species; but from his book it is clear that this should correspond to case 1. Case 3 was not considered by Poincar6. I will follow a suggestion made originally by Broucke (1965) and call it third species, by a natural extension of Poincar6's classification. Poincar6 introduced also the notion of sorte, usually translated as kind, first and second kinds are subdivisions
not to be confused with species. The
of the first species;
see
Sect. 3.1.
2.10
Species
17
The definition of
1977b):
let
p'A
species could then be formalized as follows (Perko, p"A be the minimal and maximal distance from M2 to
and
points of an orbit QIL and pO and p 0' their a generating orbit Qo; then QO is of ,
limits
as
y
-+
0 and Q A tends to
first
species if joO > 0; second species if pO 0 and =
po'
third species if
=
'
joO >
0;
0.
definitions, however,
These
(see
definition
2.4),
Sect.
do not agree with the principle of positive because they involve inequalities, and this causes
First, when a family of first species generating orbits is happen that for isolated orbits of the family the distance po falls to zero (the family "crosses" M2); these isolated orbits would have to be excluded from the family according to the above definition. This would be unnatural and inconvenient. We substitute therefore the following positive some
inconveniences.
followed,
it
can
definition: Definition 2.10.1. A
keplerian This
orbit
(i-e.
an
generating orbit belongs to the first orbit of the two-body problem).
species
if
it is
a
definition encompasses the old one, since pO > 0 implies that keplerian. In addition, a first species generating orbit can now eventually pass through M2; but its motion must then continue as if M2 did new
the orbit is
not exist.
Similarly, when a family of second species generating 0 for isolated members: the orbit can happen that joO '
=
above definition would force
family.
us
orbits is
followed,
it
shrinks to M2. The
to exclude these isolated members from the
We therefore substitute the
following positive
definition:
Definition 2.10.2. A generating orbit belongs to the second species if at least one of its points coincides with M2. This includes the
case
where all
Finally, the above definition for tained; we reformulate it as
points
third species
Definition 2.10.3. A generating orbit sists of the point M2 alone.
Generating they can differ
orbits of the third
(see
2.4.2).
Definition
in their
period T*,
coincide with M2.
belongs
species
are
is
positive
to the third
not
and
can
species if
be
ii
necessarily identical,
which is part of the definition of
an
re-
con-
since
orbit
One consequence of our new definitions is that the three species are no longer mutually exclusive. This is natural, and reflects the fact that families of
species can intersect, i.e. share a common orbit. A generating orbit belong simultaneously to the first and to the second species. A generating
different can
18
Fig.
2.
Definitions and
Properties
2.4. Set relation between
species 1, 2,
3.
species belongs also to the first and to the second species. species is represented on Fig. 2.4. We have defined precisely the three species for generating orbits. It will be sometimes useful to speak more loosely of first, second, and third species periodic orbits for y small but not zero, defined as being in some sense close to generating orbits of the corresponding species. This corresponds to Poincar6s use (1892, Vol. III, p. 364): "Nous sommes ainsi conduits h penser que les solutions de deuxi6me esp6ce existent et que, si Fon fait tendre p vers z6ro, orbit of the third
The set relation between the three
elles terident h
se
r6duire h des orbites
avec une
s6rie de chocs". In most cases,
periodic orbit is well defined in practice if /.i is sufficiently small. The general case is that of Fig. 1.1a, for which we shall naturally consider orbits of F. as being of the same species as those of F0. However, the definition becomes ambiguous in the vicinity of a bifurcation orbit where two families of generating orbits of different species F0 and Go intersect (Fig. Llc or 1.1d): for y > 0, there is a smooth transition from one species to another (dashed lines). This suggests that it is not possible to give a strict definition of species for ji > 0. For small u, orbits of the first species are only slightly different from the neighbouring generating orbits; they can be viewed as perturbed keplerian orbits. Orbits of the second species, on the contrary, change radically. The generating orbit exhibits an angle, i.e. a discontinuity of the slope, at each collision (Fig. 2.5a): it is non-analytic. A neighbouring orbit for y > 0 makes a sharp turn, approximately described by a hyperbolic motion around M2 (Fig. 2.5b); but the motion is now analytic. This qualitative change of character is again characteristic of a singular perturbation problem (Bender and Orszag 1978). Orbits of the third species change even more radically: the generating orbit is reduced to the point M2; but the neighbouring orbits for M > 0 are approximated by solutions of Hill's problem (see Chap. 5), which is a non-trivial, non-integrable dynamical system; they cannot be analysed in terms of the two-body problem, and can in fact be obtained only through numerical integration (see for instance R6non, 1969). the
species of
a
2.10
M2
M
0
2.5.
Vicinity
of
a
collision for
19
M
>
Fig.
Species
>
a
2 2
*
a
0
second species orbit.
A family of generating orbits (Definition 2.9.2) IS naturally formed by grouping orbits of the same species. We define thus a first species family (resp. second, third) as a family of generating orbits of the first species (resp. second, third). On the contrary, a generating family (Definition 2.9.3) does not generally belong to a definite species; it frequently bifurcates from one species to another (see Chap. 10).
Generating
3.
Orbits of the First
Species
3.1 Kinds We embark
now
upon
a
systematic
enumeration of the
generating
orbits of the
first species. This is a classical problem (Poincar6 1892); however, a complete classification has been achieved only recently with the work of Bruno (1976;
1980a; 1994, Chap. VII)
on
asymmetric orbits. A review of the results
up to
can Hagihara (1975, pp. 264 to 339). According to Definition 2.10.1, a generating orbit (or solution) of the first species must be a keplerian orbit (or solution). It must be periodic: this excludes hyperbolic and parabolic solutions. Let e be the eccentricity. Poincar6 distinguishes between orbits of the first kind (premi re sorie), characterized 0, and orbits of the second kind (deuxi me sorte), characterized by by e 0 < e :! 1. The latter definition, however, conflicts with the principle of positive definition (Sect. 2.4), and this produces the usual inconveniences: a 0, family of orbits of the second kind can contain isolated members with e
be found in
1975
=
=
which should then be excluded. We amend therefore the definition of the to make it
positive. periodic in rotating axes. If e > 0, then a consideration of the radial motion of M3 shows that during one period, it must describe an integral number of revolutions on its elliptical orbit in fixed axes. Therefore, after one period, M3 occupies again the same position in fixed axes. It follows that M2 describes also an integral number of revolutions, in fixed axes, during one period. These properties will be used for the definitions which we adopt: second kind
so as
The orbit must be
Definition 3.1.1. A generating orbit kind if it is a circular orbit.
of the first species belongs
to the
first
generating orbit of the first species belongs to the second if A/12 and M3 each make an integral number of revolubions in fixed axes during one peri.od.
Definition 3.1.2. A kind
Again, a consequence of this altered definition of the second kind is that the two kinds are no longer exclusive; and again, this corresponds naturally to the fact that families of the first and second kind
be sometimes useful to
can
generating given speak more loosely of orbits of the
These definitions have been
M. Hénon: LNPm 52, pp. 21 - 33, 1997 © Springer-Verlag Berlin Heidelberg 1997
for
intersect.
orbits. Here first
or
again,
it will
second kind
22
Generating
3.
Orbits of the First
for y small but not zero, defined orbits of the first or second kind.
Species
being
as
in
some sense
close to generating
3.2 First Kind A circular
keplerian
solution is
the direction of rotation
C' and
(iii)
c',
defined,
defined
+1
if the motion is
-1
if the motion is
in fixed axes,
by (i) the radius
a;
(ii)
as
direct, retrograde;
the
(3-1)
0. It will be convenient to angular position 00 of M3 at time t variable: the motion a mean replace by single n, defined algebraically as the angular velocity of M3: a
n
=
We
c'a- 3/2
(3.2)
take any value except
n can
x
=
andc'
=
in 1-2/3 cos[(n
distinguish
two
3.2.1 The Case
We have X
=
a
00
n
=
y
,
1)t
+
In
rotating
Ool
y
=
axes, the solution
in 1-2/3 sin[(n.
-
1)t
is
+
then
Oo]
.
(3-3)
cases:
stationary
Cos
-
zero.
I
solution: =
sinoo
(3.4)
infinity of solutions of this sort exist since 00 is a free parameter. However, qualify as a generating solution, a solution (3.4) must be a limit of periodic solutions of the restricted problem for p 0. Consider the five Lagrange equilibrium solutions for p > 0; in the limit y 0, each of them becomes a solution of the form (3.4), with the values of Oo given by Table 3.1. An to
---
,
Table 3.1.
Stationary solutions
obtained
as
limits of
Lagrange equilibrium points:
values of Oo.
Li
L2
L3
L4
L,5
0
0
T
7r/3
-7r/3
Therefore
we
have generating solutions for these four values of Oo. Numer-
ical studies suggest that no other values are possible; i.e. if periodic solutions tend to a solution of the form (3.4) for [t --+ 0, then Oo must have one of the four values
0, 7r, 7r/3, -7r/3. This observed fact does proved, however.
not
seem
to, have been
3.2 First Kind
00
For
0, IV13 coincides with M2
--
at all times. This
case
23
corresponds
to
generating orbits of the third species, which will be considered in Chap. 5. The period T can have any value for a stationary solution. However,
generating solution. For Oo 7r, linear vicinity of L3 have a definite period, which tends to 27r for p 0 (Szebehely 1967, Chap. 5). Thus we have here a seemingly isolated generating orbit, consisting of the point L3 associated with the period 27r; actually, however, this orbit belongs to family ET, of symmetric generating orbits ofthe second kind (see Sect. 3.3.1.2 ). For 00 7r/3, linear analysis shows that periodic orbits in the vicinity of L4 fall into two categories: shori-period and long-period orbits (Szebehely 1967, Chap. 5). The period of the long-period orbits tends to infinity for 0: this is a case where periodic orbits do not have a proper limit for p 0, i.e. there is no corresponding generating orbit. The period of the 0. So here again we have a seemingly short-period orbits tends to 27r for isolated generating orbit-, but in fact it belongs to family Ell of asymmetric generating orbits of the second kind (see Sect. 3.3.2). A similar analysis applies to the. case 00 -7r/3, not all values of T
analysis
shows that
correspond to periodic orbits
a
=
in the
-*
=
=
3.2.2 The Case
The solution
is
n
then
:A
1
circular motion in
a
parameter corresponding and
00
we can
-_
x
to
define the orbit
a
axes. Oo is now the trivial given orbit (see Sect. 2.3), arbitrary value for it, for instance
rotating
shift in time for
by choosing
an
a
0: --
In 1-2/3 cos[(n
On the other
hand,
-
1)tj
is
n
a
,
y
=
non-trivial parameter.
one-parameter family of orbits, with values except 0 and 1. In order to qualify
In 1-2/3 sin[(n n
as
a
-
I)t]
(3-5)
parameter;
(3.5)
.
defines therefore n
can
a
take all real
generating orbit, (3.5) must be a limit of periodic 0. It has been shown that this is problem for ft Birkhoff the case (Poincar6 1892; 1915; Hagihara 1975, p. 266). generally The proof fails, however, for some isolated values of n, namely for all values as a
orbits of the restricted
,
of the form
j
1
(3-6)
-1. Interestingly integer (0 and 1 excepted), and also for n orbits along the the bifurcation to correspond precisely enough, orbits families of with other i.e. intersections to generating family, (see Sects. treatment orbits 6.1.2 and 9.2-2). These bifurcation require a special (Guillaume 1969, 1974); they are in fact also generating orbits, but with properties very different from those of ordinary generating orbits.
with
j
any
these values
=
24
Generating
3.
Orbits of the First
(3.5) represents
The net result is that n, with the
exception
continuous families of 1 <
<
n
of the values
generating
The radius
oo
Species
n
a
n
=
1. Thus
we
have three
orbits of the first kind:
is then smaller than
a
orbit for all values of
generating
0 and
--
1,
that this
so
can
be called
family of direct interior circular orbits. We represent this family by the symbol 1-di. This is the family of direct exterior circular orbits, which we represent by the symbol Id, This is the family of retrograde circular orbits, which we represent by Ir. the
0 <
< I
n
<
-oo
< 0
n
These three families
are one-parameter families of generating orbits, in agreement with Proposition 2.9.1.
The minimal ing
period corresponds
to
one
revolution
the circle
(in
rotat-
27r
TO
(3.7)
-
In- 11
This is
a
continuous function of the parameter period-in-family T* is also given by
therefore the
The Jacobi constant is deduced from C
2'n- 1/3 +
=
Figure
In 12/3
a
n
inside each
minimum C
(3.8)
.
=
3 for
(Sects. 2.6, 2.9)
and
(3.7).
3.1 represents the variations of T* and C with
segments
family,
(2.4):
is monotonic inside each of the three families
has
on
axes):
n
is
-_
1),
so
that
no
n.
The variation of C
Idi, Id, Ir (the function (3.8) further subdivision into family
necessary.
3.3 Second Kind
elliptic keplerian solution is defined, in fixed axes, by the four elements: semi-major axis a; eccentricity e; argument of pericenter, W; time of passage at pericenter, to; and also by the direction of motion c, defined by (3.1). (c' is undefined for a rectilinear solution, e 1.) It will be often convenient to replace e and E' by a single variable e', which we call the co- ecceniri city, defined by An
=
el
=
C/XTI
e' takes values
-
e2
(3.9)
the range -I :! e' < 1. This eliminates the above-mentioned a rectilinear solution we have e' = 0. From e', one can go
in
indefiniteness: for back to e
=
e
and c'
V,
_
by
e,2
c'
-_
sign(e')
(3.10)
3.3 Second Kind
25
8
6
4
2
0 C -2 -5
-3
-4
1
-2
0
1
2
4
3
5
n
Fig.
3.1.
Period-in-family T* (dotted line) and
functions of the
The function
C
=
motion
mean
sign(x)
2 V"a- e'
+I
Jacobi constant C
(solid line)
as
n.
has been defined in Sect. 2.1. The Jacobi constant is
(3-11)
.
a
by (3.9) seems preferable to the variable e' (1973; 1994, Chap. IV, Sect. 2.2), because it introduces 0 (see comments on this point in Bruno, ibid.). Also, (3.9) appears in many equations; this is related to the
The variable e' defined
c'(1-e)
by singularity at e' the right-hand side of fact that the angular momentum is simply- /a_e/. A generating orbit of the second kind must be such that M2 and M3 each make an integral number of revolutions in fixed axes during one period. In the general case e > 0, this must be true for any period (see Sect. 3.1). So it is true in particular for the minimal period To. We call 1 the number of revolutions made by M2 in fixed axes during one minimal period, and J the number of revolutions made by M3 in fixed axes during the same time. I and J are positive integers. I and J are mutually prime: if this were not the case, there would exist a smaller period, contrary to definition. The angular velocity of M2 in fixed axes is 1, therefore used
Bruno
no
=
To
--
The time taken and
(3.12)
21rl.
semi-major
by M3 axis
to make
are
one
revolution is then:
27rl/J;
its
mean
motion
26
Generating
3.
Orbits of the First
J
Inj
a
-_
I
-
J
Species
)2/3
(3.13)
All
properties of an orbit should vary continuously along a family. This applies particular to I and J. However, I and J are constrained to be integers. It follows that I and J are constani along a family. Therefore To is constant-, and the period-in-family is also constant, and given by (3.12). For a given family, characterized by fixed values of I and J, a is fixed and given by (3.13b). The time of passage at pericenter is the trivial phase in
parameter, and is eliminated if We
are
to be
be
a
we
consider
orbits instead of solutions.
now
left with families of orbits with two parameters:
e
and
w.
This
seems
However, the requirement that a generating orbit should limit of periodic orbits for y 0 imposes a specific relation between e too many.
one
--
and w, The
as
will be
seen.
of symmetric and
cases
asymmetric orbits require separate
treat-
ments.
3.3.1
Symmetric Orbits
It
be shown
(Arenstorf 1963)
symmetric elliptical orbit with ratiogeneral generating orbit. The proof fails for isolated those which orbits, namely pass through M2. Here again, these excluded orbits turn out to be bifurcation orbits: they belong also to families of the can
nal
mean
motion
is in
that
a
a
second species. These bifurcation orbits will be considered in detail later. neighbourhood for M > 0 is very different from the case of ordinary
Their
generating orbits, Orbits with to be due to
a-
and requires a special treatment. e2 are also excluded from the 1
3
-
-
technicality
a
rather than to
a
proof;
but this
seems
fundamental difference of nature
for these orbits.
A symmetric orbit intersects twice the
x
axis at
right angles (see Sect. 2.7).
Each of these points corresponds to an extremurn of the distance r to the origin M1, and therefore must correspond either to the pericenter or to the
apocenter of the ellipse. We select
choosing
the
intersection
origin
points
a
of time in such at t
=
particular solution (see Sect. 2.3) by a
way that
0. Since the fixed and
M3 is in
rotating
one
of the two
axes
coincide at
that time, the fixed X axis coincides with the major axis of the ellipse; the argument of the pericenter, w, is either 0 or 7r. We shall study the families of orbits first
major
fixed
in
consider,
We
axis a
b
(where they are simpler), then in rotating axes. (X,Y), the family of orbits with given serni-
axes
and with the
major
pericenter
axis
+1
if the
-1
if the pericenter has
We define then cos
axes
in fixed
--
e'
,
an
angle sin
0
has
modulo ce
.
coinciding
with the X axis. We define
a
positive abscissa
a
negative
abscissa
(w (w
=
=
0), 7r).
(3-14)
27r, by
(3.15)
3.3 Second Kind
27
V) is given, the three quantities e, c, c' are defined, and therefore the orbit is completely defined. We have therefore a one- parameter family of orbits, with 0 as parameter, in agreement with Proposition 2.9. 1. When 0 is increased by 27r, the orbit is the same; so the family is closed upon itself. In Fig. 3.2, each individual orbit of the family is represented in the fixed (X, Y) system of axes; and the family itself is displayed in a meta-plane, with the orbits arranged on a circle and with 0 as angular coordinate. The closed nature of the family is apparent. The abscissa and ordinate of the meta-plane are respectively e' and ce, as shown by (3.15). (A similar figure appears in a different context in Deprit 1983.) If
e
Family of symmetric orbits of the second kind in fixed axes, arranged on 0 as parameter. The dot represents Mi. The circular orbit of M2 is represented.
Fig. a
3.2.
circle with
not
family includes the direct and retrograde obtained respectively for V) 0 and V) 7r. The Jacobi constant is, according to (3.11), The
--
C
=
2Va- cos 0
+
a-'
It increases towards the Let so
us
circular orbits of radius a,
=
(3-16)
.
right
on
Fig.
3.2.
define
+1
if M3 has
a
-1
if
M3 has
a
positive abscissa at time t negative abscissa at time t
=
=
0 0
(3-17) .
28
Generating
3.
The solution X t
a(so
--
-
E
Species
then be written in
can
cos
a3/2 (E
--
Orbits of the First
0)
sin
-
so sin
0
sin
where the parameter E
Y
,
E)
parametric form:
aso
=
cos
0 sin E
(3.18)
,
the eccentric anomaly, in a slightly generalized corresponds to cc" in that paper). We go now to the rotating axes (x, y). The solution is obtained from (3.18) and (2.5), and has a somewhat complicated form; but it is possible to derive the structure of the families by considering simply the two perpendicular intersection points. At time t 0, the rotating axes coincide with the fixed 0 axes, and M3 is on the x axis. Its abscissa is given by (3.18), where t 0: implies E
(see
form
H6non
1968;
is
so
=
--
--
Xo After X
xo
=
X,
with
and also
on
an
the
equals I?r, and E equals J7r; M3 given by (3.18):
t
is then
again
on
the
abscissa
a[so(-I)j
=
(3-19)
sin
-
half-period,
one
axis,
a(so
--
-sin
axis
x
0]
(this
(3.20)
,
is the other
perpendicular intersection)
with
an
abscissa x,
(-I)'+Ja[so
--
Two
cases
Sect.
5).
must
-
(-I)j sin V)]
be
now
(3.21)
.
distinguished (cf.
3.3.1.1 1 + J odd. xo and x, have
the xo
origin
of time
by
one
half-period,
Bruno
1978b; 1994, Chap. V,
opposite signs. By eventually shifting reduce the problem to the case
we can
>O,xi
--
a(I
-
sin
0)
,
xi
=
-a[l
-
(-l)j sin
(3.22)
Two subcases must be
odd
(Fig. 3.4).
They
distinguished: I odd, J even (Fig. 3.3); and I even, J figures show the variations of xo and xi along the family.
show that two orbits with different values of
different into
The
ir
-
fixed axes, are also different in 0, the same values are found for in
0 (modulo 27r), which are rotating axes. (When 0 is changed
xo and xj,
but the orbit
is
not the
a different value.) So in rotating axes we have given by (3-16) again a closed family (without reflections) with V) as parameter. This family will be represented by Ejj. The variation of C along the family, given by (3.16), is not monotonic, and as explained in Sect. 2.9 we must distinguish two 27r respectively. family segments, corresponding to 0 :! 7P :! r and 7r :! V) Using (3.22a), we define these two segments as follows:
same:
C
has
Defliaition 3.3. 1. For I+ J
of the
second kind with
odd, the family segment E,j
semi-major
2.niersection poZnt in the interval 0 :!!
axis x
<
a a.
-
(I/j)2/3
is
and
the set a
of
orbiis
perpendicular
3.3 Second Kind
Definition 3.3.2. For I+ J odd, the
of
the second kind with
2.ntersection
These
point
semi-rnajor
in the interval
family segments
are
a
:!
family segment El'j
axis x
a
(1/j)2/3
-
29
of orbits perpendicular
is the set
and
a
< 2a.
indicated
on
Figs.
3.3 and 3.4.
X0
a
0 -a
xi
0
<
Fig.
21T
Tr
-
-
x-
-
3.3. Variations of xo and x,
-
-
along family Eii
for I odd and J
even.
x0 a
0 -a
xi 0
2 -Tr
7r
1P
<
Fig.
-
-
-
E'.'Ii
3.4. Variations of xo and x,
3.3.1.2 1 + J
prime,
and
even.
e
E
Ii
along family Eli
for I
I and J must then be odd since
>
even
they
and J odd.
must be
mutually
30
Generating
1
X0
=
a(so
sin
-
b)
and x, have the
x0
families
two distinct
Species
Orbits of the First
xI
,
same
in
a(so
--
+ sin
0)
have, in the present case, corresponding respectively to so > 0
sign so. Therefore
rotating
axes,
(3.23)
.
we
E,+j
(Fig. 3.5a)
and Ejj and so < 0 (Fig. 3.5b). We call these two families respectively. In each family, if 0 is replaced by -V), the same orbit is obtained in
rotating
(3.18)
and
reached at
axes, with
(2.2). 0
=
a
one half-period; this is easily verified with family is a closed family with two reflections, 7r; and each family is entirely described by the
shift of
Therefore each
0 and at
V)
=
values of 0 in the interval 0 :! 0 :! on this interval, it corresponds following definitions:
7r.
to
Since the variation of C
one
family segment.
is
monotonic
We have thus the
odd, the family segment E,+j is the set of axis a semi-major (1/j)2/3 and two perpenof dicular intersection poinis wZth a posiii've abscissa.
Definition 3.3.3. For I and J orbits
the second kind with
--
odd, the family segment E j is the set of orbits of the second kind with semi-major axis a (1/j)2/3 and two perpendicular intersection points with a negative abscissa.
Definition 3.3.4. For I and J
--
X1 a
0
0
0
-a
0
X1
0
27
7T
1
E+
E-
1i
li
Fig.
27T
7T
3.5. Variations of xo and x,
along
families
E,+, (left)
and
ETj (right)
for I
odd and J odd.
A
case
orbit for
of special interest is I
same
orbit
as
=
J
=
1.
Family
E'1+1
0: A/13 coincides with M2 at all times.
0 Lagrange point L3 (Sect. 3.2.1) --
JA/12,
but
is
for
0
--
includes
a
Family ET,
third species includes the
0: M3 describes in fixed
diametrically opposed.
axes
the
3.3
3.3.2
Second Kind
Asymmetric Orbits
It will be convenient here to take the
origin of time in one pericenter. The corresponding solution in rotating axes parametric form by at
x
=
a[(cos E
y
=
a[(cos E
t
31
=
3/2 a
(E
-
-
e) cos(zu e) sin(zu
e sin
-
E)
t)
-
-
t)
-
e'sin E sin(w
+ e'sin E
cos(w
-
-
of the passsages is then given in
t)] t)] (3.24)
.
The time
elapsed between two consecutive passages at pericenter is 27rl/J; therefore, rotating axes, the points of the orbit corresponding to successive passages at pericenter are separated by angles -2-xllJ. Since I and J are mutually prime, these points form a regular polygon with J vertices. We select for the origin of time the vertex with the smallest angular coordinate; since fixed and rotating axes coincide at t 0, it follows that in
=
0 :!
27r zu
<
(3.25)
*
i
It is of interest to
study also the pattern of the passages at apocenters in elapsed between a passage at pericenter and the rotating axes. next passage at apocenter is 7rl/J. During that time, the angular coordinate of M3 in fixed axes has increased by c'7r. Therefore, in rotating axes, the angle between a passage at pericenter and the next passage at apocenter is 7r(c' 11J) (7rlJ)(c'J 1). This is an even or odd multiple of 7r/J, whether on I+ J is even or odd. Thus the points representing the depending in at apocenter rotating axes form also a regular J-sided polygon; if passages I + J is even, the vertices of the two polygons are aligned; if -T + J is odd, they alternate. Fig. 3.6 shows examples; many other cases are illustrated in Bruno (1972; 1994, Chap. 111). (The correspondence between Bruno's notations and mine is given by: I p, J p + q.) For given I and J, an orbit is completely defined by the values of e' and zu, i.e. by a point in the rectangle The time
=
-
-
--
I <
C'
< 1
=
0
,
27r
(3.26)
<
given by (3.24), with a and e defined by (3.13b) and (3.10a). As is easily seen, families of symmetric orbits are represented in this rectangle 0 and su by the two line segments zu r/J; when I + J is odd, these two line segments correspond respectively to the two family segments El'j and E ej; when I + J is even, they correspond respectively to the families E+ the motion
is
--
and
ETj.
Asymmetric zu
=
<
27r/J.
represented by
27r/J
-
zu);
orbits
are
represented by points with 0
The fundamental symmetry E it
a is
point (el, zu)
into
an
therefore sufficient to
(see
orbit
study
Sect.
<
2.7)
zu
<
7rlJ
or
transforms
7rlJ
an
<
orbit
represented by the point (e', the interval 0 <
zu
<
7r/J.
As
3.
32
I
Fig.
3, J
=
3.6.
of the First
Generating Orbits
1
4
=
=
Species
2, J
=
3
of orbits of the second ldnd in
Examples
1
=
rotating
1, J
axes,
=
3
showing
the
arrangement of pericenters (dashes) and apocenters (open circles).
generating gives Unfortunately, this relation involves an integral which can only be evaluated numerically. Several authors have worked on this problem; the most thorough search for asymmetric orbits appears to be due to Bruno (1976; 1980a; 1994, Chap. V11), who examined more than 1000 combinations of 1 and J. The results suggest strongly that the condition that the orbit should be
already explained, rise to
-
-
a
relation between e' and
asymmetric generating orbits
of the second kind exist
only
for J
=
I ;
for every value of I (and for J -- 1), there exists one and only one family of asymmetric generating orbits of the second kind. This family will be
designated -
w.
here
by Ejj;
(e', w) rectangle has the shape indicated by 1, Fig. (These figures have been obtained Bruno (1976); for Fig. 3.7b, the value I 6 was given by
the characteristic in the
Fig.
3.7a for I
=
from the tables
3.7b for I > 1.
=
used.) Here again we have one-parameter families of generating solutions, in agreement with Proposition 2.9.1. For I 1, the family El', includes the short-period orbits emanating from the Lagrangian point L4 (point e' 1, w ir/3), and it has a reflection at --
--
=
family ET, (point 02 on Fig. 3.7a), and it has another a closed family with two reflections. For I > 1, E11 is a closed family without reflections; it intersects twice the family EI, or E j (depending on whether I is even or odd) at the points Q, and 02- All characteristics are symmetric with respect to the line w 7r, as a consequence L4-
It intersects the
reflection at L5. Thus it is
e
a
=
of the basic symmetry E.
(3.11) shows
that the Jacobi constant C is
a
linear function of e'. Therefore
family segments, symmetrical of each other, must be distinguished We do not name them because they will not be needed. each family Ej. _r two
on
3.4
27T
Summary
2 -7T
E
L5
a
E
E-
E 7T
or
a
E
7T
2
2
L4
+
0-
33
Ell
+
0
Ell
or
0
E',, 0
e'
e'
>
Fig.
3.7. Families of
3.4
Summary
We have found the
-
-
asymmetric orbits of the second kind.
following
families of first
First kind: three families Idi,
Second
kind, symmetric:
species generating
orbits:
Id, 1,.
for every
pair
of
positive, mutually prime integers
I, J:
-
odd,
family Eii;
-
if I + J is
-
if I + J is even, two families
Second
one
kind, asymmetric:
By a natural extension, faTnihes.
E,+j, E j.
for every
positive integer 1,
one
family EIl.
these families will themselves be called first
species
Generating
4.
Species
Orbits of the Second
4.1 Arcs
According to Definition 2.10.2, a generating orbit of the second species passes through the point M2. Such a passage will be called a collision. Since the orbit is periodic, it has an infinity of collisions. (Note that there can be more than one
collision per period.) The collisions separate the orbit into pieces, which arcs. Two consecutive arcs join at a collision; their tangents at the
call
we
collision form in
an
angle, generally
different from
rotating axes, will be called deflection This chapter consists essentially in a
are
known,
it is
will be seen,
we
Proposition We call them
a
simple
zero.
This
angle,
angle. study of individual
matter to assemble them into orbits
arcs;
measured
once
arcs
(Sect. 4.7).
As
have
4.1.1. Arcs
families. Care should
arc
generating orbits,
form one-parameter families.
nor arc
At first view it
might
be taken not to confuse
families with families of natural to define
seem
generating
an arc
as
orbit which goes from one collision to the next; i.e., a finite such that its ends coincide with M2 while all other points
arcs
with
orbits.
that part of the piece of the orbit are
distinct from
piece of a keplerian orbit (see Sect. 2.10). However, this definition contains a negative element (inner points must be distinct from M2) and thus contradicts the principle of positive definition (Sect. 2.4), with the attendant consequences: as an arc family is followed, the arc may occasionally encounter M2 between its end points, and isolated members would have to be excluded from the family. Therefore we substitute the following positive definition: M2. It follows then that
Definition 4.1.1. An ends in
a
an arc
arc
is
a
is
a
finite piece of keplerian
orbz*i which
begins
and
collision.
The stricter concept will also be
Definition 4.1.2. A basic
arc
two ends.
M. Hénon: LNPm 52, pp. 35 - 77, 1997 © Springer-Verlag Berlin Heidelberg 1997
is
needed,
an arc
however:
which has
no
collisions except at its
36
Generating
4.
Orbits of the Second
Species
keplerian orbit to which an arc belongs will be called the supporiing keplerZan orbit of the arc. It will be convenient to consider that this orbit is described by a fictitious body, which we call M4. By definition, M4 coincides The
with M3 for the duration of the
Thus,
an arc
have
can
under consideration.
arc
one or more
case, the motion must continue
intermediate collisions. But
beyond
the collision
as
such
in
if M2 did not
a
exist-,
i.e., the deflection angle must be zero. A second species generating orbit can be decomposed into basic arcs in only one way. In the general case, the deflection angles between the successive basic
arcs are
non-zero; this will be called
an
must then coincide with basic arcs, and the
unique.
If
ordinary generating orbit. Arcs decomposition into arcs IS also
deflection
angle is zero, however, then the two basic arcs before belong to the same keplerian motion, i.e. they have the same supporting keplerian orbit. This is easily shown: the velocity of M3 has the same direction before and after the collision; it has also the same modulus, because of the conservation of the Jacobi constant C, given by (2.4); therefore M3 has the same position and velocity before and after the collision. The two basic arcs can then be considered as forming either one or two arcs, and there are two possible decompositions of the orbit into arcs. More generally, if n deflection angles vanish (for one period of the orbit), there are 2' possible decompositions of the orbit into arcs. As we shall see, second species orbits with vanishing deflection angles are in fact bifurcation orbits; and the non-uniqueness of the definition of the arcs in that case is naturally related to the fact that two or more families intersect there. a
and after the collision
We consider for
orbits
moment
a
of the second
-
generating period: Ul) U2 Each
solution
i
...
is
generating solutions
species.
We
defined
by
use
the
symbol
sequence of p
a
rather than generating U to represent an arc. A -
arcs
making
up
up
one
full
(4.1)
Uj in turn is completely defined by its supporting keplerian by the times tj' and tj" of the initial and final collisions. The supporting keplerian solution is defined by the position and velocity of M4 at some given time t. It is natural to choose either t tj' or t tj", since the position of M4 is then known (it coincides with M2), and only the velocity has to be specified. Thus, each arc is defined by tj, tj", and the velocity of arc
solution and
--
M4 either
at t
=
ti'
or
at t
=
We define the duration of
ti"
=
*
an arc
Uj
as
the
length
of time taken to describe
it:
2 -rj
=
t
j"
-
t
j'
(4.2)
(The 1968.)
The
pair of parameters
j
ti
known and does not have to be
coefficient 2 is introduced for
j
1,
>
t1l -
,
is
of the
preceding
arc.
t
compatibility with
.'j, tj"
We revert
can
now
the notations of 116non
be
replaced by the pair tj, Tj. For specified: it equals the final time
to the consideration of
a
generating
Arcs
37
specified
either.
4.1
orbit: then
t,
be
can
arbitrarily chosen,
and need not be
Thus, Proposition
4.1.2. A
generating orbit is completely defined if
durdion and either the iniiial
or
the
final velocZty of
each
we
know the
arc.
study of arcs has been made in an earlier work (114non 1968). supporting keplerian orbit can be an ellipse, a parabola or a hyperbola. The last two cases are quickly dealt with: they are found to correspond to a single one-parameter arc family (ibid., Fig. 3 and Table 1), which we call -oc to C Sh. C varies monotonically on this arc family from C Cp -0.720283. All members of Sh are hyperbolic arcs, except the end arc for C -oc, the arc tends towards a limiting Cp which is parabolic. For C form, which consists of a straight-line motion with infinite velocity from M2 to M, and back. At C Cp, the arc family Sh joins smoothly with a family of elliptic arcs, to be described below under the name Soo. The case of elliptic arcs is much more complex. It gives rise to an infinity of arc families, which exhibit a strikingly involved behaviour (116non 1968). These families will be studied in Sects. 4.2 to 4.6. The supporting keplerian orbit will be called supporting ellipse for short. Since arcs form one-parameter families, it is convenient to represent each arc by a point in a plane; the two coordinates are two quantities chosen among those which characterize the arc. Each arc family is then represented by a curve, again called characterishic. In 116non (1968), the two chosen quantities were r, which is one half of the time taken to describe the arc, and q, which is one half of the corresponding variation of the eccentric anomaly on the supporting ellipse. These parameters were well adapted to a study of individual arcs, because different arcs are generally represented by different points in the (r, q) plane (see Fig. A.1). Arc families are therefore well separated and easily studied. These parameters were also used in two subsequent papers (Hitzl and 116non 1977a, 1977b). In this work, however, our concern is with orbits, not arcs; and specifically bifurcation orbits, where all the difficulties of the problem are concentrated (see Introduction). As explained above, for second species orbits, bifurcation A detailed
The
--
-
=
--
..
--
---
-_
orbits appear whenever two or more consecutive arcs share the same supportrepresentation. ing ellipse. This occurence is not easily detected with the
We shall therefore introduce ment: two
point
arcs
with the
plane. given here
parameters, satisfying the following requiresupporting ellipse are represented by the same
new
same
in parameter
entirely independent of that of H6non (1968). arc Ai, Bi, Cij used in H6non (1968) will be replaced here by a new classification, more appropriate to our present purposes. Formulas for the translation of notations are given in Appendix A. The study of arcs with consecutive collisions was extended by G6mez and 0116 (1986, 1991a, 1991b), Howell (1987), Howell and Marsh (1991) to the elliptic restricted problem of three bodies. But their results will not be used The treatment
Also the classification of
is
families
Generating
38
4.
in
the present
restricted
4.2
Orbits of the Second
monograph, problem.
which deals
Supporting Ellipses
and
Species
exclusively
with the classical circular
Types
The discussion in this Section takes
explained
As
at the initial
or
convenient to
in Sect.
4.1,
final collision
an
place entirely in arc is conveniently
(and
the
duration).
It
fixed
axes.
defined is
by
the
velocity
therefore natural and
parameters for
a supporting ellipse the coordinates of points of intersection with the unit circle (the orbit of M2). So we begin with the study of these points of intersection. There must exist at least one such point for collisions to be possible. Therefore the pericenter and apocenter distances, ri and r2, must satisfy
velocity
the
< I <
ri
use
as
of M4 at
r,
one
(4-3)
.
distinguish four types according to the number We
and the unit circle
different -
of
points.
It
this type 1. -I and r2 >
If r,
-
seen
to have very
1,
supporting ellipse intersects the unit circle
or
in the strict
sense
if r, < I and r2
to the unit circle at
one
=
point. Again
(i-e.
1: the
it is
not
a
circle).
at two
We call
supporting ellipse
an
ellipse
is
in the strict
We call this type 2.
=
r2
1: the
=
distinguish -
these four types will be
ellipse
is an
If r,
sense.
supporting ellipses, and by implication for arcs, common points between the supporting ellipse
(Fig. 4-1);
If r, < I and r2 > 1, the
tangent -
for
Properties:
distinct
-
of the
supporting ellipse is identical with the unit circle. We
further:
type 3 if the motion is retrograde, type 4 if the motion is direct. The
four types are mutually exclusive: a supporting ellipse belongs to one only one type. We consider the velocity V of M4 (in fixed axes) at a point of intersection P with the unit circle. V can be defined by polar coordinates V and -f, with the velocity of X,) as origin for the angle -y (Fig. 4.2; in this figure the origin of time has been taken at the passage through P). For type 1, this would give different values depending on which intersection point is considered, so we specify further: the intersection point must be such that V either points towards the inside of the unit circle, or is tangent to it. V and 7 are then uniquely defined for any given supporting ellipse. 7 is limited to the interval and
0 < -Y <
7F
(4.4)
.
In order for the orbit to be
which
is
V 2-
0 < V <
an ellipse, V must be less than the escape at distance 1 from the central body Mj; thus
V2
.
velocity,
(4-5)
4.2
Supporting Ellipses
and
Types
39
Types 1 to 4 of supporting ellipses (solid lines), depending on number of points with the unit circle (dashed lines). For type 2 the two cases a < 1 and a > I are represented separately. For types I and 2, the direction of motion is arbitrary. Cross: Mi.
Fig.
4.1.
common
40
Generating
4.
Fig.
Orbits of the Second
4.2. Definition of the
V,
Species
-y coordinates.
V is most
conveniently represented by a point in velocity space, i.e. by a point polar coordinates (V, -y). The domain allowed by (4.4) and (4.5) is then a half-circle (Fig. 4.3). Points on the half-circular boundary correspond to parabolic orbits and are excluded. Type I corresponds to points inside the half-circle: 0 < V < V2, 0 < -y < 7r. Type 2 corresponds to points on the vertical boundary, i.e. 7 0 or -y 7r, with the exception of the two 1; Type 3 corresponds to the point V points V 1, 7 r; and type 4 0. corresponds to the point V 1, -y with
=
-
=
2 1
14
2
Fig.
4.3.
Types
I to 4 in the
(V, -y) plane (polar coordinates).
4.3
Type
1
41
Conversely, values of -y and V satisfying (4.4) and (4.5), together with point P on the unit circle, uniquely define a supporting ellipse. If P is not specified, then the values of -y and V define a non-orienied supporting ellipse, by which we mean a supporting ellipse which is defined except for its orientation; that is, a, e, and E' are given, but w is left unspecified (see Sect. 3.3). (Explicit equations are given in (4.12) and (4.13).) The expression of the Jacobi constant in the variables (V, 7) is a
C which
C
2 + 2Vcos -y
=
can
3
--
V2,
(4.6)
also be written -
(Vcos7
possible
The
-
-2v 2
-
1)2
_
values of C
(V sin 7)2 are
(4.7)
easily found
to be
for
arcs
of types I and
C
=
-1
for
arcs
of type
C
=
3
for
arcs
of type 4.
<
C
<
3
2,
(4-8)
3,
advantage that it allows a direct reading collisions, quantities of major interest. However, the characteristics of arc families in the plane of Fig. 4.3 are found to be complicated curves, spiralling asymptotically towards the semi-circular boundary, and so crowded together that the figure is difficult to use. We shall therefore simplify it by a change of variables. It turns out that the best way to introduce these new variables is to start now the study of the arcs of type I (the general case), in order to obtain explicit equations for the arc families. Looking at these equations, we shall then be able to select parameters which make the corresponding characteristics as simple as possible (Sect. 4.3.2). The
(V, 7) representation
4.3
Typ e
has the
which
of the velocities at
are
1
For type 1, the supporting ellipse intersects the points, which we call P and Q; specifically,
Definition 4.3.1. P is the
M4
point where M4 penetrates inside the from the unit circle.
unit
circle;
Q
is the
to
point which we have used to define V and 7. It is important remember this convention, which will hold throughout. An arc can begin either at P or at Q, and can end either at P or at Q.
point
Thus,
P
is
where
unit circle at two different
emis
the
We have thus two kinds of
arcs
Definition 4.3.2. A S-arc
axes);
a
T-arc
begins
We define further:
of type 1:
begins
different poinis (in fixed point (in fixed axes).
and ends at two
and ends at the
same
42
4.
Generating
Orbits of the Second
Definition 4.3.3. An
ingoing
We have therefore four -
-
-
-
then define T-arc
the
see
a
for
at
P;
an
outgoing
arc
begins
at
Q.
of type 1:
arcs
that S-arcs and T-arcs form one-parameter families. We could family as one composed entirely of S-arcs, and similarly for
S-arc
family.
definition to
cases
begins
Ingoing S-arc: PQ. Outgoing S-arc: QP. Ingoing T-arc: PP. Outgoing T-arc: QQ. We shall
a
arc
Species
This
definition, however,
(Sect. 2.4)
general type
contradicts the
and would eliminate isolated
1. Therefore
we
define,
arcs
principle
of
positive belong
which do not
loosely:
somewhat
Definition 4.3.4. A S-arc
a
tinuous one-parameter
arcs.
family (resp. T-arc family) i's formed by family of S-arcs (resp. T-arcs), plus any limit
S-arcs and T-arcs have very different
properties.
We
study them
con-
now
in
turn.
4.3.1 S-ares
A S-arc has one extremity in P and the other in Q. These two points are symmetrical of each other with respect to the major axis of the supporting
ellipse (Fig. 4.2). The S-arc may effect several revolutions around the supporting ellipse; in general it intersects the major axis 2p + I times. We call R the central point of intersection, of rank p + 1. It is either at the pericenter or the apocenter of the ellipse. The two halves PR and QR of the S-arc are therefore symmetrical of each other. They are described in equal amounts of time. We call R the midpoint of the S-arc. In rotating axes, the arc begins and ends in M2. The two halves PR and QR have equal durations and equal angles around the origin in fixed axes; therefore they have also equal angles around the origin in rotating axes. It follows that the midpoint lies on the x axis. In addition, the arc crosses the x axis perpendicularly. Therefore, in rotating axes also Proposition 4.3.1. A S-arc is of each other with respect to the
made x
of
two halves which
are
symmeirical
axis.
We derive now the fundamental equations for a S-arc. We take as origin of time the extremity in P (which can be either the beginning or the end of the arc). Since this is a collision, M3 and M4, which coincide for the duration of the arc, have at time t = 0 the coordinates x = 1, -0, and also X = y
Y
=
0
(Fig. 4.2).
velocity
The
V of M4 at
supporting ellipse
P, i.e. by the
the conditions which the
is then
completely defined by
1,
the
parameters V and -y. We derive now supporting ellipse must satisfy for an S-arc to exist. two
Type
4.3
X
a
[(cos E
Y
a
[(cos E
t
to +
--
M4 is given in general by
motion of
elliptic
The
e) cos zu
-
e) sin
-
A(E
sin
e
-
43
1
e' sin E sin zu]
-
+ e' sin E
zu
E)
cos
zu]
(4.9)
,
where a, e, zz7, to are the usual elliptic elements (see Sect. 3.3); e' is the coeccentricity, defined by (3.9); and E is the eccentric anomaly. We have also
introduced here the A
a
--
new
notation
3/2
(4.10)
The quantity A will play a fundamental role. It is equal to the period of the elliptic motion divided by 27r, or to the inverse of the absolute mean motion. For t X
I
=
Y
,
Substituting
in
0
--
(4.9),
k
,
we
--
1962, Sect. 6.12). The results
(2- V2)
--
-1
VI
=
A
V2(2
-
--
0
-
'
,
V
=
Cos
-Y
(4.11)
.
relations, from which the four elements be computed (see for instance Danby
can
are
(2- V2)-3/2
being equivalent
the third relation e
-Vsin7
--
obtain five
and the value Eo of E at t
a
must be
0, the position and velocity of M4
--
to
e'
=
V
v/r2
V2
cos
7
,
(4.12)
(see (3.10)) c'
V2) cos2
=
(4.13)
sign(cos -y)
further, V2
V2 COS2 sin zu
Cos zu
cos
-y sin 7
(4.14)
=
e
cos
V2
Eo
-
I
sin Eo
VV2
as
expression (4.13a)
functions of V and -y; and
to
=
-A[Eo
V2 sin
(4.15)
e
e
where the
-
-
+ V
V/2
-
V2
of
and Eo
should be substituted to obtain
e
finally
(4.16)
sin
where the value of Eo given by (4.15) should be substituted. (4.14) defines zu only modulo 27r, as is natural. Noting that sin 7 is
positive, zu
--
we
obtain
as one
sign(cos 7)
Similarly, Eo
V2 COS2 arccos
always
particular determination I
(4.17)
-
e
is defined modulo 27r
by (4.15), and
we
have the
particular
determination
Eo
V2 arccos
1
(4.18)
44
Generating
4.
The relations
V
=
(4.12) 1
(V, 7)
and
can
/a
be inverted into
ael -y
'
arccos
=
a
Thus, we
the
could have used
(a, e) a
V"2_ a
-
and e' to
(4.19)
I
relation, and describe non-oriented supporting ellipses; as
sets of variables
explained above, however (Sect. 4.2), in the present problem. a
Species
Orbits of the Second
are
in
a
one-to-one
the variables V and -f
are more
natural
The necessary and sufficient condition for the existence of a S-arc is that place at Q, the other point of intersection of the ellipse with
collision takes
circle, at some time t. (t can be positive or negative; if it is positive, we obtain an ingoing arc PQ; if negative, an outgoing arc QP.) P and Q are symmetrical with respect to the major axis of the ellipse (Fig. 4.2); therefore the angular coordinate of Q is 2w (modulo 27r). Successive passages of M2 through Q happen at times the unit
2
20 + 27ra
=
where t20
is
(4.20)
,
the time of
We could take 120
=
passage, and a is an arbitrary integer. when an arc family is followed, W defined However,
particular
one
2w.
by (4.17) presents discontinuities: it jumps from
through
7 passes
eliminated
t20
--
by
2w
a
-
the value
different choice Of 120,
27rsign(cos -f)
Substituting (4.17), t20
=
7r/2.
we
a
v
to
real
-7r or
vice
versa
discontinuity;
it
when
can
be
namely
(4.21)
.
obtain I
-2sign(cos -y) arccos V
A.
This is not
I
-
V2 COS2
V2(2
-
-
(4.22)
V2) COS2,y
The variation Of t20 with 7 is represented on Fig. 4.4, for various values of (It will be convenient to use A instead of V. The two variables are related
by (4.12b).) This figure shows that
our
definition Of t2o has the
following
interpretation: for a direct orbit (,y < 7r/2), t20 is the time of the last passage of M2 through Q before t 0; for a retrograde orbit (7 > 7r/2), 20 is the =
time of the first passage of M2 through Q after t As a consequence, from (4.20) we have that -
-
-
If
a
>
If
a
<
If
a
--
0, the S-arc is ingoing. 0, the S-arc is outgoing. 0, the S-arc is ingoing for
--
0.
retrograde orbit, outgoing
a
for
a
direct
orbit.
Similarly, t4
=
successive
t40 + 27rA,3
passages of M4
through Q happen
at times
(4.23)
,
where t40
is
the time of
Since JW4
is
in
one
P at time t
particular --
0, and
passage, and 3 is
at the
pericenter
an
arbitrary integer.
at time t
--
to,
it will
4.3
Type
45
1
CP.,
0.5
0.9 CN
1--,
0.7
0
0.5 0
-------------------
------------------
A
0.3535
...
-0.5
iT/2
0
IT
'Y
Fig.
4.4.
t20/27r
limits for A
-
oc
as
a
function of
and A
--+
2 -3/2
for various values of A
=
a3/2.
Dashed lines:
46
Generating
4.
be
Q
in
Orbits of the Second
2to because of the symmetry of the elliptic motion (Fig. 4.2). 2to. Here again, however, a different choice
at time
We could therefore take 140 proves to be
t40
--
more
2to
-
=
convenient:
2-r, AH (I
-
A)
we
define
(4.24)
,
H(x) is defined (4.13a), we obtain
where the step function
using (4.18) t40
-
and
2(2
Species
V2)-3/2 arcta..
-
with the convention that
in Sect. 2.1.
Vv/'2---V2 sin -y V2
arctan(oo)
2V
sin
-y
and
(4.25)
2- V2
1
-
Substituting (4.16)
+7r/2.
=
The variation Of t40 with 7 is represented on Fig. 4.5, for various values of A. This figure shows that our definition Oft4o has the following interpretation: if A 2! A <
1) t40 is the time of the first passage of M4 through Q after 11 t40 is the time of the last passage of M4 through Q before t
t
0; if
=
0.
0.5
0
-00
0.3535
A 0
t --
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
...
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0.5
7
-0.5
0.9
7T/2
0
Fig.
4.5.
A
oc
-
As -
t40/27r
and A
a
If 3 >
as a
,
function of
2 -3/2
.
0, the S-arc
is
for various values of A. Dashed lines: limits for
Dotted line: limit for A
consequence, from 4.23
ingoing.
7T
we
have that
---,
1-.
Type
4.3
-
-
0 0
If If
0, the S-arc is outgoing. 0, the S-arc is ingoing for A 2 1, outgoing for A
< --
There will be
only
if and
12
if
t4
:::::::
by
1 2
to the other end of the
(4.20)
and
(4.23)
we
S-arc,
have
(4.27)
by
(t20
t40)
-
-sign(cos -y) arccos
=
V2 COS2
V-1
Vv/2---V2 sin 7
V2)-3/' arctan
_
V2
I
-
V2(2
-
+
-
V2) COS2 (4.28)
2-V2
given by (4.12b). Therefore,
definite function of V,
a
and 3 in
ce
a
-
-(2 A is
Q, corresponding
also be written
can
with Z defined 7rZ
proper choice of
(4.26)
13A
--
collision at
< 1.
-
This condition Z
a a
47
1
for any
given pair
of integers a, 0, (4.27) and (4.28) define a relation between V and -Y, which is the equation of a S-arc family. It can be verified that (4.27) is equivalent
(30) in 116non (1968), when the appropriate changes of variables are made (see Appendix A). This relation is also equivalent to the "timing condition" given by Perko (1974, Equ. (3)); the following relations give the correspondence between Perko's notations (on the left) and my notations (on the right): to relation
a
a
--
T2Q1 rn
=
C
,
C,
,
-6/120
-
a
e
=
T1P2
0+
k
,
(Perko distinguishes unnumbered can
equation
1
-
140 +
H(l
The
(1) 120
(11)
--+
'
A
20 Z
following
A
-,
0 ,
1/2,
t40
'
0
4-y
-
2
(a
,
(iii)
A
=
I
(a
values for the
3 -x
(1
1, V
and
(4.28)
will be useful:
(4.30)
-
v/2-): 3
+ 2
--
0
--
2 3/2
V2-=
(4.22), (4.25) 0):
-+
z
)
140
-
7r
V
oo, V
27r, 1
-4
(4.29)
-
.
simply by allowing negative
limit forms of -+
oo
a)
second case, with a different timing condition (ibid., near the end of page 203); however, this second case
2-3/2 (a 1
a)
-
k.)
and
m
-
27rAH(I
a
be included into the first
parameters
,
(1+2 COS2 -y) sin -y
cos2-y) sin -y
1),
and A
--
,
(4.3'1)
.
1+
(a
--*
1+,
V
--+
1+):
48
t20
Orbits of the Second
Generating
4.
=
2-y
t40
7r
-
7r
=
Species
2 sin -y
-
(-y
Z 7r
(iv)
A
120
-- -
1
-4
27
(a
-
-
-4
t40
7r
2 sin 7
-
Z 7r
S-arc, seen in rotating point, reached at
A
intersects the
arc
axis
x
t
midpoint, which will be needed. coordinates in rotating axes: =
a[(cosE
y
=
a[-(cos
e) cos(t
-
E
-
e) sin(t
-
The time of passage at the t
=
zu
According t
=
From E
to
7rsign(cos -y)
-
rAH(l
-
(4.9c) =
(4.23)
to
we
=
-7rH(I
-a
4.3.2 The
-
A)
+
+ e'sin E
ira
-
A)
+
7ro
to
(A, Z)
axis
now
axis.
(and
the
the abscissa of obtain the
we
,
(4-34)
.
(4.20)
and
(4-21)
have also
we
7rA,3
(4-36)
.
anomaly
at the
midpoint:
(4-37)
.
-
x
(4.35)
values of t and E into
(-1)' [(-1)0 (1
to the
x
(2.5),
and
zu)]
-
the
.
2H(I
-
(4.34),
A))
-
we
e]
obtain the coordinates of
Y
=
0
(4.38)
.
Plane
We notice then that the relation
quantities
cos(t
on
zu)]
-
midpoint is, according
(4.24),
and
esinEsin(t
+
zu)
-
(4.9a,b)
From
find then the value of the eccentric
Substituting the the midpoint x
+
w)
lies
=
-
(4.33)
+sin -y)
symmetric with respect
axes, is
this
x
(-y
t2/2 t4/2, perpendicularly). We compute
Its middle
(4.32)
1
-
I-):
1-, V
7r
+ sin -y)
A and Z.
(4.27)
is
a
linear relation between the two
we use the (A, Z) plane to represent Therefore, families, all characteristics would be simply straight lines! To see if this works, we examine the relation between the (V, -y) and (A, Z) variables. First, there is a simple, monotonic relation between A and V, given by (4.12b); thus, A is essentially equivalent to V as a parameter.
if
could
S-arcs and S-arc
The range of values
2-3/2
_< A <
Next, for
a
This relation is
oo
(4.5)
for V is translated into
a
range
(4-39)
.
given V (or A), we given by (4.28). It
the relation between Z and 7. is shown by Figs. 4.6 to 4.8, for various examine
values of A.
1) Fig.
4.6 shows that for A
tonic. When -y is
equivalent
increases
to -y
as a
>
1, the relation between Z and
from 0 to 7r, Z
increases
parameter. There is
a
7
is
mono-
from -I to +1. Therefore Z
one-to-one relation between the
4.3
Type
1
49
0.5
cp
0
...........
..........
A
1
-0.5
7T/2
0
Fig.
4.6. Relation between Z and
Dotted line: limit for A
-
I+.
for A > 1. Dashed line: limit for A
Tr
00.
50
4.
Generating
Orbits of the Second
Species
0.9 z
7
0.8 0 0.7 F 0.6
0.5
0.5
54
0.4
7A= A
0
... ..
--
-TT12
0
Fig.
3 -55 0.3535
---------------------------------------------------------
4.7. Relation between Z and
Dotted line: limit for A
for A < 1. Dashed line: limit forA
Tr
2-3/2
1-.
0
z
-0.5
v
0
Fig.
7T/2
4.8. Relation between Z and -y for A
=
1.
7T
4.3
half-annulus defined of the
(A, Z) plane
1 < A <
which
oo
Fig. 4.3 by by
on
1
51
0 < -/ < 7r, and the domain
defined
-1 < Z < I
,
(4.40)
,
(Fig. 4.9).
call domain D,
we
V2,
1 < V <
Type
Each
point of Di represents
one
well-
defined non-oriented supporting ellipse. (Note, however, that the actual computation of -y for a given Z requires the inversion of (4.28), which can be done
only numerically.)
I
--------------
r
z
D
0
21
D,
D3 -
0
-
-
-
-
1
-
-
-
-
2
-
-
-
-
3
-
4
A
Fig. 4.9. Domains in the (A, Z) plane. corresponding domains.
2)
For A <
Dashed boundaries do not
1, unfortunately, things
are
not
so
belong
simple: Fig.
to the
4.7 shows
that the relation between Z and -y is no longer monotonic. When 7 increases from 0 to -x, Z starts from 0, increases to a maximum, and then decreases to 0.
Thus, there is
two-to-one relation between -y and Z. We call Z .. (A)
a
the maximum of Z for
a
given A,
and -y,,, (A) the value of -y for which this
maximum is
reached. Then the half-circle defined
0
mapped
< -y <
7r
2-3/2
is
A
<
<
1
twice
0
,
on a
<
Z:!
This will be called domain D2
domain of the
Z,(A)
(Fig. 4.9).
.
Fig. 4.3 by 0 < V (A, Z) plane defined by on
<
1,
(4.41)
We shall consider that it consists of
and Dsheets, D+ corresponding respectively to the domains 0 < V < 2 2 1, 0 < < r. The two sheets have in common 7,,, and 0 < V < 1, their upper boundary, defined by: Z Z,(A), 2 -3/2 < A < 1, and should two
Y
=
be considered in
Appendix
between the
is
shown
simplifies
along
B. With that
(A, Z)
3) Finally, it
as sewn
for A
by Fig.
into
and
boundary will be called F; it is studied convention, we have a one-to-one correspondence it. This
(V, -/)
domains.
1, the relation between Z and -y is again monotonic; 4.8. In that particular case, in fact, the relation (4.28) --
52
Z
7r-, (-Y
-
Orbits of the Second
Generating
4.
+ sin
-Y)
I
-
Species
(4.42)
.
When y increases from 0 to 7r, however, Z increases from -I to 0 only. There is a one-to-one relation between the half-circle defined on Fig. 4.3 by V -- 1, 0 < -f < A
and the line segment in the
7r
1
--
-1 < Z < 0
,
(A, Z) plane
defined
by
(4.43)
.
This line segment will be called doTnain D3 (Fig. 4.9). We have thus, for all values of A, a one-to-one correspondence between
(V, 7)
the
D
and
Di
=
(A, Z) representations; D+ 2
U
consisting
of the domain
(4.44)
D3
U D_ U 2
(A, Z)
Therefore the
the latter
variables
can
ellipses and arcs of type 1. Figs. 4.6, 4.7 and 4.8 show representation across the line A
be used to represent non-oriented support-
ing
I.e.
e*)
a
by
used
Bruno
are in
the
(A, Z)
to be
close relation to the variables
real,
(N
(1973; 1994, Chap. IV):
1
A -,
N
1. This
--
consequence of our The variables (A, Z) used here
it is not
a discontinuity in discontinuity appears particular choice of variables.
that there is
e*
7rz -
sin
e*slgn(P)
2 =
ZZ
cos
-
2
for
a
> 1
for
a
< I
(4.45) .
and negative in domain D-. The quantity P is positive in domain D+ 2 2 characteristics thus sinusoids of the in to correspond straight-line (A, Z) plane the (N-', e*) plane (Bruno 1973; 1994, Fig. IV.14). In the case a < 1, Bruno introduced also two variables (x, y) which give a straight-line representation of characteristics (ibid., Fig. 15); they are related to my variables by The
A x
i so
y
z
=
A
-
that the =
(13
-
Y
)
(4.46)
=
I-A
equation (4.27) of
a)x
-
oz
a
characteristic becomes
(4.47)
.
4.3.3 S-are Flamilies
simple matter to find all S-arcs and S-arc families. As shown above, for a given S-arc, the values of a, #, A, Z are uniquely defined. They satisfy D(4.27). Conversely, if we specify two inte 9ers a and #, a domain D1, D+, 2 2 or D3, and a point (A, Z) of that domain such that (4.27) is satisfied, then we have defined one and only one S-arc. The values of V and -y are defined, It
is now a
therefore t
=
we
know the
0. The other end
the duration of the
is
arc
velocity at t
is
It2 I
at the end of the
t2, with t2 given by
-
-
arc
which
(4.20)
and
corresponds
(4.22);
to
therefore
4.3
The domain D3
can
be eliminated: for A
--
Type
1
53
1, (4.27) shows that Z
must
integer; but (4.43) shows that there are no integer values of Z inside D3. For given 0 and a, (4.27) is the equation of a straight line in the (A, Z) plane, which we call A. We specify also a domain, and we call S,"p that part of A which lies inside the domain. Then every point of S,,6 represents a S-arc. We have thus a one-parameter family of S-arcs, in agreement with Proposition 4. L L We call it arc family S,,3. We find all families by considering the straight line A for all values of a be
and 3. 4.3.3.1 Domain
following
(1) 0
results
D1. This domain has
are
easily proved: exists only
S,,o
0: then
a
simple shape (Fig. 4.9), and
the
for a 0. The family Soo is represented Fig. 4. 10, case 1. Its left end is the point (1, 0) of the (A, Z) plane, on the left boundary of Di; Its right end is at infinity. (The dots and the symbols So, S+, S- in Fig. 4.10 will be explained in Sect. 4.6.) (ii) 0 > 0: then S,,o exists only for a > 0. The intersection of A and D, is a segment of straight line. There are two particular cases: for a fl, the left end is the point (1, 0) (case 2 on Fig. 4.10); for a # + 1, the left end is the point (1, -1), which is a corner of D, (case 3). The general case is a > 0 + 1: the left end is then on the lower boundary of D1, defined by: A > 1, Z -1 (case 4). The right end is always on the upper boundary of D1, defined by: --
=
on
--
=
--
A > 1, Z
=
(111) 0 a
1.
< 0: the situation is the
changed (this
respect
to Z
=
is
0).
same as
in
(ii),
with the
signs
of
0 and
consequence of the symmetry of the domain D, with S,0 exists only for a :! 0. There are two particular cases: a
0, the left end is the point (1, 0) (case 5); for a 0 1, the left end 1: the left end is then is the corner (1, 1) (case 6). The general case is a < 0 on the upper boundary of D, (case 7). The right end is always on the lower boundary of D1.
for
a
=
=
-
-
4.3.3.2 Domain
curved
(i) (ii)
boundary --
0:
no
D2. Here the analysis is IF. The
following
results
families exist in this
complicated because proved in Appendix B.
more
are
of the
case.
> 0: S,,a exists only for 2 -3/2# < a < fl. The left end is on the boundary of D2, defined by: 2 -3/2 < A < 1, Z 0; the right end is on IF (case 8 on Fig. 4.10). -3/ 'P. A particular case is a < 2 (iii) 0 < 0: Sp exists only for the end in is the a which is a corner of D2 (case 9). 3; right point (1, 0), The general case is 0 < a < 2 3/20 (case 10); the right end is then on the lower boundary of D2. The left end is always on the curve IF. and D-. Therefore, each arc family S"'P D2 consists of two sheets, D+ 2 2 consists of two pieces, which are represented separately in Fig. 4.10 (cases 8 to 10). We observe that in every case, these two pieces have one point in common on F. One can pass continuously from one sheet to the other across F; therefore the two pieces should be considered as forming a single
lower
--
=
54
Generating
4.
S
so
0
S-
S
0 L
0
-
Species
Orbits of the Second
-
-
-
2
1
L
-
3
0
0
-
0 -
1
-
-
-
so
-
3
2
0
S
0
1
2
2
3
0
2
1
3
3
4
D,
o
0
S
0
S
so S 0
0 S+
S++
-
0
2
1
3
0
D
ds+0-1'
+
S
2
1
0
2
/l SO
0
0
D-
AS-.,IlI
S
0
3
7
X-S",I'l
O
0
-
-
01 SO, S
0
0 0
1
0 0
8
9
segments
(see
(position 4.6)-
arcs
Sect.
not
1
0
4.10. Cases I to 10 of S-arc families. The abscissa
Z. Dots: critical
-
0
Sol
2
Fig.
3
6
S+1
0
2
1
5
-s
s-
S
10
is
A and the ordinate is
accurately indicated); S-, So, S+:
arc
family
Type
4.3
1
55
continuous
arc family. This arc family S,,,q begins on the lower boundary D+ the corner (1, 0) of D+ in the particular case 9), rises until it at (or 2 2 crosses IF, then enters D2 goes down and ends on the lower boundary of D2
of
,
(or
at the
corner
(1, 0)
Summary.
4.3.3.3
of
An
in the
D2
arc
particular
family Sp exists
case
9).
for any
pair
integers (a,
of
which satisfies
(13=0
a--O)
A
V
(0>0
A
a>2 -3/2/3) V
(#
<
0 A
ce
< 2
-3/20)
.
(4.48) In most cases,
and
# define
single S,,p family, because A intersects only particular case # < 0, a 0, A Actually there is a discontinuity I in the (A, Z) plane, as already mentioned; therefore we have across A here, for given a and 0, two separate arc families. This slight ambiguity in our notations will be removed when we consider arc family segments (Sect. 4.6). In all cases 2 to 10, the characteristic S,,o is an open segment of straight line: the two end points do not belong to domains D, or D2, defined by (4.40) and (4.41). In case 1, the characteristic is an open half-line, and again the end point does not belong to domain D1. The reason for this is simply that as we tend towards one of these end points on a S-arc family, the S-arc tends towards a limit arc which does not belong to type 1. These limit arcs should be included in the corresponding families, according to Definition 4.3.4; this a
a
of the two domains D1 and D2 In the intersects both D, and D2 (cases 5 and 9). one
=
-
=
will be done
in
Sect. 4.5, after
Data for the 10
column
"Bruno"
Fig. IV.18). -
-
-
In In In
cases
we
have studied types 2, 3, and 4. are collected in Table 4.1. The
of S-arc families
refers to
a
similar classification in Bruno
The last three columns will be
explained
(1973; 1994,
in Sects. 4.5 and 4.6.
1, a # 0, the S arcs are ingoing (because A > 1; see Sect. 4.3. 1). there is a > 0, 0 > 0, and the S arcs are ingoing (ibid.). cases 2, 3, 4, 8, cases 5, 6, 7, 9, 10, there is a < 0, P < 0, and the S arcs are outgoing.
case
-_
-_
Many examples of S-arcs are shown in fixed axes in H6non (1968), rotating axes in Bruno (1973; 1994, Figs. IV-1 to IV.11).
and in
4.3.4 T-arcs and T-are Families
The two end
points
of a T-arc coincide. Therefore it must consist of an
number J of revolutions of M4 on the supporting ellipse. time, M2 has also made an integral number I of revolutions It follows that the duration of the
arc
the
and the
supporting ellipse a
=
is
JnJ
=
J11;
is
semi-major
integral
the
same
the unit circle.
mean
motion of
axis is
(4.49)
J
now a
on
27rl; the absolute
(1)2/3
We state
During
fundamental proposition.
56
Generating
4.
Table 4.1. The 10
families
Orbits of the Second
cases
Species
(see Fig. 4.10)
of S-arc families
and the 3
Types of Case
Arc
cases
of T-arc
(see Fig. 4.11)_
Domain
Bruno
Definition
end
arcs
Number of critical orbits
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S S
D, D, D, D, D, D, D,
> 0
1
> 0
I > Ce
0
=
+ 1
=
+ I <
2-3/20
0 <
9
D2 D2
10
D2
0 > 2
T
1
T
2
T
3
D, D2 D3
Proposition
3
p
IV
3
2
1
11
4
2
0
11
2
2
0
111
3
2
1
3
2
2
0 > 0
a
0
=
-3/2#
< 0
# 0 <
< 0
<
#
< 0 >
2
2
2
V
2
2
2
VI
4
3
0
VI
2
2
0
2
2
0
2
2
0
4
3
0
< 0 a
Ce
p
>
I > J I < J
I
=
J
=
1
ordinary generating orbit of the second species identical T-arcs of type I in succession.
4.3.2. An
not contain two
I
can-
A proof of this proposition requires some tools which will only be devetoped in a sequel to this volume, devoted to the quantitative study of bifurcations. Here we give only the flavor of the proof, in the form of a heuristic argument. Assume that there exists an ordinary generating orbit Q0 which contains a sequence of n identical T-arcs, with n 2: 2; we call this a T-sequence. Either all these arcs begin and end in P, or they all begin and end in Q. The deflection angles between the T-arcs are zero. We assume first that the T-sequence constitutes only a part of the generating orbit. Then the deflection angles at the two ends of the sequence do not vanish; if they did, then the next arc would be again an identical T-arc and the sequence could be extended.
Each T-arc has the
origin take place
of time at the
with i
--
values of
beginning
at times ti
I to
same
=
1, J, and a given by (4.49). We take the T-sequence. The successive collisions
of the
21rli, with i
--
0 to
n.
We number also the T-arcs
n.
We consider
small value of p. By definition, there exists a periodic orbit. Each T-arc is slightly perturbed; it generating Q. is approximately replaced by an arc of ellipse with semi-major axis a + Aai, where Aai is a small quantity. The period of this elliptic motion is orbit
2r(a
now a
close to the
+
Aai )3/2
-
27r
I J
+
37rv/a-Aai
(4-50)
4.3
The times at which the orbit
cross
the unit circle
are
also
Type
57
1
perturbed
and
become 27r1i + yi, where the yi are small quantities. The difference between two successive values of yi equals the perturbation in the duration of the T-arc: yi
yi- I
-
37rJv _aAai
--
(4.51)
.
Each collision in Qo is replaced in Q,, by a passage close to M2, which can approximated by a small arc of hyperbola, resulting in a deflection angle. For i I to n 1, this angle must be small since Q/_, is close to Q0. The deflection is inversely proportional to the distance of passage, and therefore to yi. The deflection produces a change in the semi-major axis of M3. A detailed computation shows that, approximately
be
=
-
Aai+l where
v
-
4ya
Aai
2
(4.52)
--
Vyi
is the modulus of the
velocity
of M3 in
a
collision,
in
rotating
axes.
1. are small; therefore yi > y for i = 1 to n Aai+l On the other hand, for i = 0 and i = n, i.e. the two ends of the T-
and Aai
-
sequence, the deflection
angle keeps essentially
had in Q0. It follows that yo = 0(p), y,, be neglected in the equations (4.51). We n
-
multiply and
1,
now
(4.51) by Aai
add the
we
equations.
=
for i
0(p),
=
the
non-zero
value which it
and therefore yo and y"
I to n,
(4.52) by
yi for i
--
can
1 to
All terms cancel in the left-hand side and
obtain
we
n
0
=
37rJVa- )7 Aa 2
+
positive
for
n
2
V
But the first term in the term is
4pa
-(n
-
1)
(4.53)
.
right-hand side
is
positive or zero, and the second an impossibility. Neighbour-
> 1. We have thus reached
ing
orbits cannot be constructed for small p, and therefore the orbit Q0 is
not
generating.
Another way to obtain this result is as follows. Suppose for instance that Aal > 0. Then from (4.51) we have y, > 0. From (4.52) we have Aa2 > Aal.
Then from
(4.51)
for i
=
I
we
yi and Aai constantly increase arrive at a final value Y,, = 0.
A similar demonstration
T-sequence
have Y2 > yl; and so on. The perturbations along the T-sequence, and it is impossible to
can
be made in the
particular
case
where the
constitutes the whole orbit.
We return
study of T-arcs. It follows from Proposition 4.3.2 mutually prime; otherwise the given T-arc could be decomposed into a succession of two or more shorter, identical T-arcs, and therefore it could not be part of a generating orbit. At the difference of S-arcs, T-arcs are not symmetric, either in fixed or now
to the
that I and J must be
in
rotating
axes.
inwards from
one
This is seen, for instance, from the fact that the end point, and outwards from the other.
arc
lies
58
4.
We
Generating
the
can use
Orbits of the Second
Species
(A, Z) plane to represent
T-arcs also. From
(4.49)
we
I
A-
J
have
(4.54)
,
S-arcs, we use the (A, Z) plane to find all T-arcs and T-arc given T-arc, the non-oriented supporting ellipse is uniquely defined, and therefore the numbers 1, J, A, Z are uniquely defined. A satisfies (4.54,) Conversely, if we specify two positive integers I and J, mutually prime, a domain D1, D+, D-, or D3, and a point (A, Z) of that domain such that 2 2 (4.54) is satisfied, we define a unique non-oriented supporting ellipse; however, there are two possible T-arcs, one running from P to P and the other from Q to Q, i.e. an ingoing T-arc and an outgoing T-arc. We distinguish between these two arcs by naming them T' and T', respectively. (The superscripts i and e refer to "interior" and "exterior", i.e. the region into which M4 is moving at each extremity of the T-arc.) For given I and J, (4.54) is the equation of a vertical line in the (A, Z) plane. We specify also a domain, and we call T1j that part of the vertical line which lies inside the domain. Then every point of T,,j represents two T-arcs. Therefore TIi represents two families of T-arcs, which we call respectively arc family Tl'j and arc family Tl'j. (Actually, we shall see later that these two arc families are joined into a single closed arc family when the end points are added.) These are one-parameter families, in agreement with Proposition 4. L L We find all T-arc families by considering the vertical line (4.54) for all values of I and J, positive and mutually prime; i.e. for all positive rational values of A. The situation here is simpler than for S-arcs: we have only three cases to distinguish: (i) I > J, i.e. A > 1: T1j always exists and is a vertical segment in D1, with its two ends on the lower and upper boundaries of D1, respectively (case 1 in Fig. 4.11). (ii) I < J, i.e. A < 1: Tli exists only if As in the
of
case
families. For
a
-
IIJ It
is a
pieces,
the
-3/2 > 2
vertical segment
(4-55)
IF,
D2. Each family Tj'j and Tl'j consists itself of two D+ and D-, with a common point 2 2
in
lying respectively
in the sheets
the case of the S-arc families. Thus, each T-are family boundary of D-2 , rises until it crosses I, then goes down and ends on the lower boundary of D2 (case 2 in Fig. 4.11). 1: Til is a vertical segment, coinciding with the (iii) I J 1, i.e. A whole domain D3; it begins in (1, -1) and ends in (1, 0) (case 3 in Fig. 4.11). The characteristics of T-arc families are again open segments, the two end points not representing arcs of type 1; these end points will be added in on
begins
curve
on
as in
the lower
=
=
-_
Sect. 4.5. Data for T-arc families
are
collected in the lower part of Table 4.1.
4.4
D
Types 2, 3,
0 0
0
0
1
3
2
59
2+ 0
D,
4
D3
D-
0
2
0
2
Fig.
3
4.11. Cases I to 3 of T-arc families.
4.3.5 Overview
We have found that -
-
family S,,p for
a
two families
Tj'j
arc
families of type I consist of
any pair of
and
T'j (I, J) satisfying (4.55).
integers (a, 13) satisfying (4.48); pair of positive, mutually prime integers
for any
gives a general view of the arc families in the (A, Z) plane. (The explained in Sect. 4.6.) As is easily shown, the characteristics are dense everywhere in the domains D, and D2 (this is true also for the S-arc families alone, and for the T-arc families alone). It is therefore impossible to 3 and represent all families- in Fig. 4.12, only S-arc families with -3 T-arc families with J < 3 have been represented. For the arc families S12 and S23, one of the points of intersection with the curve F seems to be the point (1, 1). In reality these points of intersection are slightly below, as shown by the enlargement of Fig. 4.13. The same situation exists for all arc families Sp-i,p (see Appendix A): as 8 increases, the point on IF approaches the point (1, 1) but never coincides with it. This is because the curve IF has an infinite slope in (1, 1) (see Sect. B.1).
Fig.
4.12
dots will be
4.4
Types 2, 3,
The
study
4
of these types is much
simpler.
2, the supporting ellipse arc must begin with that of T-arcs similarity integral number P of revolutions For type
is tangent to the unit circle at one point, which and end at P. Therefore the situation has some
(preceding Section):
the
arc
consists of
an
we
call P. An
60
4.
Generating
Orbits of the Second
Species
F
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
0
N
Fig.
4.12. Characteristics in the
(A, Z) plane.
Dots: critical
arcs.
4.4
Types 2, 3,
4
61
0.98
S12 S23 V
0.96 0.96
0.98
Fig.
4.13. Enlargement of a part of Fig. 4.12. Open circles: intersections of characteristicsS12 andS23 with 17. Dots: critical arcs.
of M3
on the supporting ellipse, integral number P of revolutions the supporting ellipse is given by
during
the
same
the unit circle. The
time M2 makes
semi-major
( )
(4.56)
-
J*
There
an
axis of
2/3
1*
a=
and on
important differences, however: for type 2, P and P do not have mutually prime for the orbit to be a generating orbit. On the other hand, they must be different: P 0 P, since a : 1 for supporting ellipses of type 2 (see Fig. 4.1). We call m the greatest common divisor of P and P: are
to be
P
=
MI
with I and J a succession
P
,
=
mutually
of
m
basic
MJ
(4-57)
,
prime arcs
integers. Then the
(see
Sect.
4.1);
arc
considered consists of
each basic
arc
consists of the
supporting ellipse described J times. (Care should be taken not to confuse the three numbers J, P, and m.) If a > 1, the supporting ellipse is tangent at its pericenter: 1. a(l e) If a < 1, the supporting ellipse is tangent at its apocenter: 1. + a(l e) Therefore e is always given by -
e
_
co
la- 11
(4-58)
=
a
The motion
c'
--
-1
can
be
direct,
(see (4.13b)). Thus,
with 7 an arc
=
0, c'
_-
of type 2
+1, or retrograde, with 7 7r, is completely specified by the =
Orbits of the Second
Generating
4.
62
Species
mutually prime, different m is an arbitrary positive integer; 27rml. Since this definition involves only and c' 1. Its duration is 27rl* integers, arcs of type 2 do not form continuous one-parameter families; they
c', where I and J positive integers, satisfying IIJ > 2- 3/2;
four numbers
1, J,
m,
--
=
are
two
are
isolated from each other. For type
3, the supporting ellipse
is
the unit
circle, described in the
retro-
grade direction. M4 encounters M2 twice per revolution. Therefore, an arc of type 3 consists of a succession of m basic arcs; each basic arc is a half-circle (in fixed axes), described is completely specified by Its duration is
integer.
For type
M4
sense.
Arcs of type 3
supporting ellipse
the
4,
the
m7r.
retrograde direction. Thus, an arc of type 3 single number rn, which is an arbitrary positive
the
in
must have collisions with
with M2 at all times.
Thus,
an arc
also isolated from each other.
are
circle, described in the direct M2; it follows then that M4 coincides of type 4 is completely specified by its is the unit
arbitrary positive real number. Arcs of type 4 form a single one-parameter family, with r as parameter. 3 for type 4, and for that (4.8) shows that the Jacobi constant is C type only. On the other hand, C is the same for all the arcs composing a generating orbit. It follows that if one arc in a generating orbit is of type 4, then all arcs are of type 4, and the whole generating orbit coincides with M2: it is a generating orbit of the third species (see Chap. 5). duration
2,r,
which
be
can
an
=
4.5 Ends of S- and T-arc Families We have
points
now
found all
possible
arcs, and
we
return to the
of the characteristics of the S- and T-arc families
study
of the end
(type 1)
found in
arcs will corresponding 4.3.4, be included in the corresponding families. (1) We begin with T-arc families, for A > 1. We consider a given arc family Tj'j. If we tend towards the lower end point (Z -1), Fig. 4.6 shows that -y tends towards 0. On the other hand, A > 1 implies V > L Therefore in the 1 (see Fig. 4.3). The limit we have a supporting ellipse of type 2, with c' limit arc of type 2 is easily identified: for all arcs of the family, M2 makes exactly I revolutions and M3 makes exactly J revolutions; this must hold in J, and the limit, and therefore the limit arc is characterized by P 1, P the c' 1, T-family, m 1; or equivalently, by the same values I and J as in
limit
and the
Sects. 4.3.3 and
arcs.
These limit
--
=
=
=
--
=
and c'
=
The
1.
same
limit
In the
same
towards 7r, and c'
-
Tiej
-
in
--
the arc is
-1.
family T'j; this is a consequence of the Q become confused in the limit. we tend towards the upper point (Z 1), -y tends limit we have a supporting ellipse of type 2, with
is obtained for
way, if
1. The limit
1, and c'
m
arc
points
fact that the two
P and
_-
characterized
Again
this limit
by
the values I and J of the
arc is common
T-family,
to both families
Tl'j
and
4.5
(ii)
Ends of S- and T-arc Families
63
For T-arc families with A <
1, the results are similar, the only difarc of type 2 with c' being +1 corresponds now to the end point on the lower boundary of D+ (Z while the limit arc of 0), 2 -1 corresponds to the end point on the lower boundary of type 2 with E' ference
that the limit
=
=
=
D2 (Z
=
0
again).
Table 4.2 summarizes the values of c' in the various
cases
of limit
arcs
of
type 2. Table 4.2. Values of c' for type 2 limit
arcs.
C'
Domain
Z
D, D,
+1
-1
-1
+1
D+ 2
0
+1
D2
0
-1
With the inclusion of the two limit arcs, the two families Tj'j and Tj'j can as forming a single closed family, which we call T[j. We define
be considered
7*
the
angle between the velocities of M2 and M4 at the terminal collisions for a V arc (since the collisions happen then at T-arc; thus, and -y* point P), -7 for a T' arc (since the collision happens at Q, and because of the symmetry of the ellipse). The value of 7* specifies an arc of the family. In Fig. 4.14, individual arcs of an arc family Tjj are represented as
of the
--
(X, Y) system of axes (assuming that the initial collision takes 0); and the arc family itself is displayed in a meta-plane, with the arcs arranged on a circle and with 7* as angular coordinate (the positive vertical direction is taken as origin). c' is +1 in the upper half of the figure, in the fixed
place
at t
-1
the lower half. The left half
in
--
to T' arcs; the two limit
and
7*
arcs
corresponds
of type 2
to
are on
T'
arcs, and the
the vertical
right half 0 axis, for
7r.
=
There
superficial similarity between a family of T-arcs (Fig. 4.14) and family of symmetric orbits of the second kind seen in fixed axes (Fig. 3.2). In both cases, the family is closed, and is characterized by two mutually prime integers, I and J, which are the numbers of revolutions of M2 and M3, respectively. But there are also essential differences. In Fig. 3.2, the eccentricity varies over the whole interval 0 :! e < 1, and the family includes two circular orbits; in Fig. 4.14, the eccentricity varies only over the interval is a
a
C0
:!
with eo
(iii) Z 7
=
are
e
< I
(4-59)
,
given by (4.58),
and the circular orbits
For the T-arc families with A
-1 and Z
0 and
7r.
=
0.
Fig. I
=
J
=
are never
two end
points correspond to corresponding limit values of limit supporting ellipses are of types 4 and 3, I on the T-arc families, and the duration of a --
4.8 shows that the
Therefore the
respectively. We have
1, the
reached.
64
4.
Generating
Orbits of the Second
Species
Cx x
x
x
T
x
1
(E)/ x
4.14. A closed arc family Tri, arranged in a circle with y* as parameter. Cross: Mi. Dashed line: orbit of M2. Dot: initial and final collisions. Arrow: initial
Fig.
and final
velocity of M3.
4.5 Ends of S- and T-arc Families
T-arc is 21r.
Therefore,
the limit
arc
in the
point (A, Z)
--
(1, -1)
is
65
an
arc
4, characterized by a duration 27r; the limit arc in the point (1, 0) is of type 3, characterized by m 2 (i.e. formed of two basic arcs). Here
of type an arc
--
again, the two families Tj'j a single closed family.
(iv)
We consider
and
Tj'j, together
with the two limit
points,
form
the end
points of the S-arc families. We consider points (A > 1, Z 1 in Dj; D+ in Z 0 and that the limit values and show 4.6 4.7 1, Figs. D-). 2 2 that the limit arcs belong to type 2. c' is given by are again 0 or 7r, so
first the end A <
now
the four horizontal boundaries
on
--
--
of 7 Table 4.2.
We compute
the parameters P and P of
now
a
limit
arc.
The duration
It2l
S-arc is the absolute value Of t2 given by (4.20); the limit value of therefore equal to 27rI*. Substituting cosy -- c' in (4.22), we obtain
of
a
P
ja -,E'H(A
=
with H defined
-
I)l
(4-60)
,
by (2.1). Next, P
can
be obtained from
P
A
--
(4-61)
-
J*
where
is
value, given by the equation (4.27) of the S-arc
for A the limit
we use
family: + Z
A
with Z Z
=
--
(4.62)
-,E' if A
-c'H(A
>
-
We obtain thus the J
-
(v) to
1131
*
1)
--
0 if A < 1
(Table 4.2);
this
can
be written:
(4.63)
.
simple
result
(4-64)
.
point (1, 0) in the domain DI, corresponding Fig. 4.10. From (4.27) we have: a 0, and 0 can take shows that of is the 4.6 limit value 7r. Therefore the Fig.
We consider next the end
cases
1, 2, 5
on
all integer values.
limit
1, Z
supporting orbit is of type 3. expressions (4.22) and (4.25)
The
I and -y
V
=
m
of the limit
=
7r;
arc.
so
cannot
we
But
Fig.
use
for t2o and t4o
them
directly
are
indeterminate for
to find the
4.4 shows that when 7
--*
7r
parameter 1+
and V
-*
[7r, 27r]. Similarly, Fig, 4.5 shows that the limit Of 40 lies in [0, 7r] (this can also be seen from (4.25)). On the other hand, we must have Z 0. Then (4.28) shows that there must simultaneously,
the limit Of t20 lies in the interval
--+
be
t20
-'
7r
40
The duration of the
and
we
-*
Ir
arc
is
-
(4-65)
given by (4.20) or (4.23), taken in absolute value, m of the limiting arc of type 3:
obtain the parameter
66
Generating
4.
m
11
--
+ 2a I
Orbits of the Second
11
--
201
+
Species
(4.66)
.
(vi) End point (1, 1) in D, : this corresponds to case 6 in Fig. 4.10. From (4.27) we have: a P 1; and 0 can take all negative integer values. Fig. 4.6 =
-
shows that the limit value of -y is 7r. Therefore the limit supporting ellipse is again of type 3. As in the previous case, the limit Of 120 lies in [7r, 27r] and the limit Of t40 lies
t20
-4
27r
[0, 7];
in
t40
,
-2(a
=
+
1)
0
'
-20
--
(4.28)
1. From
--+
have then
we
(4.67)
-
limiting
The parameter of the m
but here Z
of type 3
arc
is
(4.68)
.
(vii) End point (1, 0) in D2 0, and (4.27) we have: a
corresponds to case 9 in Fig. 4.10. From negative integer values. Fig. 4.7 shows that the limit value of is ir: the limit supporting ellipse is of type 3. 7r and V I-, the limit value Of t20 lies in [0, 7r] Fig. 4.4 shows that for -y be from seen (this can also (4.22)); Fig. 4.5 shows that the limit value Of t40 lies in [-7r, 0] (this can also be seen from (4.25)). On the other hand, we must have Z 0; and from (4.28) we find --
:
this
take all
can
-*
0
t20
t40
'
--+
(4.69)
0
and m
-2a
--
-20
=
(4.70)
.
(viii) End point (1, -1) From (4.27) we have a
and
=
0
0
can
in D,
:
this
take all
t40
cases
does not work here: for -Y
---+
-27r, which does
limiting expressions. Computations indicate that in the limit-, therefore we write I + q-y
=
=
27ro
(4.23)
Fig.
4.10.
(4.25)
0 and V
[-27r, -7r],
-
I and y
I+,
-+
Fig. 4.5 -I gives:
120 and t40- In order
arc, it is necessary to
V
--
and
are
of the
expand same
the
order
(4-72)
+ 2 arctan
and
-+
the condition Z
,
where q is assumed to remain finite (4.20), (4.22) and (4.71) give
t2
[0, 7r];
in
not restrict the ranges Of
to obtain the duration t of the
V
3 in
positive integer values. Fig. 4.6 shows that the limit value supporting ellipse is now of type 4. The method used
shows that the limit value Of t40 lies in -
case
(4.71)
4.4 shows that the limit value Of 120 lies
Fig.
to
+ 1
of T is 0: the limiting in the three previous
120
corresponds
I
2q
give
+
2(3q
2
4q2
the limit. Note that -y > 0 and q > 0.
in
+
1)
+ 1
0(_Y2)
(4.73)
4.5 Ends of S- and T-arc Families
4
1
2 arctan
--
+0(_ 2)
+
2q
This is
6q
I arctan
2q
+
dividing by 2-y
after
07r
2
0
=
duration of the
0,
,
we
=
an
=
is 27-
arc
arctan
Values of
+
2q
also
-r are
equation
t2
=
=
and
This
T
for
t4; from
has
+
1)
+ I
to the limit
and
one
I to 10
(4.73)
are
or
07r
0:
only
one
solution for
listed in Table 4.3. The
(4.74), going
to the limit
(4.76)
.
given
in Table 4.3. From
(4.75)
(4.76),
one can
obtain
(4.77)
4
equation, and an
explained
its solutions given earlier study of Hill's in
Chap.
Table 4.3. Solutions of the
q
and
r:
=
countered in will be
4q2
going
3Ttan
2
find 1
7-
2(5q -
(4.75)
implicit equation for q. It fl; solutions for 0
an
67r#q
.
each positive value of
-y
2q
+
(4.74)
1 arctan
+
.
(4.26) gives then, 3q
27r#
67
by Table 4.3, have already been enproblem (116non 1969). The relation
5.
implicit equation (4.75).
T __
1
0.150851
2
0.086781
7.682131
3
0.061311
10.873562
4
0.047475
14.042501
5
0.038757
17.201401
6
0.032752
20.354942
7
0.028362
23.505281
8
0.025012
26.653555
9
0.022371
29.800418
10
0.020235
32.946275
4.419371
(1x) End point (1, 0) 0; (4.27) we have: a
in
D+ 2
this
corresponds to case 9 in Fig. 4.10. From # negative integer values. Fig. 4.7 shows that the limit value of 7 is 0: the limit supporting ellipse is again of type 4. The computation is similar to that of the previous case. Here q < 0. Equations (4.73), (4.74), (4.75) are found to be still valid. (4.75) has one and only one solution for each negative value offl; this solution can be read from Table 4.3 simply by changing the signs of # and q. The duration of the arc is 27 -t2 -4, and (4.76) must be replaced by --
=
=
and
can
take all
68
4.
Generating
Orbits of the Second
I arctan
T-
Values of
(x)
-
-
2q
(4.78)
irg
again given by
r are
Species
Table 4.3.
point of arc family Soo at infinity (case Appendix A, this end consists in a Fig. 4.10). which at Soo joins smoothly with the family Sh of hyperbolic parabolic arc, arcs. (Sh cannot be represented in the (A, Z) plane, because A and Z are defined only for elliptic arcs. Attempts to extend the definitions to hyperbolic arcs result in a complex value for A.) The last
case
to be considered is the end
I in
As shown in
The types of the limit
collected in Table 4.1, columns 6 and 7. 4, except for the limit parabolic arc in paragraph (x), which is indicated by a 'p' in the table. We have now completed our inventory of arc families. The possibilities
These types
are
arcs
always 2, 3,
are
or
are
(i) Particular case: the family of arcs of type 4 (Sect. 4.4). (H) Other particular case: the semi-infinite arc family Soo, consisting of arcs of type 1, except at its end in A 0, which is of type 3 (Sects. 1, Z 4.3.3 and 4.5). (iii) The general case: a finite arc family S,p, TI'j) or T,j, consisting of arcs of type 1, except at its two ends, which are of type 2, 3, or 4 (Sects. 4.3-3, 4.3.4, 4.3.5, and Table 4.1). --
4.6 Extremums of C: Arc
--
Family Segments
was done for families of periodic orbits (Sect. 2.6) and families of generating orbits (Sect. 2.9), we divide now each arc family into segments: an arc family segment is defined as an interval in which the variation of C is monotonic. Each segment will receive a different name. An arc is then uniquely identified by the name of the arc family segment to which it belongs and the
As
value of C. Note: this does not
concern
the
family
of
arcs
of type
4, for which C has
the constant value 3. First
we
must find the extremums of C
on
arc
families. An
sponding to an extremum of C inside an arc family will be arc (Hitzl and 116non 1977a, 1977b). (Note: "critical" is used meaning by Bruno 1994.) As mentioned
arc
called with
a
a
corre-
critical
different
Sect. 4.1, the variation of C is monotonic in the
hyperbolic arc family Sh, which therefore forms a single segment. The case of elliptic arcs is much more complex. C is given by (4.6) as a function of V and - . It is therefore implicitly also a function of A and Z: this function QA, Z) is obtained by eliminating V and y between (4.6), (4.12b), (4.28). Curves of constant C
are
in
represented
A tedious computation
in Fig. 4.15 for the three domains D1, D+, D2 2 gives the slope of these curves at a given point:
4.6
Family Segments
Extremums of C: Arc
69
-2
2 0
-7.6
L
0
1.5 2 2. 5
2.98
2.9 4
3
2
1
A
1
1 C) Q),
z
0.5
0.5
4tD
C
C3
0
1
0
1
0.5
0
:
0
1 1
Fig.
A
4.15. Curves of constant C in the
Lower Lower left: domain D+. 2
1
0.5
A
right:
(A, Z) plane. Upper figure:
Dotted line: limit domain D-. 2
domain Di.
curve
IF.
70
4.
7r
Generating
(OZ) aA
=:
C
+(-V8
[(-2V6
+
+(-2V5
Orbits of the Second
7V6
+ 8 V3
-
V2(2
_
-
-
4V4) COS4 7 + (2V7
11V4
_
V2_ 3V[1
+
Species
_
-
80
+ 8
V3) COS3,Y
V2) Cos2
8V) COS 7
(_V6
+
+
4V4 -3 V2
V2 arctan
V2) COS2,y] Siny
Vv/2-
+
4)]
V2
-
V2
Sin
-Y
1
(4.79) We consider first the
case
given by (4.27).
teristic
of
a
the characteristic is tangent to
( 0z) OA
family S,,,p,
S-arc
An extremum in C a curve
C
with its
occurs in _-
Const.,
this
straight-line characarc family whenever
i.e. when
(4.80) C
can be verified that this equation is identical with the one given by Bruno (1973; 1994, Equs. (IV.44), (IV.45)). (Bruno's k corresponds to my 0.)
It
(4.27)
(4.80)
and
form
a
system of two equations for the two unknowns V
and -y; solutions of this system The system can only be solved Hitzl and 116non
(1977a)
(see Appendix
A for the
for each of the
cases
give the critical arcs for the given family S,,O. numerically; solutions have been tabulated by
for all
families with -9 S
arc
correspondence
distinguished
on
of
notations).
Fig. 4.10,
:!
a
9,
-9 :!
It turns out
there is
0 :! 9 that,
fixed number of
a
critical arcs; this number is shown in Table 4.1 and Fig. 4.10. This fact was first conjectured by Bruno (1973; 1994, Chap. IV), then supported by the numerical results of Hitzl and H6non (1977a), and finally proved by Bruno
(1978b; 1994, Chap. V). Table 4.4 lists all critical mains
Di, D+, 2
arcs
with
Column I indicates the D-. 2
Table 4.1. Columns 2 and 4 will be the values of maximum or
with the old
Fig.
:!
jai
5, 1#1 :! 5, in the three dodefined in
Fig. 4.10 explained below. Columns 3, 5, 6 case as
and give
A, Z, C. Column 7 indicates whether the extremum in C is a a minimum. Finally, column 8 establishes the correspondence name
of the critical
4.12 shows the
position
arc
in Table I in Hitzl and H6non 1977a.
of the critical
teristics. In domain D1, several critical
arcs on
the
displayed
charac-
-1; actually they are slightly above that line, as shown by Table 4.4, and also by the en1 (taken as an example) 2, Z largement of the vicinity of the point A in Fig. 4.16. Similarly, in domain D2, many critical arcs lie slightly above Z 0. Other special cases are the vicinity of the point A 0, shown 1, Z 1, shown in Fig. 4.13. enlarged in Fig. 4.17, and the vicinity of A 1, Z Because of the existence of these extremums, each arc family must be subdivided into one, two or three segments, depending on the case. We call arcs seem
=
=
to lie
Z
=
-
--
=
--
S,' ,3,
The relative
or
-
for the various
=
=
S1,6, S,,,,,
these segments for
0, +,
on
cases.
or generically S,*,,,,O, where the asterisk stands position of these segments is shown on Fig. 4.10
(This figure
is
schematic: the critical
arcs
are
not at
4.6
Table 4.4. Parameters for critical
case
left
A
right
Family Segments
Extremums of C: Arc
arcs
with
z
jal
: :- 5,
1#1
:! 5.
old
C
name
Domain D, I
so00
1.674613
S '
0.000000
-0.399131
max
2
S101
1.007823
S'_'
0.007823
-0.936944
max
1*00, 791
S_22
0.003582
-0.975496
max
1.000678
ST3 SZ4 S"
0.002033
-0.986987
max
0.001307
-0.991935
max
0.000910
-0.994512
max
o
S22 S30`3 S40'4 So.55 3
SaoCO
4
S'O'13
5
6
1.000182
(Ce
-
I
>
0)
I >
>
0)
A,(-I) A2 (- 1) A3(-l) A4 (-1) A5 (- 1)
SO-1-1
1.999615
S+1_1
-0.999615
2.970940
max
Ao (1)
so so so
2-2
1.499892
S+ -2-2
-0.999784
2.987426
max
A, (1)
3-3
1.333289
S+3-3
-0.999866
2.992986
max
4-4
1.249977
S+ -4 -1
-0.999909
2.995528
max
S05-5
1.199987
S+5_5
-0.999934
2.996900
max
1.596827
SO-2-1
0.403173
-1.439479
2.999747
S+ 2-1
0.999747
2.945910
0.352208
-1.283380
0.999807
2.970936
1.225423
So-3-2 S+ -3-2 S0 4-3
0.323732
-1.212991
1.666619
S+4-3
-0.999858
2.981729
S-5-4 SO-5-4
1.173978
S
0.304088
-1.172029
1.499973
S+5-4
-0.999892
2.987425
S-3-1
2.339389
SO-3-1 S+3-1
0.660611
-1.785103
3.999826
-0.999826
2.929162
3.177181
So-4-1
0.822819
-2.002284
4.999873
S+4-1
-0.999873
2.917267
S_ -5-1
4.098798
S0
0.901202
-2.141393
SO'5_1
5.999903
S+5_1
-0.999903
2.908345
S-4-2
1.713640
S-4-2
0.572721
-1.544550
so
2.499922
S+4-2
-0.999844
2.957106
S-2-1 0
S
-
2-1
S_3-2 S
0 -
3-2
S-4-3 S
7
1.000327
-
Ao (- 1)
S
0 4-3
0 3-1
S_4-1
S
0 4-1
4-2
1.999904
0 5-4
-
5 -1
0
2.131384
So-5-2
0.737233
-1.739035
2.999937
S+5-2
-0.999874
2.945908
S_ 5-3
1.492200
0.523399
-1.422497
S05-3
1.999957
So-5-3 S' 5-3
-0.999872
2.970935
S_5-2
S
0 5-2
min
B, (- 1)
max
B, (-2)
min
B2
max
B2
-
1.323896
A2(1) A3 (1) A,, (1)
min max
min max
1) 2) B3(-l) B3(-2) B4 1) B4 2)
min
A, (-2)
max
A, (-3)
min
B, (1)
max
Bi (2)
min
Ao (2)
max
Ao (3)
min max
min max
min max
A2(-2) A2(-3) B2 (1) B2 (2) A3 (- 2) A3 (-3)
71
72
4.
Table 4.4.
Generating
Species
(continuation)
left
case
Orbits of the Second
right
A
z
old
C
name
Domain D+ 2 8
S,+,
0.503491
S102
0.006982
2.874117
max
C12(1)
S21 + S24 + S21 + S14
+
0.667039
0.001118
2.971299
max
C23 (1)
S,+5
0.600231
S23 S20'4 0 S215 0 S34 0 S35
S45
0.800045
S0,5 4
0.500885
0.401887 0.750107
SQ,3
9 10 --
-S,0"8
<
(0
> 2
0
0.003538
2.872598
max
0.009436
2.641314
max
0.000428
2.987469
max
0.001155
2.947564
max
0.000223
2.992996
max
C24 (1) C25 (1) C34 (1) C35 (1) C45 (1)
0)
-3/2p
>
>
0,
0)
Domain D2
S12 S 3 S 4 S2-1 S;4 S ,-5
0
-0.870656
min
C1 2 (2)
0.963281
min
C23 (2)
0.324006
0.074233
min
0.126840
0.914061
min
C24 (2) C2,5 (2)
0.996268
-0.982706
min
C34 (2)
0.307440
min
C3,5 (2)
-0.989951
min
C4,5 (2)
0.965171
0.683973
S12 0 S23 S24 0 S2,5 S30U4 0 S35
S45
0.999582
S()5 4
0.997908
9
S-ja
(0z
<
10
S,2,a,3
(0
-3/2 > 2 0 >
8
-
0.982585 0.997181 0.581001 0.425368 0.999067
0.991544
0.419866
-
-
0) >
0)
-0.9995
-
-
-
-
1.9995
Fig.
4.16. Detail of
Fig.
4.12.
-
-
-
-
-
-
-
-
2
4.7 Orbits
73
0.01
0
1.01
Fig.
4.17. Detail of
their exact
of the
arc
Fig.
position.)
4.12.
Table 4.4 indicates also in columns 2 and 4 the
family segments
right of a larger values of A, respectively). In critical orbit; the arc family segment coincides
at the left and at the
(A, Z) plane, respectively (smaller cases
and
3, 4, 9, and 10, there is no arc family. Its generic
with the whole In
cases
8 and 10, the segment
Fig. 4.10).
The
For
T-arc
family;
a so
is indicated in Table 4.4.
name
SO,,o
is
partly
in
D-2 and partly
in D2
is true for segment S,,,,, in case 9. family, things are much simpler. A is constant
(see
same
(4.6)
is V.
names
critical orbit in the
the
a
of 7. From Fig. 4.14, we see segments, which are identical with the two separate families
originally distinguished
along
monotonically decreasing function that the closed arc family Tjj is formed of two
shows then that C is
Tj'j
and
T,j
in Sect. 4.3.4.
zzz
4.7 Orbits Now that the
form A
study of
generating given orbit
...
arcs
has been
orbits of the second
I
U- 2
;
consists of U- 1
)
where each Ui represents
an
UO some
)
completed, species (see
we
infinite sequence of
U1
)
U2
arc, and the
)
must assemble them to
Sect.
4.1).
arcs:
(4.81)
...
subscript
i takes all
integer values.
This sequence is not arbitrary; it must be periodic since the orbit is periodic. Also, all arcs must have the same value of C, which is the value for the orbit. Each arc can be hyperbolic or elliptic. Also, the arcs can belong to different
types.
74
4.
Generating
Orbits of the Second
Species
Conversely, if we choose a value of C and a sequence of arcs having that value, we define an orbit. It is obviously sufficient to specify a finite subsequence corresponding to one period of the orbit. In particular, a finite subsequence corresponding to one minimal period To of the orbit (see Sect. 2.3) will be called a rninimal sequence. A minimal sequence has no sub-periods, it cannot consist of two
1.e.
or
identical smaller sequences, since the
more
orbit itself would then have
a period period of the orbit contains p arcs, then p distinct minimal sequences can be written, depending on which arc is taken as the first one. Conversely, sequences which can be deduced from each other by circular permutation represent the same orbit, and they will be considered as equivalent. If p 1, the orbit will be called simple; if p > 1, coTnposite. Marco and Niederman (1995) have obtained results on the density of composite generating
If
one
smaller than To.
minimal
=
orbits made of two S-arcs. We must
given sequence be obtained as
distinguish
We
(1)
periodic orbit which has been built from a of arcs is always a generating orbit, i.e. whether it can always 0 of periodic orbits of the restricted problem. a limit for p
now
ask whether
a
,
two
cases:
All deflection
angles
are
different from
zero.
It
seems
reasonable to
generated for p : 0, point is replaced by by "patched orbit distance to of the small arc a near M2, M2 being adjusted to hyperbolic the and between collisions is slightly each arc correct deflection angle; produce modified so as to meet the hyperbolic arcs at both ends. This procedure can be applied iteratively. The method is standard practice in astronautics, and works well; convergence is fast for small values of p, and the patched conics orbit is a good approximation to an exact orbit. However, today there is no mathematical proof in the general case that an ordinary orbit of the second species is generating. Proofs have been achieved in particular cases, notably by Perko (1974, 1976b) and Henrard (1980) for symmetric orbits with a small number of arcs p. (ii) One or more deflection angles are zero. In that case, the answer is not so simple. The above intuitive approach fails, because for /I :A 0 it is impossible to devise an arc of hyperbola giving a zero deflection angle. The orbit is definitely not generating in certain cases; according to Proposition 4.3.2, an ordinary generating orbit of the second species cannot contain two identical T-arcs of type I in succession. Once these cases are excluded, the evidence accumulated so far suggests that all remaining orbits are generating. But here again there exists no proof for the general case. Perko (1977a, 1981a) has given proofs for particular cases, again corresponding to symmetric orbits with small p. Therefore we conjecture that in this case, means
of
neighbouring procedure:
conics"
a
orbits
can
be
each collision
state
Conjecture two
4.7.1.
An orbit
identical T-arcs in
of
the second species which does not contain
succession
is
a
generatzng orbit.
4.8 Second
In the
Families
75
defined
by its
Families
Species
4.8 Second
Species
each
preceding Section,
arc
was
considered
as
being
intrinsic parameters, for instance the pair (A, Z). However, an arc be defined by the name of the arc family segment to which it belongs,
with the value of C
species by
of the second -
-
list of
a
value of C.
arc
a
the
Therefore
following
we can
two
also define
a
also
together generating orbit
things:
family segments;
a
Consider
(Sect. 4.6).
can
parameter plane where the abscissa is C; the choice of the
ordinate is immaterial. The p
arcs
which constitute
a
given
orbit of the second
line (Fig. 4.18). p points lying on the same vertical species Each of these arcs belongs to an arc family segment, represented by a curve. Let us now vary C: each arc moves on its characteristic, and we generate orbits (in a continuous one-parameter family of second species generating second call a species farmly. agreement with Proposition 2.9.1), which we Actually we generate a family segment, since for each value of C there is only are
represented by
orbit.
one
C
Fig.
4.18. A
composite orbit, formed by
lines: characteristics of
arc
3
arcs.
Each dot represents
an arc.
Solid
families.
species family segmeni is thus defined simply by a list of p arc 1, we family segments. By convention, this list is enclosed in braces. If p 0 L have a composite have a simple family segment, for instance IS53 1; if P > 1, we family segment, for instance MO, So 2-1) S Oj. The interval of variation of C for the second species family segment is the intersection of the intervals of variation of C of the arc family segments. (Clearly the list must he such that A second
--
this intersection is
non-empty.)
76
Generating
4.
4.9 Ends of
Orbits of the Second
Species
Family Segments
Second species families do not generally obey a principle of natural termination similar to the principle valid for families of periodic orbits (Sect. 2.5,
Proposition 2.5.1): they may come to an end, from which one can move in only one direction along the family. In fact, a second species family segment ends whenever one or more of its constituent arc family segments end. There are two cases.
(i)
One
For
C
-+
hyperbolic
-oo
This is
arcs, there exists
(Sect. 4.1).
can
then contain
Jacobi constant has one
particular end which corresponds to the orbit also tends toward infinity. (Sect. 2.5). The second species family
one
velocity along
The
of natural termMaiton
a case
segment
reach the end of their families.
or more arcs
hyperbolic
only hyperbolic
lower bound C
a
--
family segment Sh,
arc
arcs, since for
-2v 2- (see (4.8)). the second
elliptic
arcs
Since there
the
only species family segment is is
fShlIn all other cases, i.e. for the other end of the
arc family segment Sh and elliptic arc family segments, the arcs involved in the end are of type 2, 4, or parabolic arcs, as has been shown in Sect. 4.5 (see Table 4.1).
for all
3,
or
Most of these ends
are
in
a sense
artificial,
and due to the
use
of the
(A, Z) coordinates: with other coordinates, such as the (,r,,q) coordinates (see Appendix A, Fig. A.1), the arc families are naturally continued past these
points. It might seem indicated to erase these artificial boundaries, and join the arc families as defined here so as to re-create the larger families
to
Aj, Bj, Cij. It turns out, however, that this operation is unnecessary. Later (Chap. 9), we will need to divide the families of generating orbits into
on
smaller
pieces, or fragments, separated by bifurcation orbits. As will be seen Chap. 6, most of the present end points correspond in fact to bifurcation orbits. Thus, the present division into families is not so artificial after all; in
it will be useful later on, and
allowed
some
we
keep
it.
(We
can
do this because
looseness in the definition of families of
we
have
generating orbits;
see
2.9). (ii) One
Sect.
or more arcs reach an extremum in C. The simplest case is when only one arc is involved, as in Fig. 4.19. The second species family can then be naturally continued: the particular arc which reaches an extremum moves beyond it into a new segment of its arc family, represented by the other branch of the characteristic, while all other arcs reverse the direction of motion on
their characteristics
(and
thus remain in the
same arc family segments). This by the sequence of orbits 1, 2, 3 on Fig. 4.19. The value of C on the second species family goes through an extremum: another second species family segment is entered. Things are not so simple if two or more arcs reach an extrernurn simultaneously. (Examples of this situation will be found in Chap. 10.) Let j be the is
indicated
by
the
number of these
segments when
arrows
arcs.
we
and
For each of
form the
them, we can choose among two arc family list; therefore we obtain 2i different possible lists.
4.9 Ends of
Family Segments
77
:
2
T 1
1
13
2
1 1
1
1
1
1
1
1
2 C
Fig.
4.19. Effect of
an
extremum of C in
an arc
family.
corresponds to 21 branches joining at the extremum; and for j > 1, it is possible to decide without further examination which branch continues into which, for p 0 0. Therefore we make This not
Restriction 4.9.1.
simulianeously An ation
exception
can
The
cases
where iwo
or more
arcs
reach
an
exiremum
will not be considered. to this restriction will be made in
be saved
one
by symmetry considerations (family
case, where the situa, Sect.
10.2.1).
Generating
5.
These
Orbits of the Third
orbits in which M3 coincides with M2
are
2.10.3).
In the
rotating
(x, y),
Species
at
all times
(Defini-
generating orbit thus reduces to a point. The period T can probably take any positive value (see below). Thus, generating orbits of the third species can be formally considered as forming a single one-parameter family, which we call the ihird species family. This family is of a peculiar kind: all orbits are identical in shape since they reduce to the point M2; the parameter is the period T. (It is not possible here to define a minimal period To, nor a period-in-family T*.) The Jacobi constant has the same value C 3 for all members of the family. tion
system of
axes
the
=
v-generating
5.1 It
Orbits
thus at first view that the
study of generating orbits in this case is help for classifying and explaining periodic orbits. Numerical studies, however, suggest that while shrinking towards M2, a periodic orbit tends towards a definite shape. As an example, compare in Szebehely 1967 the figures 9.6(a), 9.17(a), 9.21(e), which represent orbits of family g for ft 1/2, 1/11, 0.012155, respectively. Moreover, for a given small y > 0, a sizable part of a family may consist of orbits of small dimensions around M2, exhibiting a striking variety of shapes; see for instance Broucke 1968, Fig. 33, orbits 6 to 48, and Fig. 34, orbits 49 to 148. These observations suggest that the (x, y) coordinates are not appropriate any more and that we should look for another representation. A natural idea is to enlarge the vicinity of M2 by a factor depending on /.t. Specifically, we introduce the change of coordinates seems
trivial,
and not of much
--
X
where
-_
I +
,
Y
-
/1,77
(5-1)
,
M2 is in the origin, positive number. In the new system -oo. Substituting (5.1) in (2.2), we obtain new rejected at equations of motion for and 77. We expand in powers ofy the terms which represent the attraction of M1. The equations become v
and M,
is
a
is
2 +3 -
=
ill-3v W + q2)37-2
+ 0
M. Hénon: LNPm 52, pp. 79 - 94, 1997 © Springer-Verlag Berlin Heidelberg 1997
V)+0(/')'
80
Generating
5.
C and
-
=
=
a
4/1
+p
3 2
2/1
+
0(11")
+ /I
00')
+
(5.2) write
we
2Y
(5-3)
1-3v -
2
2
2
_
+
limit of
distinguish a
2
V+
0(1.1v)
---+
(x, y) system,
for
+
of variable is also necessary for the Jacobi constant:
by (5. 1). The generating the
q2)3/2
generalize now Definition 2.9.1: 0 periodic orbits for p
We orbit
-
+
Species
obtain
we
IF
3
77
-
(2
change
A
1-3v
Y
-2
Orbits of the Third
given
a
in the
new
v,
we
call
coordinates
v-generating
( ,,q)
Chaps. 3 0-generating orbits when there is v-generating orbits. They correspond to V
=
defined
and
4, in
a
need to
0
(except
origin).
5.1. 11.
again,
(5.4)
will be called
As will be seen, and
Here
O(M)
orbits which have been discussed in
them from
shift of the
Proposition
for
+
we
as an
extension of
Proposition 2.9.1,
we
have
v-generating orbits form one-parameter families.
must
distinguish
between
families of v-generating
and v-generafing families (see Sect. 2.9). As in Sect. 2.9, we define a family segment of
v-generating
orbits
orbits
as
a
maximal interval in which the variation of IF is monotonic. We ask
now:
what is the relation between the
orbits? More
0-generating the v-generating cannot be
a
v-generating
orbits and the
generally, may ask: what is the relation between orbits for two different values v = v, and v = V2? There
direct
we
identity because
change
the
of coordinates
(5.1)
becomes
meaningless p
of
0
for y = 0. So we must follow an indirect route through the 0. Assume vi < V2- Consider a V2-generating orbit Q2: there exists
periodic
( ,,q)
orbits
Q,,,,
defined in
system defined by
(5.1)
cases a
set
interval 0 < p < po, such that in the = V2, there is Qp --+ Q2 for y --+ 0. The
some
with
I/
Q,, orbits, seen in the ( , 77) system with v vj, are scaled down by a factor /,1"2-"1; therefore they tend to the origin for p 0. In other words, define orbit which a vl-generating coincides with the origin M2. The they 01, two orbits Q, and Q2 are in some sense equivalent since they are obtained as limits of the same set of periodic orbits. We have thus same
=
--+
Proposition
5.1.2.
If
v,
vl-generaitng orbit reduced As
a
particular
Proposition
case,
5.1.3.
we
V2,
to the
any
V2-generating orbit corresponds
to
a
to
a
point M2.
have
Any v-generaiing orbit (with
0-generating orbit reduced orbit.
<
to
the point
M2,
i.e.
a
v
>
0) corresponds
third species generah.ng
5.1
v-generating
Orbits
81
However, this correspondence is somewhat trivial and not of much interest, because the orbit Q, is highly degenerate. In the case of 0-generating orbits, for instance, several families include a third species orbit (see Sect. 5.2); since that orbit is unique (apart from changes in the period), Proposition 5.1.3 cannot be used to sort between these families and put them individually in relation with families of A better
generating
v-generating
orbits.
to ask is: what is the relation between
question
orbits for two different values
Consider
such
one
family
for
v
--:::
v
=
v, and
V2, which
V
we
-
families
of
V-
1/2?
call F2. Consider the
vicinity of that family for p > 0. Locally, the situation is in general as in Fig. 1.1a: there exist families F,, of periodic orbits, close to F2. The orbits Q,, which define a v-generating orbit 02 of F2 belong to the corresponding families F,,. Thus, speaking loosely, we may say that locally, F. tends to F2 for y , 0. Consider
now
another value
v
=
vj, and
a
family F,
of
vl-generating
Suppose that for p > 0, in the vicinity of F1, we find families F,' which are the same as above (only seen in a different system of coordinates). orbits. It
seems
then that the two families F, and F2
are
related in
a
strong
sense.
We cannot say that they are identical, since the change of coordinates (5-1) becomes meaningless for p = 0; however, they become the same family as soon as ft differs from zero. We recall that the main aim of the present study is
to understand what
happens
for p small but non-zero;
generating
orbits
only tools to that end. In any such application, F, and F2 will be used to describe pieces of the same family (see examples in Chap. 10). We will say are
F2, and conversely. thus in continuation, then for small
that F, is the continuation of If F, and F2
orbits in y
a -
Q,-,
are
appear in the
ratio
for
v
=
P,
and
v
-
P2
Assume vi < 1/2; then this ratio becomes
0. This has
Proposition
( , 71) systems
an
important
5.1.4.
y, their
defining
with dimensions zero
in the limit
consequence:
If two families of v-generating
orbits F1 and F2
for
two
values v, < 1/2 are in continuation, then this continuation corresponds to orbiis of F, whose dimensions tend to zero, and to orbits of F2 whose dimensZons
tend to
infindy.
proceed now to a detailed study of P-generating orbits. Inspection equations (5.2) shows that the value v 1/3 plays a critical role. consider three different 0 (see we Accordingly, limiting problems for A Bruno 1978a, pp. 261-262; Bruno 1994, p. 6): 0 < v < 1/3, v 1/3, and At will of families the the also discuss continuation v > 1/3. same time, we We
of the
--
--
=
between various values of
v.
82
Generating
5.
Orbits of the Third
5.2 Continuation from
begin by making
We
a
v
=
Species
0 to
> 0
v
list of families of
0-generating
continued to families of v-generating orbits with with
v
>
orbits which
0. For
Proposition 5.1.4, the family of 0-generating orbits M2, i.e. a third species orbit.
this,
can
be
in accordance
must include
an
orbit
reduced to
5.2.1 First
Species
We go back to Chap. 3 to find which first third species orbit. For orbits of the -first kind
species family segments
(Sect. 3.2),
for
n
-4
a
the three families
not include the direct circular orbit of radius
period becomes infinite
include
Idi, Id, 1, do 1; (3.7) shows that the minimal
1. Therefore this kind is excluded.
For orbits of the second
I J kind, there should be a 1, therefore I families and (see (3.13)). Only symmetric Ej+l E , and the asymmetric family Eal are candidates. For symmetric orbits (Sect. 3.3.1 there should be e 0 and c' 0 (see (3.9) and (3.15)). The equations +1, therefore of the motion in fixed axes are then given by (3.18): =
=
=
the
-_
X
=
=
SO cost
Y
,
=
so
sint
(5-5)
.
This coincides with the motion of M2 if so E+ conta z'ns a third species orbZt. For
+1. Thus: the
asymmetric orbits (Sect. 3.3.2), there should be e'
shows that these values is
=
are
not
--
approached by family Ell.
1,
family segment zu
=
0; Fig. 3.7
Therefore this
case
excluded. Thus the
only
case
for first
species is the family segment period-in-family is 27r.
orbits of the second kind. The
5.2.2 Second
E&
of symmetric
Species
We go back to Chap. 4 to find which second third species orbit.
species family segments include
a
The third
species orbit,
must consist in
a
sequence of
ments which include
and
seen
an
arc
as
arcs
member of the second
a
of type 4
of type 4
are
(Sect. 4.2).
given by
The
species family, arc
family segFig. 4.10
Table 4.1 and
are
1)
The
arc
family segments
Sio+,,i + z
with i >
0, in domain Dl. The end
arc
of type 4 is discussed in Sect. 4.5, paragraph (viii). The duration of this arc is 27i, where 7i is given by Table 4.3, column 3, line i. 2) The arc family segments S-j'_j with i > 0, in domain D2. The end arc is discussed in Sect. 4.5, paragraph (ix). The duration of this arc is again 27j, with
given by Table 4.3. arc family segments Tj', and Tj'j, in domain D3. The end 3) discussed in Sect. 4.5, paragraph (iii). The duration of this arc is 27r. ri
The two
is
arc
5.3
v-generating
Any family segment of v-generating quence of arc family segments from this must not have two
arcs
Tj'j
Orbits for 0 <
orbits built from
list
in succession
(with two
or
v-generating
Orbits for 0 <
v
an
<
1/3
83
arbitrary
the restriction that
arcs
Proposition 4.3.2) includes a third species orbit. The tained by adding the durations of the arcs.
5.3
v
seone
Tj'j in succession; period-in-family is
see
ob-
1/3
<
gravitational terms in (5.2), representing the attraction M2, negligible in the limit, except at the origin. Accordingly, we proceed as we did in Sect. 2.10 for the original (x, y) coordinate system, and we divide v-generating orbits into three spectes, depending on what happens In that case, the
of
become
when y ---+ 0: 1) All points of the orbit remain at case, the effect of
a
M2 vanishes in the limit
finite distance from M2. In that p
0 and the
--+
equations reduce
to
2
2)
+
3
-2
,
F
,
=
3
2
_ 2_ 2
(5-6)
of the orbit tend towards M2 while others remain at a finite distance. In that case, the v-generating orbit is made of a succession
Some
points
of arcs, which we call v-arcs; a v-arc is defined as a part of the V-generating orbit which begins and ends in M2. Each passage through M2 will be called a collision. The effect of M2 vanishes in the limit y --+ 0 for all points of the orbit except the collisions; therefore each v-arc is a part of a solution of the
equations (5.6). 3) All points of the orbit tend towards M2. Then the v-generating orbit reduces to the point M2. As in Sect. 2.10, it is preferable to define species not on the basis of the above three cases, but with positive definitions: Definition 5.3.1. A v-generating orbit isfies the equations (5-6).
to the
belongs
v-generating orbit belongs points coincides with M2.
Definition 5.3.2. A least
one
of
its
v-generating orbit belongs of the poZni M2 alone.
Definition 5.3.3. A sisis
again, species in
Here
each
the three more
species
detail.
are
not
first species if
to the second spectes
if
at
species if it
con-
exclusive. We consider
now
to the third
mutually
it sat-
84
Generating
5.
Orbits of the Third
5.3.1 First
Species
The
solution of the
general
COS
K2
t +
sin
K1, K2, K3, K4 3K32- K 1
2
I'
=
origin
K,
=
The
of
time,
cost
period is
(see
is
for instance 116non
COS
t
1969,
,
constants of
are
(5.7)
3K3t + K4
-
integration.
The Jacobi constant is
2
-
Periodic orbits the
2K3
t +
-2K, sin t + 2K2 where
equations (5.6)
follows)
called H5 in what
K,
Species
(5.8)
K2
are
obtained for K3
the
equations reduce 71
,
0. With
=
-2K, sin t + K4
--
appropriate
an
choice of
to
(5.9)
.
27r.
To find out which of these orbits
v-generating, we consider the continuAccording Proposition 5.1.4, this continuation corresponds to orbits whose dimensions tend to infinity; this is realized for K1 :Loc. The terms representing the attraction of M2 have vanished in the equations (5.6); therefore it seems intuitively clear that the families of orbits which we are now considering should be continuations of families of 0-generating orbits of the first species, in which the effect of M2 also vanishes. In fact, the equations (5.6) describe the keplerian motion of M3 around M, in rotating axes and in a small vicinity of M2, and (5.7) is the well-known epicycloidal motion ation to
v
=
0.
are
to
--+
found in that situation. It
E1+1 be
was
found in Sect. 5.2.1 that the
symmetric symmetric with respect of
only possible
to the
axis. This
family
continuation is
orbits of the second kind. Therefore the orbits
implies K4
=
(5.9)
0. The
should
equations
reduce to
K, This is
cost
q
,
-2K, sin t
--
family of elliptical being Ki. It 5.1.1. Proposition a
can
in fact be shown
is
by
continuation of the
El+,.
axes
so
family by (3.18), with a 1,
X
--
COS
E
-
Y
sin
--
2
(5-10)
-K 1
orbits around the
the parameter It
IP
origin, with a 2:1 axis ratio, one-parameter family, in agreement with
a
direct
a
computation that this family is a family El+, is described in fixed
An orbit of 1:
cososinE,
t
=
E
-sinosinE
.
(5.11) the parameter
is
0. We X t
=
=
expand
cosE
E
-
0
-
sin
along
the
family. The 0:
third
species orbit corresponds
to
to first order in
b + 0( b 3) E +
0(o 3)
Y
--
sin E +
0(0 2)
(5.12)
5.3
Going
rotating
to
=I
X
_O
We define K,
0
=
y
(2.5), =
Orbits for 0 <
v
<
1/3
85
obtain
we
20sint
+
0(02).
(5.13)
by
Ki y 1/3
-
K,
COS
(5.14)
( , 77)
Therefore the
(5.1),
coordinates with
0(02)
t +
which reduces to call this
t+0(02)
COS
and going to the =
coordinates with
v-generating
(5.10)
77
,
in
is
obtain
-2K, sin t+0(02)
--
the limit
family (5.10)
we
0. 0 v-generating.
For
extremum at
K,
(5.15)
,
simplicity,
we
continue to
Ej+j.
family
The Jacobi constant has
an
=
0; thus there
are
two
family segments, corresponding to K, !! 0 and K, : 0, respectively. However, changing the sign of K, is equivalent to making a shift of ir in the origin of Thus, the family El+, comes back family segment to consider.
time. one
5.3.2 Second
over
itself,
and
we
have in fact
only
Species
In that case, the
v-generating orbits
include the
point M2.
It
seems
intuitively
clear that these families of orbits should be continuations of families of 0-
generating
orbits of the second
species, which also include the point M2;
and moreover, that each v-arc family is a continuation of one of the arc families studied in Chap. 4. The possible arc families have been enumerated in
Sect. 5.2.2. It will be
Proposition
5.3.1.
seen
v-arcs
that,
as a
generalization
of
Proposition 4.1.1,
form one-parameter families.
The discussion is similar to that of
part of
a
solution
studied in
H5;
we
Chap. 4, but simpler. (5.7), beginning and ending in M2. These
of time in such
a
a
,r. This i
K,
V-arc
is
a
have been
>
0).
set of 4
(5.7)
such that
equations
We choose the
place --
'q
at times =
0 for
for the 5 unknowns
i
cosr
2K2
(-r
v-arc
solution of
produces the following Ki, K2 K3 K4) T: =
a
way that the two end collisions take
7-. We must therefore look for t
arcs
recall here the main results.
We call 27- the time taken to describe
origin
Each
COS
The second
+
2K3
0
K2 Sin
K4
0
2K, sinr + 3K3,r
7+
equation
can
7=
0 --
0
.
be satisfied in two different ways.
(5.16)
86
Generating
5.
Orbits of the Third
5.3.2.1 S-ares. The first
case
Species
is: K2
equations of
0. The
the
V-arc
are
then found to be
K, (cos t 7
must be
-
r)
cos
71
(5.17)
7
solution of the
a
t
2sint + 2-sin7
K,
=
implicit equation
37 tan
T
4
shape
The
(5.18)
=
of these
arcs
The Jacobi constant IF
K
--
(4 3
2 1
Cos2
is shown in
H5, Fig. 8, for the
first three values of
T.
is
(5.19)
7-1
For each value of r, the v-arcs form a continuous one-parameter V-arc in agreement with Proposition 5.3.1. The parameter is K1. The Jacobi
family,
constant has
an extremum at K, 0; therefore, for each value of 7, there are family segments, corresponding to K, 2 0 and K, :!- 0, respectively. The equation (5.18) has already been encountered in (4.77); the first ten solutions for -r are listed in Table 4.3, column 3. This, added to the fact
two
=
v-arc
that the
v-arcs
(5.17)
are
symmetric, indicates that these
V-arcs
are
the
continuation of the families of S-arcs identified in Sect. 5.2.2.
velocity
The initial
K, sin We call the is
seen
a v-arc
2 7
,
3
successive
Ki
Thus, the
=
v-arc
(5.17)
(Sect. 4.3.1).
Sin
solutions for
(- I)i
On the other
(5.17)
outgoing the us
Table 5.1. Continuation of
S20,1 S30,2 S4,3 S'',4 ...
7-1
+
T2 73
+
...
...
71, 72,
7-:
...,
7i,
(as
in Sect.
5.2.2).
It
that
to
arc
Kl(-I)'
0, ingoing in the opposite if P > 0, outgoing if 0 < 0 find, for each S-arc family segment, the value
arcs
if
Sp
are
family segment
7
S-1-1 S-2-2
72
S
73
3-3
...
>
ingoing
family segments
S_4-4
14
(5.20)
(5.21)
is
hand,
This allows
sign(Ki)
is
.
7
7
-,r
--
7
(4.76)
and the sign of K, for the v-arc the results are shown in Table 5. 1. of
at time t
7
from Table 4.3 and from
Sign(Sin 7j) case.
of
74
v
0 to
v
sign(Ki)
7-1
...
from
which is its continuation;
+
+ ...
T, I T, 1
> 0.
7-
sigii(K2)
7r
+
7r
5.3
simplicity
For
we
use
the
v-generating
name
which is its continuation. 5.3.2.2 T-arcs. The second
equations
of the
K2
--
Sin
I
S,*,,o
also for the
is: sinr
case
Orbits for 0 <
=
v-arc
<
v
1/3
87
family segment
0. We obtain then
-r
--
7r.
The
v-arc are
q
i
--
2K2 (1
+
1)
COS
(5.22)
and the Jacobi constant is F
=
2
-K 2
This is
(5.23)
again a one-parameter family,
in agreement with
Here the parameter is K2. The Jacobi constant has
therefore there
two
are
v-arc
an
Proposition
extremum at
family segments, corresponding
to
5.3. L
K2
-
0;
0 and
K2
K2 :! 0, respectively. The
v-arcs
(5.22)
are
asymmetric with respect to the axis. This, added 27r, indicates that these V-arcs are the
to the fact that their duration is
continuation of the families of T-arcs identified in Sect. 5.2.2. The initial and
final velocities
K2
=
tions of the
both =
i
Therefore the
designate
are
0
equal
to
(5.24)
-
0 and K2 family segments with K2 family segments T1'1 and TIl, respectively.
v-arc
arc
them with the
:! 0 For
are
continua-
simplicity
we
same names.
5.3.2.3 Orbits. As in the
case of 0-generating orbits (Sects. 4.1, 4.7), a periodic orbit is defined by a sequence of v-arcs with the same value of r. The discussion of v-generating orbits is similar to that of Sect. 4.7. Since
the families of T-arcs considered here
are
continuations of the families of
T-arcs in
0-generating orbits, we have the same rule as in Sect. generating orbit of the second species cannot contain two identical succession
Next
4.7:
a
V-
T-arcs in
(Proposition 4.3.2). discuss the deflection
angles. For a S-arc, the initial velocity at time t --r is given by (5.20). The slope is -2/37 < 0. Similarly, we find that the final velocity at time t +7- has a slope +2/3,r > 0. It follows that the deflection angle never vanishes between two S-arcs. For a T-arc, both the initial and final velocities are given by (5.24): the slope is 0. It follows that the deflection angle never vanishes between a S-arc we
=
--
and
a
T-arc.
The T-arcs.
only case where the deflection angle vanishes is between two identical However, this case has been excluded. In the remaining orbits, all
deflection as
angles
are non-zero.
in Sect. 4.7, based
on
the
We
Conjecture
5.3.1. A sequence
cal T-arcs in
succession
i's
a
can
patched
then
conics
of v-arcs
use
the
same
procedure,
intuitive argument
to make
which does not contain two identi-
v-generaiing orbit.
Orbits of the Third
Generating
5.
88
Thus,
v-generating orbit
a
Species
consists of
an
arbitrary
sequence of
v-arcs
from
the set
S-1-1
S2,1 S3,2 S4,3 )
with
no
also
one
two
arcs
family,
For IF
Tll
or
and
two
can
K,
S-2-2
Tj'l
be used
-oo, there is
--+
arcs
)
;
Tii 1 Tel 1
S-3-3
in succession. Each such sequence defines for the family.
as a name
-*
oo
K2
or
--+
oo for each arc, and the
corresponds Proposition 5.1.4.
dimensions of the orbit become infinite. This towards
v
=
0,
in agreement with
v-generating
5.3.2.4 Families of
than for variation
0-generating along every
v-arc
Orbits. Here
(Sect. 4.8).
orbits
(5.25)
,
to the continuation
things
are
much
The parameter r has
family segment,
so
that
no
a
simpler
monotonic
extrernurn in IF is
ever encountered. Moreover, the range of the parameter I' is the same for all v-arc families: -oo < I' :! 0. Therefore a family of v-generating orbits is
simply by
defined
sequence of
Tli 1, Tel I
a
v-arcs.
P-arc family segments, or equivalently by a 4.8, this list is enclosed in braces, for instance:
sequence of
As in Sect.
_
All orbits of
a
given family
can
be deduced from each other
by
a
change
of scale.
Species
5.3.3 Third
In all first and second
species
families of
v-generating
orbits found in Sects.
0. As shown by 5.3.2, the orbits shrink to the origin M2 for F Proposition 5.1.4, this indicates a continuation to larger values of v. This will 5.3.1 and
--+
be confirmed in Section 5.5.
5.4
m-generating
This is the most
of
magnitude.
Orbits for
v
=
1/3
interesting case: the terms in (5.2) are then of the same order changes of variables (5.1) and (5.3) take the particular
The
forms + y
x
1/3
Y
=
Y
1/3
(5.26)
77
and
C In the
3
4y
+p
2
+ Y 2/3]p
limit, the equations reduce
2 and
-
+
3
( 2 +,q2)3/2
(5.27) to
-2
2
+
772)3/2
(5.28)
5.4
r
2
3
-
This
is
-
,,vFV
Hill's
1986).
and Petit
772
+
problem (see
for
v
=
1/3
2
2
+
v-generating Orbits
89
(5.29) Szebehely 1967, p. 609; 116non defined by (5.26) will be called Hill's
for instance
The coordinates
,
71
coordinaies. We recall that in addition to the usual symmetry E (2.8) with respect to the horizontal axis, Hill's problem also has a symmetry with respect to the vertical
axis:
( ) 771 7 ) 0 Hill's
(_C
_,
t)
(5.30)
can problem only through non-integrable: integration or by perturbation methods. They form again oneparameter families, in agreement with Proposition 5.1.1; we call them Hill families. The five families a, c, f, g, g' containing simple-periodic symmetric orbits (i.e. orbits which intersect the axis only twice) were computed in 115; their characteristics are reproduced in Fig. 5.1. A point in this figure corresponds to a perpendicular crossing of the axis in the positive direction, i.e. a point of a periodic orbit such that
solutions
is
be obtained
numerical
q
0
=
-_
,
0
,
> 0
(5.31)
.
No numerical exploration appears to have been done for asymmetric families, nor for multiple-periodic symmetric families (except for a part of a triple-periodic family called 93 in H6non 1970). Chauvineau and Mignard (1990, 1991) have published a detailed Atlas of surfaces of section of Hill's problem, in which many chains of islands can be seen, indicating the presence of multiple-periodic orbits. In fact, since the problem is non-integrable, an infinite number of families of periodic orbits must exist and there can be no hope of a complete classification. Hill orbits can have collisions with M2; there is an instance in family g' for C 3.68448. However, in the present coordinates, M2 has an apparent mass I instead of 0. As a consequence, these collisions are of a completely different nature from those which we have encountered previously in second species orbits. In particular, the orbit exhibits a cusp rather than an angle at M2. This singularity can be completely eliminated by one of the standard regularizations for two-body collisions (see for instance Szebehely 1967, Chap. 3). This suggests that all periodic orbits of Hill's problem are v-generating. A partial confirmation is provided by the observation that the known Hill families can indeed be numerically continued for small p (see Chap. 10). The Jacobi constant r varies monotonically along each of the families a, c, f, g. Therefore each of these families is also a family segment, which we call Hill-a, Hill-c, Hill-f, Hill-g, respectively. Family g', on the other hand, has an extremum in IF, in the critical orbit gl (Fig. 5.1). We must therefore distinguish two segments. Following Perko (1982b), we call Hill-g'+ the "left" segment, in which the abscissas of the two points of intersection with the --
axis decrease
istic in
as we move
Fig. 5.1;
away from the critical orbit
orbits at the left of
Fig.
5 in
115),
(descending
and
Hill-g'-
character-
the
"right"
90
5.
Generating Orbits
of the Third
Species
L2
9 g -
-
-
-
-
-
-
-
-
-
-
-
-
-
f
L, C
0
Fig. 5.1. 1969.)
2
Characteristics of Hill families in the
6
4
(r,
plane. (Adapted
from
H6non,
5.5
Continuation from
segment, in which the abscissas increase
(ascending
v
<
1/3
to
Y
=
1/3
91
characteristic in
Fig. 5.1; right Fig. H5). For C < 3.68448, family g' becomes double-periodic. An orbit of the Hill9/_ segment has then two perpendicular crossings in the positive direction, and is therefore represented by two points on Fig. 5.1. Conversely, an orbit of the Hill-g+ segment has then two perpendicular crossings in the negative direction, and is therefore not represented any more on Fig. 5.1. The period-in-family T* varies continuously along each fan-lily. It tends to 0 at one end of the two families f and g. It has an upper limit for each of the five families computed in H5; the range of periods covered by these five families is 0 < T*/2 < 7.682131. However, it seems likely that there exist multiple-periodic orbits with larger periods, and it can be conjectured that periodic orbits exist for every positive period 0 < T* < oo. orbits at the
of
5
in
...
5.5
Continuation from
v
<
1/3
to
v
=
1/3
All Hill families
computed in H5 have branches of orbits whose dimensions in-oo (see Fig. 5.1). According to Proposition 5.1.4, indefinitely, for IP this corresponds to a continuation towards v < 1/3. By inspection of the shape of the orbits, this continuation can be determined. crease
5.5.1
-4
Family f
Fig. 3 in H5 shows that for IP -oo, the orbits of family f tend toward elliptical shape with a 2:1 axis ratio, which is well described by the equations (5.10). This strongly suggests that family f is the continuation of the family E1+1 (Sect. 5.3.1). Perko (1982a) has given a mathematical proof of the existence of a family of periodic solutions of Hill's problem which are close to the form (5.10) for large values of K1. He also showed that no other first species family of symmetric periodic solutions exists for large K1. --+
an
5.5.2
Family
a
The
-oo has a single close approach asymptotic shape of family a for r Therefore it should be the continuation of a second (H5, Fig. 2). of a single arc S from the set (5.25). This arc can be species family consisting identified by looking at the asymptotic value of the period of the Hill orbits (H5, Table 2) and by noting that it is an ingoing arc ( < 0 after the close --
per orbit
approach).
We thus find that Hill
species family fS2,11.
family
a
is the continuation of the second
92
5.
Family
5.5.3
This
Generating
family
is
Orbits of the Third
Species
c
symmetrical
continuation of the second
of family a with respect species family IS-1-11.
to the 77 axis. It is the
Farnilly g'
5.5.4
has two branches g+ and g' emanating from the critical orbit gl (Fig. 5.1). The orbits again exhibit a close passage to M2 for IF --+ -00 (H5, Fig. 6). Proceeding as for family a, with the help of the period found in This
family
H5, Table 5, we find that the IS3,21, respectively.
two branches
are
continuations Of
IS-2-21
and
Section 2) has given a mathematical proof of the existence periodic solutions of Hill's problem with large dimensions and a single close passage near M2, which correspond to the continuation of second species families formed of a single S arc (either a Si+,,i arc or a S-i,-i arc, with i > 0). This covers in particular the asymptotic branches of families a, Perko
(1982b,
of families of
c,
g+'
and
5.5.5
g'-.
Family
Family
g
g has two close
approaches
of two
Each of the two
-oo (H5, Fig. 4); species family consisting
per orbit for 17
therefore it should be the continuation of
a
second
--+
is
asymmetric; therefore it must be a T arc. only possibility species family Tl',Tl',. Corroborative facts are: the two arcs are symmetrical of each other; one of them is ingoing, the other is outgoing; and the period of each arc appears to approach 27r asymptotically (H5, Table 4). Perko (1982b, Section 3) has proved the existence of a family of periodic solutions of Hill's problem with large dimensions and two close passages near M2, symmetrical with respect to both the and the 77 axis, which corresponds to the continuation of the second species family f Tj'j, TIl arcs.
5.5.6
arcs
then is the second
The
Recapitulation
The results
Perko
are
recapitulated
(1983)
in Table 5.2.
also showed that the families of
symmetric periodic orbits and more generally the families corresponding to a single g'-) g, g+, S arc, can be analytically continued to families of periodic solutions of the restricted problem for small M > 0. This proof goes directly from v 1/3 to v 0, and thus bypasses the study of the case 0 < v < 1/3 made in Sect. 5.3. a, c,
--
=
v-generating
5.6
Table 5.2. Continuation between
v
1/3
<
second
v
=
1/3
family segments
for
v
Orbits for
<
1/3
v
and
>
v
1/3
93
1/3.
T*/2
species
E+
f
7r
1 S211
a
4.419371
C
4.419371
11 0
IS3021 IS-72-21 I Til I, Tir-i 1
g-
7.682131
g+
7.682131
9
2r
5.5.7 Other Families
likely that any other second species family made of arcs from the (5.25) is similarly the continuation of a family of periodic orbits of Hill's problem. This latter family can only be traced through numerical integration. These families were briefly discussed by Perko (1982b, Section 4), who noted that their existence could be proved in principle, but "the work necessary to complete this task would be monumental". It
seems
set
Y-generating
5.6
Orbits for
Y
>
1/3
In this case, the right-hand members of (5.2) become infinite in the a further change of variables for the time is necessary: we write
limit,
and
t
=
/,,(3v-1)/2j
In the limit 1-t
(5.32)
.
0, the equations reduce
-*
d2 dj2
-
(2
+
We must also make IF
and
_-
d77 62
q2)3/2 a
change
to
77
W
+
(5.33)
q2)3/2
of variables for the Jacobi constant:
/-I 1-3vf
(5-34)
obtain in the limit
we
2
(g ) (d7l)
2
2
(5-35)
_
di
Here
we are so
equations
di
close to M2 that the effect of M, has become
describe
simply
the
negligible.
The
keplerian
motion of M3 around M2. The axes do not rotate anymore; the rotation has slowed down to zero as a consequence of the change of variable (5.32). All
elliptic
and circular solutions of this
However, only the circular orbits
are
two-body problem are periodic. v-generating. This is stated without
Generating
5.
94
proof by
(1994,
Bruno
small but not
zero.
small but not
zero.
p.
7).
It
can
intuitively. Consider
be understood
Y
The rate of rotation of the axes, in the time unit i, is also Therefore an elliptic orbit around M2 precesses slowly. In
closed,
order for it to be
Species
Orbits of the Third
be commensurate with that of M2
period should
its
around Ml: in fixed axes, M3 makes J revolutions around M2 while M2 makes I revolutions around M, (compare Sect. 3.3). In the original coordinates t,
X, Y, the period of the elliptic orbit is then 27rI/J, and
its
semi-major axis
is
2/3 a
=
y
1/3
(5.36)
continuously tends to zero, the parameters I and J of an orbit, 1/3 In the being integers, cannot change. Therefore a is proportional to p 1/3-Y : the orbit coordinates, the dimensions of the orbit are of order M Now
as
p
.
does not have
Thus there
a
finite limit for
only
are
direct circular orbits and the
v-generating orbits for retrograde circular orbits. The
is
f
=
The variation of r
is
monotonic
Jacobi constant
1/3.
>
v
two families of
11p,
where p = on each
+7
V
Y
1/3:
>
the
value of the
is the radius of the orbit.
family, which
consists therefore of
a
single family segment.
5.7 Continuation from
v
==
1/3
to
>
v
1/3
1/3 can Proposition 5.1.4, families of v-generating orbits for v be continued to larger v only if they include orbits which shrink toward M2. +oo. As shown in H5, this is the case for the two families g and f, for r and direct toward tend orbits these that show and H5 3 4 in clearly Figs. the into continued Therefore are around orbits circular they M2. retrograde According
to
=
,
two families found in
Sect. 5.6 for
v
1/3.
>
The other end of these two families radius. Here there is
no
corresponds
continuation; the families have
to orbits of a
vanishing
natural termination.
5.8 Conclusions
Looking
back
over
the various
sufficient to consider the
cases
case v
=
for v,
1/3,
we
1/3 is equivalent to an asymptotic description large orbit dimensions (left of Fig. 5.1). It is of second species families for C 3, i.e. all --+
dimensions of the orbit tend to consider that
a
---+
+oo and
in the
problem.
The
case
of Hill families for I'
also
an
-+
0 <
V
-co
and
<
asymptotic description
become of type 4 and the y) plane. We can therefore
arcs
(x,
species family continues straight into a Hill family. case v > 1/3 is an asymptotic description of Hill families vanishingly small orbit dimensions (right of Fig. 5. 1). It is
second
Similarly, the for r
zero
observe that in practice it is
i.e. Hill's
thus included in Hill's
case.
6. Bifurcation Orbits
We have
find the
now
inventory
finished the
bifurcation orbiis,
of
which will
generating orbits. play a fundamental
Our next task is to role
explained
as
in
the Introduction.
We define first
Definition 6.0.1. A branch is
nating
in
a
given
diredion
from
a
family of generating given generating orbZt.
part of
a
a
orbits
ema-
given family, two branches emanate from a generating orbit which lies inside the family (general case), and one branch only from a generating orbit which lies at one end of the family (particular case). Note the difference with families of periodic orbits, for which there are always two branches (Proposition 2.5.1). Here we deal with families of generating orbits, which are more loosely defined (Sect. 2.9), and which can have Inside
a
ends. We consider
through
or
end
now a
in
particular generating orbit, and all families which
that orbit. We state
Proposition 6.0.1. orbit is always even.
The number
a
pass
fundamental property:
of branches emanating from
a
generating
proof of this Proposition will not be offered here. It follows intuitively principle of natural termination (Proposition 2.5.1), and the fact that generating orbits are obtained as limits of periodic orbits for M 0. A rigorous proof could perhaps be built along these lines. Proposition 6.0.1 is also supported empirically by the fact that no exception has been found in A formal
from the
--+
hundreds of examined
cases.
We define next
Definition 6.0.2. A bifurcation orbit is
a
generatz.ng orbit from which
more
than two branches emerge. The generic situation is that of Fig. 1.1a: two branches emanate from a orbit Q. These two branches are then trivially joined. In the case
generating of we
a
bifurcation orbit,
on
the other
must solve the non-obvious
M. Hénon: LNPm 52, pp. 95 - 124, 1997 © Springer-Verlag Berlin Heidelberg 1997
hand, there are 4 branches problem of how these branches
or
are
more, and
joined.
Bifurcation Orbits
6.
96
periodic orbit (and by extension of a generating orbit) includes the period-in-family (Sect. 2.4). In order to qualify, a branch must have the same period-in-family as the given generating orbit. Thus, the situation of Fig. 2.1b is noi a bifurcation. A necessary condition for a generating orbit to be a bifurcation orbit is obviously that it belongs to more than one family. (This condition is necessary but not sufficient. A generating orbit may be the end point of two different families. In that case, there are only two branches; it is not a bifurcation, but a continuation from one family to the other.) Therefore an exhaustive search for bifurcation orbits can be made by considering every possible pair We recall that the definition of
a
of families in turn.
Species
6. 1 First
and First
We search for bifurcation orbits
species
first
is
Species species families. The kind, and the second kind is asymmetric orbits; therefore we have 6
common
to two first
divided into first and second
symmetric and
itself subdivided into sub-cases to consider.
6.1.1 First Kind and First Kind
The three families of first kind orbits no
orbit since
common
algebraic
mean
motion
Idi, Id, 1, found in Sect. 3.2
they correspond
to three
disjoint
can
have
intervals for the
n.
6.1.2 First Kind and Second
Kind, Symmetric
family of symmetric second kind orbits contains two circular orbits, 0 and 0 7r (Fig. 3.2). The period-in-family is given corresponding to 0 by (3-12). On the other hand, the period-in-family for first kind orbits is given by (3.7), with n --,E'Jll. The requirement that these periods should be equal Each
=
=
gives
lei
-
il
from which C
I -
-
-_
we
(6.1)
1
derive
(6.2)
+1
Thus, only the direct circular orbit with
0
can
correspond
to
a
bifurca-
tion. We obtain two infinite sequences of bifurcation orbits: -
Families Idi and
EJ,J+1 (I
circular orbit with -
Families Id, and
mean
EJ,1_1 (I
circular orbit with
mean
!
1)
motion 2!
2)
motion
have n
--
have n
a common
(I
+
orbit, which is the direct
1)/I.
a common
orbit, which
is
the direct
First
6.1
Species and First Species
97
Each of these bifurcation orbits lies inside the
family
orbit is also
Ei'j
family I and inside the (In the Eli family, the bifurcation of the points where the two segments
E; thus 4 branches emanate from it.
and
an
extremurn in
C,
and
one
Ejej meet.)
6.1.3 First Kind and Second
Kind, Asymmetric
Fig.
3.7 shows that the only circular orbits (e' 1) belonging to a family asymmetric second kind orbits are the equilibrium points L4 and L5, for family E,', These orbits do not belong to a continuous family of circular orbits --
of
.
(see
Sect.
3.2.1),
6.1.4 Second
and therefore
they
Kind, Symmetric
qualify
do not
and
as
bifurcation orbits.
Symmetric
Two different families of second kind orbits
can have a common orbit only they have the same IT and J, since these numbers are determined by the semi-major axis a. If I + J is odd, there exists only one symmetric family Eli; therefore this case is excluded. If I + J is even, there exist two families E+ and E but Fig. 3.5 shows that these two families have no common
if
member.
6.1.5 Second
Kind, Symmetric
and
Asymmetric
Family El', of asymmetric second kind orbits has two common orbits with family EIl (for I even) or E l (for I odd) of symmetric second kind orbits (see Sect. 3.3.2) (except for I 1 where there is only one common orbit with ET1). The period-in-family, given by (3.12), is 27rl in both families, so these orbits qualify as bifurcation orbits. This, however, is one of the rare cases where for y 0 0 we have the situation of Fig. 1.1b: each family is displaced, but preserves its identity; no bifurcation from one to the other takes place. The reason is that symmetric second kind orbits remain symmetric for y 0 0 (and of course, asymmetric orbits remain asymmetric). The symmetry of the orbits cannot suddenly change into asymmetry as one follows a family; therefore a branch of symmetric orbits cannot be joined with a branch of asymmetric orbits. (This principle will be used more extensively in Chap. 7.) Thus, the existence of these bifurcation orbits has no effect, and we can --
ignore them.
6.1.6 Second
Kind, Asymmetric and Asymmetric
Two diffferent families of of
1, and therefore
asymmetric second kind orbits have different values
cannot have
a common
orbit.
98
Bifurcation Orbits
6.
6.2 Second
Species
and First
or
Second
Species
It will be convenient to
belongs each
to at least
arc
study these two cases together. The bifurcation orbit species family. It is therefore made of arcs, and supporting keplerian orbit (Sect. 4.1). We introduce a
one
is part of
a
second
fundamental distinction between the two
1)
following cases: angles between successive arcs vanish. Then the belongs to a single supporting keplerian orbit, with
All deflection
bifurcation orbit
it coincides in fact. We call this the
if
case
total
a
6ijurcation. This
is
whole which
necessarily
of the families
belongs to the first species. One deflection or more 2) angles do not vanish. The arcs making up the bifurcation orbit belong to more than one supporting keplerian orbit. We call this case a partz.al bifurcation. Only second species families can participate in case
that
one
case.
of
In each of these two cases, types 1, 2, and 3, depending
involved in the bifurcation will be
seen
-
We
identify
For
given
a
For
a
total
between
bifurcations
(Sect.
properties. analysis will proceed as follows.
the bifurcation orbits.
bifurcation
it. Each of these -
distinguish
the type of the supporting keplerian orbit 4.2). We have thus a total of 6 cases, which
on
to have very different
In each case, the
-
will also
we
arcs
orbit,
arcs which can belong to family, which we also identify. identify the first species families passing through we
is part of
bifurcation,
we
enumerate the
an arc
the bifurcation orbit. -
We
identify the
second
orbit. This involves the -
We
study
species families passing through following steps.
the variation of C
along
each
arc
family,
the bifurcation
in
the
the bifurcation orbit. This is necessary to determine which combined into an orbit. -
We define
vicinity of arcs
can
be
simple symbol for each arc. This allows a condensed represpecies orbits. Even more important, this symbolic will allow us to describe the results in each of the 6 cases representation with a single table, valid for all bifurcations (which are in infinite number a
sentation of second
for types I and -
are -
formation rules, which specify which sequences of symbols acceptable, i.e. correspond to actual second species families.
The results for all orders
table, -
in the form of
a
n
up to
a
maximal value
are
collected in
a
list of branches.
bifurcation, symbols are also defined for the first species families, corresponding branches are included in the table. Finally we study the number of branches, as a function of the order n. For
a
total
and the -
2).
We enunciate
6.2 Second
Species
and First
or
Species
Second
99
6.2.1 Total Bifurcation
As has been mentioned in the
previous section, in that
case
the bifurcation
orbit coincides with the supporting keplerian orbit. The bifurcation orbit is a generating orbit, and therefore periodic. It follows that the supporting
keplerian
orbit cannot be
a hyperbola or a parabola; it must be an ellipse. We bifurcation ellipse. It is of type 1, 2, or 3. (Type 4 corresponds to the third species and has been considered in Sect. 5.2.2.) We shall correspondingly say that the bifurcation is of type 1, 2, or 3. The bifurcation orbit is made
call it the
of
a
definite number
of basic
n
arcs
(see
Sect.
4.1)-
we
call this number the
order of the bifurcation. A total bifurcation is thus characterized and its order. We will thus 2T3, for
use a
short
instance, represents
designation
of the form
by its type
T;
the total bifurcation of type 2 and order 3.
6.2.1.1 F'kee Branches and Rooted Branches. So far
have
generally periodic orbits, periodic solutions; the origin of time was unspecified (see Sect. 2.3). When we join two branches at a total bifurcation, however, it is necessary to specify their relative phase, i.e. which basic arc of the second branch is associated with a given basic arc of the first branch. We consider first the considered
case
and
we
rather than
of types 2 and 3, in which all basic
6.2.1.4).
There
then
arcs are
identical
(see
Sects. 6.2.1.3
a branch possible free and rooted branch branch of a periodic orbits, origin unspecified; with a specified origin. To one free branch there correspond periodic solutions, n rooted branches, derived from each other by shifts in time. We will consider only rooted branches in what follows. The advantage is that a joining of two rooted branches is completely specified. The drawback is that we have more branches to consider. Also, essentially the same junction We call T the shift of the origin by one basic arc. is obtained several times. If two rooted branches H, and H2 are joined, then the two branches PH, and Ti H2, with j an arbitrary integer, are also joined since the same shift is applied to both branches. Two cases are possible. (i) H, and H2 correspond to two different free branches. The n rooted branches Ti H1, with 0 < j < n, are joined to the n rooted branches V H2
of
are
choices. We call
n
branch
with the
-
These
origins
(ii) many
(0
<
with i
=
n
junctions
are
essentially
and the same, viewed with
one
n
different
of time. It
also
can
happen
examples i < n). Thus, Hi
that H, and H2
of this will be
seen.
correspond
to the
We have then H2
--
same
free
T'Hi, for
branch; some
i
T'H1. PH, is joined joined T2'Hl; and therefore H, coincides with T2'Hl. This is possible only if n/2. The family comes back over itself with a shift of one half-period.
This is
is
instance of the
with
It follows that
refleciion phenomenon, described in Sect. 2.5. complicated. There are two different kinds of basic arcs, which alternate along the orbit (and therefore n must be even) (see Sect. 6.2.1.2). Therefore there are only n/2 possible choices for the phase of the second branch relative to the first. To one free branch there correspond an
Type
I is
slightly
more
Bifurcation Orbits
6.
100
n/2
rooted branches. T is
now
We still have the two
arcs.
defined
as
a
shift of the
origin by
two basic
cases:
correspond to two different free branches. The n/2 rooted TiH1, j < n/2, are joined to the n/2 rooted branches These TIH2. n/2 junctions are essentially one and the same, viewed with n/2 different origins of time. (ii) Hi and H2 correspond to the same free branch. We have then H2 T'H1, and reasoning as above we find that i n/4. This is again a reflection.
(i) H,
and H2
branches
with 0
=
6.2.1.2
Type
1. In fixed axes,
a
bifurcation
ellipse
of type I intersects the
points P and Q. The bifurcation orbit cannot have only at Q, because the families emanating from it
unit circle at two distinct
collisions
only
at P
or
would then consist of
a
succession of identical T arcs, which is not allowed there are collisions both at P and at Q. We take
(Proposition 4.3.2). Therefore
origin of time in one of the P collisions. Since the orbit is periodic, there an infinity of other collisions in P. Consider the first of these collisions for t > 0. It corresponds to an integral number I of revolutions of M2 and an integral number J of revolutions of M3. Therefore it takes place at time t 27rl, and A must be of the form the
are
--
A
I =
(6-3)
*
J
mutually prime (otherwise there would have been a previous place at times 27rpI, where p takes all collision). in Q are separated by an interval successive values. collisions Similarly, integer times and 27rl take t* of time + 2rpl-; we choose t* so that place at I and J
are
The collisions in P take
0 < t* < 27rl
(6.4)
.
The sequence of collisions is
P
Q
P
t*-2-7TI 0
t*
27TI
Q
P
2 -irI
t
k
Fig.
represented by Fig.
-
1
finity
of
arc
families.
S-arcs,
the collisions in
(A, Z) plane,
since
point
arcs
are
below). Q
P
t*+27TI 47TI
t* + 47TI
5
3
bifurcation
by joining
All these
this
a
will be defined
Q
ellipse
of type 1.
It follows that the bifurcation
obtained
Q.
(k
1
6.1. Successive collisions for
Arcs and
6.1
ellipse supports
the collision in P at t
=
an in-
0 to any of
represented by the same point in by the bifurcation ellipse. Each
is defined
the arc
Species
6.2 Second
by
is characterized
two numbers
the time of the collision in
t* +
27rpl
Therefore a
t20 + 27ra
and
a
ao
-_
-
0
0
-
or
Second
Species
0; from (4.20) and (4.23)
we
101
have for
Q: :'--
t40 + 27rAO
(6.5)
.
by
related to p
are
+P1
and
a
and First
(6.6)
00 +PJ,
with 1* a0
Each
00
27r
belongs
arc
Z
120
-
-_
=
OA
a
-
to
01
Z
-
t40
(6.7)
27rA
S-arc
family, with
a
characteristic
(6.8) and
(6.6)
001
aj
__
-
.
(4.27), (6.3),
From
an
t* =
-
we
have
a0J
-
L
J
J
(6-9)
-
J
integer (Note: L and J are not necessarily mutually prime). 1/2. 3/2, Z example, consider the point with coordinates A Fig. 4.12 shows four S-arc characteristics passing through this point (inclined 2, 3, J 1, 1 1, #o lines). Their indices are given by (6.6) with ao of values other For 1: and p are S43. they S-5-3, S-2-1) S11) -2, -1, 0, p, an infinite number of other characteristics passing through the point are where L As
is an
an
=
=
=
=
--
=
--
obtained.
ellipse supports also one T-arc obtained by joining two P, and another T-arc obtained by joining two successive collisions in Q. Correspondingly, the point (A 1/2) belongs 3/2, Z also to the characteristic of the two arc families Tj'j and Tj'j (vertical line). (One word of caution: it does not follow that a given bifurcation orbit belongs to an infinity of orbit families; on the contrary, this number is always finite, as we shall see by enumerating these families below.) A bifurcation ellipse of type I is thus characterized by three integers 1, J, L. Conversely, any given triplet (1, J, L), with I and J positive and mutually prime, corresponds to a bifurcation ellipse of type 1, with A and Z given by (6-3) and (6.9), provided that the point (A, Z) lies inside D1, D2, or D3. All characteristics with a, 0 satisfying The bifurcation
successive collisions in
=
=
01 pass
has -
-
aJ
=
(6.10)
L
through the point (A, Z); an infinity of solutions.
If I >
(4.40)), 2J
-
and since I and J
J, the point (A, Z) lies
ILI
in domain
are
mutually prime, (6-10)
Di provided that
Therefore for given I and J, L 1. 1 values from -J + 1 to J or
< J.
-
can
IZI
<
I
(see
take any of the
102
-
Bifurcation Orbits
6.
If I
J
=
But this are no -
=
1, the point (A, Z) lies in domain D3 if -I happen since Z is an integer in this
can never
<
Z < 0
case.
(see (4.43)).
Therefore there
solutions.
If 2 -3/2j < I <
j, the point (A, Z) lies in domain D2 if 0 < Z ::' t Z, (A) (see (4.41)). Therefore, for given I and J, L can take any value from I to L, or on [JZ, (A)]. The point (A, Z) can be either on the sheet D+ 2 =
the sheet
D2
so
-,
the number of solutions is 2L,,,.
Table 6.1 gives the number of solutions for L for all couples with 1, J < 9.
Table 6.1. Number of bifurcation
ellipses
for all
< 9.
1, 1
1
2
1
0
1
2
0
4
1
1
3
3
2
5
4
1
3
4
5
0
4
7
2
8
0
1
4
1
1
5
5
7
7
with
(6.4)
and
9
1 3
5
7
9
9
9
11
8
10
13
13
6
12
6
12
useful
inequalities
some
acceptable (I, J) couples
15 14
for ao and
go.
From
derive
we
14027rA
<
/3o
<
t40
J
27rA
always (see Fig. 4.5)
But there is
t40
1
27rA
Therefore, 0 :!
now
8
3
9
4
We derive
The
1
7
3
8
9
1
6
6
6
(6.7b)
5
acceptable (I, J)
00
(6-12) 00 and J
since
<
00
i
are
integers,
(6.13)
.
J is
possible only
if 140 <
0, i.e. A < I (Fig. 4.5). But in that integer, and Z can take no integer value inside D2 (see (4.41)). Therefore Oo J is not possible and we have case
case,
(6.9)
=
gives Z
--
I
ao
-
=
an
=
0 <
00
<
j
(6-14)
-
Substituting (6.14) -Z :!! ao :!! I
-
A
(6.9),
into -
Z
.
we
derive
(6.15)
Species
6.2 Second
But there
always
is
Z < I and Z > -A
0 :!! ao < I
ao
uniquely
(4.12b).
and
we
103
have for
6. For
a
go ellipse, by (6.9), (6.14), and (6.16). V is obtained from (6.3) found by solving (4.28) numerically; there is a unique solu-
and
of
defined
-y is
Z(-I)
is monotonic
computed
from
(4.6).
then be
therefore
ellipses in domain D, for I :! by a triplet (J,I,L), the values
defined
tion since the relation can
Species
(6-16)
bifurcation
are
(see Fig. 4.9);
Second
.
Table 6.2 lists all bifurcation
given
or
(6.14):
relation similar to
a
ao
and First
Sect. 8.2.1. Lines
are
(Fig. 4.6).
quantities K by J, then by 1,
The
ordered first
The Jacobi constant C
and A2 will be defined in and
finally by increasing
C. Table 6.3 lists the bifurcation
in domain D2 for J :! 6. The value
ellipses
and DSect. 4.3.2) is found by solving -y, separating the two sheets D+ 2 2 (see these two sheets are then found by values of to (B.1). The two 7 belonging
solving (4.28) numerically (see Fig. 4.7). By convention, of L with a superscript + or to indicate whether the -
in
D+ 2
=
0
a
,
_-
ao
13-00
,
QP will be called second P
-
bifurcation orbit is
-1
a
,
-
ao
I
9
,
(6.17)
PQ
arcs
to
(6-17)
-
basic arc; it -
alternating basic arc; it corresponds
sequence of
a
QP (Fig. 6.1). PQ will be called first basic P
write the value
D_ 2
or
The bifurcation orbit is made of and
we
corresponds
-
180
-
i
to
-
(6-18)
first basic arc corresponds non-negaizve slope 0. From (6.16) we that this is also the characteristic with the smallest non-negative a. The parameters a and fl of the second basic arc, given by (6.18), verify
(6.6), (6.14),
From
and
we
find that: The
to the characteristic with the smallest see
-1 :5;
0
<
0
-1:!
,
Thus: The second basic ative
slope
(smallest
3
largest negative
the
a
< 0
belongs
arc
in absolute
(6.19)
.
to the characteristic with the
value).
integer k, k
(6.6)
=
I +
neg-
a.
identify by
It will be convenient to
odd
largest
This is also the characteristic with
related to p
each S-arc of the sequence
(6.6) by
an
(6.20)
2P.
becomes k
a
=
ao +
I
2
0_00+
k
1
2
(6.21)
We also introduce m
--
IkI
.
(6.22)
104
6.
Bifurcation Orbits
Table 6.2. Parameters of bifurcation
ellipses
in domain D, for 1:!
I
i
L
go
ao
2
1
0
0
0
1.170487
2.029584
-0.406767
0.988607
3
1
0
0
0
1.232579
1.958335
-0.450861
0.986363
4
1
0
0
0
1.266155
1.927339
-0.487019
0.989460
5
1
0
0
0
1.287635
1.909584
-0.513880
0.991899
6
1
0
0
0
1.302746
1.897928
-0.534364
0.993614
3
2
1.112141
5
4
5
5
6
2
3
3
4
5
v
'Y
C
K
1
1
1
2.954024
-1.422126
-2.549387
0
0
0
2.123027
-0.403687
1.008158
-1
1
2
0.924770
2.102200
-0.120423
1
1
2
2.834460
-1.758363
-0.361481
0
0
0
1.985024
-0.428800
0.992735
-1
1
3
0.985946
1.875714
-0.049091
2
2
2
3.054298
-1.333769
-9.137151
1
1
1
2.907199
-1.282752
-0.739806
0
0
0
2.185084
-0.423816
1.016965
-1
2
3
1.207315
1.596095
-0.403472
-2
1
2
0.636217
2.568912
0.214538
1.207111
1.083752
2
1
1
3.008443
-1.538875
-2.104653
1
2
3
2.799492
-1.427408
-0.651011
0
0
0
2.081899
-0.399138
1.000170
-1
1
2
1.226256
1.478221
0.284008
-2
2
4
0.684201
2.470728
-0.408823
1.135175
3
3
3
3.090164
-1.269159
-18.246032
2
2
2
3.022092
-1.256763
-4.549367
I
1
1
2.887029
-1.203216
-0.046521
0
0
0
2.230854
-0.446564
1.021154
-1
3
4
1.382644
1.260881
-0.543841
-2
2
3
0.890458
2.204024
-0.083376
-3
1
2
0.489346
2.745112
0.393685
4
4
4
3.107497
-1.224575
-29.805877
3
3
3
3.066741
-1.219891
-9.944386
2
2
2
3.006032
-1.206432
-2.632404
1
1
1
2.876200
-1.151883
0.299039
1.066877
1.055676
0
0
0
2.266736
-0.468053
1.023196
-1
4
5
1.507113
1.019915
-0.628980
-2
3
4
1.063438
1.911391
-0.258872
-3
2
3
0.713208
2.482294
0.116994
-4
1
2
0.398739
2.831267
0.505806
6.
A2 -
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6.2 Second
Table 6.3. Parameters of bifurcation
flo
ao
1-
2
1
1+
2
1
1
1
L
2
3
3
3
4
5
4
5
5
6
Species and First
Second
ellipses in domain D2 for J:! C
v 0.830439
or
Species
6.
K
A2 +
2.884769
-0.296034
4.702546
1.083763
2.087673
0.143434
-
1-
3
2
3.034581
-0.554474
12.243632
+
2-
2
1
2.849926
-0.489624
2.708950
+
2+ 1+
2
1
1.364912
1.574496
0.420187
3
2
0.702138
2.567360
-0.210325
0.888024
-
-
1-
2
1
2.940282
-0.104933
4.553158
+
2-
4
2
2.577249
0.103001
0.470401
+
2+ 1+
4
2
1.598948
1.362324
-0.224300
2
1
0.821215
2.456192
0.429005
0.770895
-
-
1-
4
3
3.081612
-0.668904
22.394477
2-
3
2
3.001922
-0.654354
6.816408
+
3-
2
1
2.839521
-0.589224
2.001254
+
3+ 2+ 1+
2
1
1.532927
1.229779
0.559909
3
2
0.966630
2.201452
0.086668
4
3
0.526583
2.744731
-0.392206
0.916298
+
1-
5
4
3.102988
-0.735653
35.053079
+
2-
4
3
3.056791
-0.730337
12.581128
+
3-
3
2
2.987683
-0.714983
4.725887
+
4-
2
1
2.835453
-0.650269
1.658279
+
4+ 3+ 2+ 1+
2
1
1.651306
0.979150
0.645508
-
3
2
1.140924
1.907027
0.262335
-
4
3
0.760251
2.481672
-0.115857
-
5
4
0.422880
2.831130
-0.505108
-
0.933143
105
Bifurcation Orbits
106
6.
Figure
6.1 shows the value of k for various end
points Q of the S-arc. It the Q collision, if all collisions are
is apparent that k
is simply the rank of consecutively, with rank 0 attributed to the P collision taken as JkJ basic origin of time. Thus: the S-arc corresponding to k i's made of m arcs. It is an ingoing arc if k > 0, an outgoing arc if k < 0. In particular, the first and second basic arcs correspond respectively to k I and k -1. basic is made of that each the two T-arcs of note two arcs. we Finally
numbered
=
=
=
species families. Having found the S-arcs and T-arcs supported by the ellipse, we must now find the families of generating orbits to which a given bifurcation orbit belongs. The order n of the bifurcation orbit (the number of basic arcs) must be even: starting from P, one must be back in P after one period. We ask first whether the bifurcation orbit can belong to a first species family. The bifurcation ellipse is of type 1, therefore not circular; so it can only belong to a second kind family. The semi-major axis is given by (6-3); therefore the second kind family is characterized by I and J, and the period2: the bifurin-family is 27rI (see Sect. 3.3). It follows that the order is n First
bifurcation
--
cation orbit is made of
one
first basic
arc
and
second basic
one
arc.
It is then
symmetric (see 4.3.1) family of second kind symmetric orbits. We determine now the family segment to which the bifurcation orbit belongs. The points of perpendicular intersection with the x axis are the midpoints of the two basic arcs. Their abscissas are given by (4.38), with a and,3 given by (6.17) and (6.18). Three cases must be distinguished.
symmetric
since both
belong
does indeed
-
to
The
a
same
bifurcation
(_I)L sign(A
-
and therefore it
Sect.
are
If I is odd and J is even, both points are at the 1) is positive, at the apocenter in the opposite that L has the
-
arcs
parity
flo. Looking
as
at
pericenter case.
Also
Fig. 3.3,
if (-I) we
we
belongs to the family segment 1) equals +1 or -1, respectively. orbit
o sign(A(6.9)
find from
obtain:
E'jj
or
El'j if
point is at the pericenter and the other at point at the pericenter. Its abscissa has the of shows that L has the same parity as ao. Looking at -(-1)'0. (6.9) sign obtain: we Fig. 3.4, If I is
even
and J is
odd,
one
the apocenter. We consider the
The odd -
bifurcation orbit belongs even, respectively.
If both I and J
are
-(-I)'- 00 sign(A parity The
to the
family segment
El'j
or
Ej'j if L
is
or
ao +
as
odd, the
1).
bifurcation -
orbit
sign of the abscissas of both
We find from from
flo. Looking
(-l)L sign(A Finally,
-
at
Fig. 3.5,
belongs
1) equals
-1
to the
or
we
(6.9)
same
obtain:
family segment
Ej+j
or
Ejj if
+1, respectively.
possible that the bifurcation orbit belongs also to a family of I (Sect. 3.3.2). asymmetric orbits? This would require that J
is it
second kind
points is:
that L has the
=
and First
Species
6.2 Second
Therefore, from what has been said above, I
corresponds
shows that this
to 7 >
7r/2,
or
>
e'
1, L
< 0
or
Second
0, and
--
Species
Z
--
0.
107
Fig.
(equation (4.12c)).
4.6
On the
hand, e' is always positive for second kind asymmetric orbits (Bruno 1976; 1994, Chap. VII) (see also Fig. 3.7). Thus the answer is no. In summary: a bifurcation orbit of type I belongs to a first species family 2. It belongs then to one of the families E1j, Ej+j, or only if its order is n
other
=
E_
ii,
species families. We consider now the second species families passing through the bifurcation orbit, for a given order n. Each family is specified by a minimal sequence of arcs, corresponding to one minimal period (Sect. 4.7). Each arc must belong to the set identified above: the infinite set of S-arcs, each characterized by its odd index k, plus the two T-arcs Tl'j and T,j. Bruno (1973; 1994, Chap. IV, Sect. 3.4, Problem 2 and Theorem 3-3) showed that, provided that Schanuel's hypothesis is true, an extremurn in C cannot coincide with a bifurcation of type 1. An empirical confirmation is provided by the computation of many bifurcations (of which only a small Second
part appear observed. It
this
in
seems
Conjecture arc family.
monograph), therefore very
6.2.1. A
in which
likely
no
such coincidence has
that the
bifurcation of type
I
is
following
not
ever
been
is true:
exiremum in C in the
an
point of its arc family (see Sect. 4.5), it follows that a bifurcation of type I always lies inside an arc family segment. Thus, every family passing through the bifurcation orbit provides two branches, with AC > 0 and AC < 0, respectively, where AC is the variation Since
of C
an arc
of type I
as one moves
is never an
end
away from the bifurcation orbit.
shorthand, a symbolic representation arc in represented by positive number, which is simply the number of basic arcs which it contains. Thus, a S-arc is represented by the number m, and a T-arc by the number 2. Given this information, one can determine the sequence of arcs without ambiguity: an odd symbol m corre-m for a QP arc; a m for a PQ arc and k sponds to an S-arc, with k
Symbols.
We
use
which each
here,
as a
convenient
is
a
--
--
symbol a QQ arc. m
--
2
corresponds
to
a
T-arc, which is
Tl'j
for
a
PP
arc
and
Tl'j
for
The sequence of symbols is written, starting from the origin. It may hapThis arc is thus cut pen, however, that the origin lies inside a S or a T arc.
beginning and at the end of the sequence, respectively. We represent each part by a number, equal to the number of basic indicate arcs which it contains, and with a preceding or following dash to 6 and the that it is a part of an arc. To illustrate, consider the case n free branch represented by the sequence 51. It corresponds to three rooted branches, which we represent by: 61, -312-, -114-. The two branches corresponding to opposite values of AC will be dissymbols in Table 6.4. It will be tinguished by a sign + or -; hence the into two
parts, lying
at the
--
108
Bifurcation Orbits
6.
adopt the convention that this is not the sign of AC but rather A2AC, where 2 is a quantity which can be computed for each
convenient to
the
sign
of
bifurcation orbit of type I
(see
8.2.1).
Sect.
given sequence following rules:
Formation rules. A
satisfies the
1) 2)
Each number must be odd
or
of
numbers is
positive
acceptable
if it
2.
succession, since this would correspond to on the corresponding branch, which is made of ordinary generating orbits, and would violate Proposition 4.3.2. (Also the sequence must not begin and end with a 2.) 3) The sum of the numbers must equal n, the order of the bifurcation. There must not be two 2's in
two identical T-arcs in succession
4)
The sequence of a repetition of
consist of
correspond
to
a
arcs
must be minimal
(Sect. 4.7),
i.e. it must not
smaller sequence. Such a subsequence of arcs would subsequence of numbers with an even total; hence the rule: a
the sequence of numbers must not consist of the repetition of a subsequence with an even total. For example, the sequence 3131 is forbidden, because it contains the since
the total
same
arcs.)
subsequence 3+2 is odd,
31 whose total is 4.
the two
(However,
subsequences
3232 is allowed:
32 do not
represent the
Results. Table 6.4 lists all rooted branches up to n -- 6. Thick horizontal lines separate the sets of branches corresponding to different values of the order
heading (e.g. M) (it is the name of the bifurcation as defined in Sect. 6.2.1). Only branches belonging to the same set can be joined to each other. The n/2 rooted branches corresponding to each free branch are written in consecutive lines, in a group delimited by blank lines; they are deduced from each other by circular shifts of an even n.
The
name
of each set is indicated in
number of basic
a
arcs.
species branches. The second kind family has also been listed in Tais represented by the symbol E. The bifurcation orbit is not an end point, nor an extremurn of C on this family (see Sect. 3.3.1); therefore is provides also two branches, which are distinguished with according to the First
ble 6.4; it
above convention. Number
of branches. The number of second species branches en is tabulated Appendix C, Table CA, as a function of n. Appendix C shows also that for 1.839... For all values of n, the number large n, e,, grows asymptotically as n of branches equals 4 or more, and therefore the orbit is indeed a bifurcation in
.
orbit.
Finally,
we
point out the remarkable fact that, with the symbolic
rep-
resentation which has been
bifurcations of 6.2.11.3
Type
introduced, Table 6.4 is the same for all total type 1, independently of the particular values of 1, J, L. 2. The discussion is much
results of Sect. 4.4.
simpler
in this
case.
We
use
the
6.2 Second
Table 6.4. Total
explanation
bifurcation, type
Species
and First
or
Second
1: rooted branches for
n
:!
Species 6
(see
109
text for
symbols).
of
-2211-
-21111-
213
1113
132
1311
+312
2121
1T6
M
51
E
+
-312-
11
-1141T4
+-411-
-213-
31
15
-112-
+231
-11211-
-2121-
21111
33
-211-
-132-
13
+321
112
123
11112
-1311-
11211 -111111-
3111
-231-
211
1212
-1122-
-11112-
11121
1131
12111
-1212-
_1111-
-1131-
121
Arcs and cation
arc
families. We begin again with
ellipse
tangent
is
a
to the unit circle at
search for the one
point
arcs.
The bifur-
P. There is
a
single
basic arc, extending from one collision in P to the next; it has a duration 27rl and corresponds to J revolutions on the bifurcation ellipse; I and J are
prime.
The bifurcation orbit is characterized
by
the three numbers
1, J, c'
(Sect. 4.4). again,
Here
each of them consists of
Using is the end
rn
basic arcs, with
the results of Sect.
point
+MJ
ellipse supports an infinite number of arcs; m taking all positive values. 4.5, point (iv), we find that each of these arcs families, whose parameters are respectively
the bifurcation
,
of two S-arc
a
+ml + c'H
a
-ml + c'H
I
1)
(6.23)
1
(6.24)
and
-MJ H
is
the step function defined
In
addition,
families
Tl'j
and
by (2.1).
1) is the end point Tj'j (Sect. 4.5, points (i) and (ii)).
the basic
arc
(m
=
of the two T-arc
now for the families of generating orbits to given bifurcation orbit belongs. We ask first whether the bifurcation orbit can belong to a first species family. The bifurcation ellipse is not circular; so it can only belong to a second kind family. The second kind family has the same semi-major axis as the supporting ellipse. Therefore it is characterized by the same values of I and J, and the period-in-family is 21rI (see Sect. 3-3).
Firsi
which
species families. We search a
110
6.
Bifurcation Orbits
It follows that the order is It
n
=
1: the bifurcation orbit is made of
then symmetric since this
one
basic
is
symmetric and therefore it does indeed belong to a family of second kind symmetric orbits. If I + J is odd, this is the family Ejj. There is a perpendicular intersection point in M2 with a positive abscissa x0 1. Therefore, if I > J, i.e. a > 1, Fig. 3.4 shows that the bifurcation orbit belongs to the family segment Ej'j; if I < J, conversely, the bifurcation orbit belongs to the family segment E'j. If I + J is even, using again the fact that there is a perpendicular intersection point is in M2, we find that the bifurcation orbit always belongs to family Ej+j (see Fig. 3.5). Is it possible that the bifurcation orbit belongs also to a family Ejal of second kind asymmetric orbits? The answer is no: if I+ J is odd, since a > I for asymmetric orbits, the bifurcation orbit would have to be common to Ej'j and Eal; but these families do not intersect (see Fig. 3.7). Similarly, if I + J arc.
is
arc
-
El+j
is even, the families
Second species
through
'families.
the bifurcation
and
do not intersect.
Ejal
We consider next the second
orbit,
for
given
a
order
n.
species families passing family is specified by minimal period (Sect. 4.7). Each
minimal sequence of arcs, corresponding to one Each arc must belong to the set identified above: the two infinite sets of a
S-arcs, given by (6.23) and (6.24), plus the two T-arcs T,j and T,j. Here a complication arises with respect to type 1, because the arcs entering the bifurcation orbit are end points of the corresponding arc families. One can therefore move away from the bifurcation, on a given arc family, only in one direction: each family provides only one branch. This corresponds to a definite sign for the variation of the Jacobi constant, which we represent by AC. Now in a composite orbit, all arcs must have the same value of AC. It follows that we can only associate arcs corresponding to the same sign of AC. We must therefore determine this sign for each arc family. C is given as a function of A and Z, or equivalently as a function of V and Z (A is a monotonically increasing function of V; see (4.12b)), by eliminating -/ between
approaches ac -
(9z
(4.6) an
__+0
(4.28) (see
and
Sect.
end point of type 2
ac
V_
,
,
2c'
-
2V
4.6). Computations
on a
S-arc
family,
show that
--
-,E'sign (#)sign
I
(6.25)
.
Using the equation (4.27) of the characteristics and the geometry plane, one finds that the sign of AC, as one moves away from an along a S-arc family, is given by
sign(AC)
as one
there is
of the arc
(A, Z)
of type 2
(6.26)
Thus, for any given value of m, (6.23) and (6.24) correspond to two branches with opposite signs of AC. (This is related to the fact that these two families can in a sense be considered as continuations of each other; see Appendix A. L) For the two T-arc families, things are much simpler; there is (Sect. 4.6)
sign(AC)
-_
-E'
.
(6.27)
Species
6.2 Second
Symbols.
and First
or
Second
Species
ill
family of generating orbits passing through the biby a symbolic representation, in which each S-arc is represented by its m value, a T, j arc is represented by the letter i, and a Tj'j arc is represented by the letter e. In addition, the sequence of symbols is preceded by the sign of the quantity EAC. With the help of (6.26), this sign determines which of the two descriptions (6.23) and (6.24) should be used for the S-arcs: if IIJ > 1, a sign corresponds to (6.23) and a + sign to (6.24); if IIJ < 1, the opposite holds. We represent
a
furcation
-
Formaiion rules. A
following
1) 2)
given
symbols
sequence of
is
acceptable
if it satisfies the
rules:
i and
e can
be present
only
There must not be two i
correspond
if the or
sign
two
e
is
in
to two identical T-arcs in succession
succession, since this would on the corresponding branch
(Proposition 4.3.2). (Also the sequence must not begin and end begin and end with a e.) 3) Counting i and e as 1, the sum of the numbers must equal
with
a
i,
or
n, the order
of the bifurcation.
4)The
sequence of
arcs
must be minimal
(Sect. 4.7).
Results. Table 6.5 lists all rooted branches up to n -- 4. The n rooted corresponding to the same free branch are written together; they
branches are
deduced from each other
by
circular shifts.
First species branches. The second kind family is also listed; it is represented by the symbol E. This family traverses the bifurcation orbit, giving two branches. The bifurcation orbit is not an extremum of C ((3-16) shows that I.e.
a
family of the second kind has extremums in C for orbits), so that both signs of AC are present.
0 and
7r,
circular
of branches. The number of second species branches e,, is tabulated in Appendix C, Table C.5, as a function of n. Appendix C shows also that for large n, en grows asymptotically as n'. For all values of n, the number of branches equals 4 or more, and therefore the orbit is indeed a bifurcation Number
orbit. Here again, thanks to the all total bifurcations of type
symbolic representation, Table 6.5 is valid for 2, independently of the particular values of 1,
J, 6'. 6.2.1.4
Type
retrograde
3. The bifurcation
ellipse
is the unit
direction. At the difference of types I and
circle, described in the 2, we have here a single
bifurcation orbit. arc families. There is a single basic arc, of duration -T. The bifurellipse supports an infinite number of arcs; each of them consists of M basic arcs, with m taking all positive values. Using the results of Sect. 4.5, points (v), (vi), (vii), we find that these arcs are end points for the following Sp arc families:
Arcs and
cation
112
6.
Bifurcation Orbits
Table 6.5. Total bifurcation, type explanation of symbols).
2Tl
2T3
E
4
-21-
-31-
+-12-
-22-
M
-11-
31
12
-211-
li
i2
-li2-
-
i3
e2
ei
-
-
l2e
-
2ei
-
-leil-
-
ei2
-
i2e
-
Illi
-
Me
-
llel
-2il-
-lel-
ie
e12
-
3i
2e
el
-lell-
-
13
-iii-
le
2el
-
-112-
2i
il
-
-ill-
11i
3e
-2el-
-
lell
-
elll
-
llie
-
iell
-
elli
-
llei
-
eill
-
ille
-le2e3
-
lel
-
ell
211 -1111-
liel
112 -
lie
-
iel
-
eli
-
lei
-
eil
-
ile
+
121
21i -11il-
leil
li2 -
lile
-
ilel
i2l -
2le
-
-llel-
-
le2
-
e2l
-
leli
-
elil
-
-
2il -lill-
ieil
-
eili
-
ilie
-
12i -
2ie
ie2
e2i
liei
-
i12
-liel-
n
-
-13-
21
2
-
2T4
3
1
-
2: rooted branches for
leie eiel
-
iele
-
elei
4
(see
text for
6.2 Second
(i)
For each odd value of rn
m:
Species
the two
and First
arc
or
Second
Species
113
families in domain Di with
-
a
(6.28)
2
and
13
(ii)
For each M
a
(6.29)
2
__
even
value of
13
1
-
-
2
and the
=)3
a
family
arc
and
arc
(6.30)
2
in domain
with
D2
(6.31)
(Sect. 4.5, point (iii)),
families
a
can
T1'1
and
period-in-family
is
major axis a
rn
=
2 is the end
point
of
now
generating
for the families of
orbits to
belongs. species family. Here the bifurcation ellipse is circular can belong to a first kind family. Indeed the family 1,
first
a
it
7r
consider a
with
according
mean
(3.7).
to
one
motion
n
--
-1
(Sect. 3.2).
It follows that the order is
basic
arc.
n
The --
1:
The two branches of 1, have
of AC.
opposite signs we
arc
We ask first whether the bifurcation
the bifurcation orbit is made of
include
the
T,',.
orbit of radius 1, with
an
Next
domain D, with
--
bifurcation orbit
includes
in
2
given belong to retrograde; so
orbit
family
M =
species famihes. We search
First
which
arc
_M
=
In addition
the two
the
rn:
=
a
second kind
1, therefore
circular
retrograde
I
-_
J
family. That family -_
must have
1. Indeed both families
orbit of radius 1. The
E1+1
a
semi-
and
period-in-family
ET,
is 27r;
2. In each family, the bifurcation orbit is an exn C, and both branches correspond to AC > 0. -1 and therefore Finally, the bifurcation orbit has a co-eccentricity e' cannot belong to a family of second kind asymmetric orbits (see Fig. 3.7).
therefore the order is tremum
=
of
--
Second species
families. Each second species family is specified by a minimal corresponding to one minimal period (Sect. 4.7). Each arc
sequence of arcs, must
belong
to the set identified above.
Computations show that AC
< 0 in
case
(6.30),
AC
>
0 in all other
cases.
We represent a family of generating orbits passing through the biby a symbolic representation, in which each S-arc is represented by its rn value, a Tl'j arc is represented by the letter i, and a Tl'j arc is represented by the letter e. In addition, the sequence of symbols is preceded by the sign of the quantity AC. For m odd, this is not sufficient to distinguish between (6.28) and (6.29), since AC is positive in both cases; therefore we distinguish the case (6.29) by writing a prime after the value of m. Table 6.6
Symbols.
furcation
114
Bifurcation Orbits
6.
illustrates the the
names
use
of the
symbols more concretely. This table specifies also family segments, derived from Fig. 4.10 and Table 4.1;
of the
arc
this will be useful later. If the
usual;
origin
if the S
becomes
+
31
e
arc,
is inside
arc
+
S arc, we represent the two parts with dashes as is primed, both parts are primed. For instance, the sequence
-2111
-
represent the
we
a
by two
a
shift of the
parts by
Table 6.6. Bifurcation of type 3:
symbol *
1
SOOO
+2
*
3
so
+4
S202
+
+5
SO-,-, (Di) SO-2-2 (Di) So 3-3 (Di)
+
+31 +
5'
Formation rules. A
following
If the
origin
or -e
and
is inside
a
i
or
e-
6
symbol
arc
(D2 ) (D2 ) S-3-3 (D2 ) S -1
2
S -2-1
S-3-2 S-4-3
6
sequence of
-
symbols
is
arc
i
Tii I
+ e
Tie,
+
-,
S_ -2-2
-4 -
given
i-,
symbols.
symbol
arc
origin.
-i and
acceptable
if it satisfies the
rules:
1) i, e, and odd numbers can be present only if the sign is +. 2) There must not be two i or two e in succession, since
this would
correspond to two identical T-arcs in succession on the corresponding branch (Proposition 4.3.2). (Also the sequence must not begin and end with a i, or
begin
and end with
3) Counting
a
i and
e.) e as
2, the
sum
of the numbers must
equal
n, the order
of the bifurcation.
4)
The sequence of
arcs
must be minimal
(Sect. 4.7).
Results. Table 6.7 lists all rooted branches up to
n
=
4.
1, E1+1, E l are also listed. The family 1, represented by the symbol I. The two branches of El+, in fact correspond to the same orbit, with a shift of a half-period (this is an instance of the reflection phenomenon discussed in Sect. 2.5); they are two rooted branches corresponding to the same free branch. Since they both have AC > 0, we must distinguish them by two different symbols. When we move away from the bifurcation on family El+,, the perpendicular crossings of the x axis corresponding to the pericenter and apocenter takes place at the left and right of M2, respectively (Fig. 3.5, left; the bifurcation orbit corresponds to 0 7r). First species branches. The families
is
--
The orbit is
correspond
as
shown
to the
Fig. 6.2a. Thus, the two perpendicular crossings junctions between the two basic arcs (see Sect. 7.3.2). We on
6.2 Second
Species and First
or
Second
Species
115
represent by E++ and E+- the rooted branches for which the crossing
at the
origin is at the right and the left of M2, respectively. Similarly, the two branches of E j correspond to the same orbit; they both have AC > 0, and we must distinguish them by different symbols. In that family, the passages through pericenter and apocenter correspond to the midpoints of the two basic arcs. In the vicinity of the bifurcation, the orbit has the shape of Fig. 6.2b. There are two non-perpendicular crossings of the x axis in the same point, which lies at a small distance to the left of M2 (the point M2 has been slightly displaced to make this distance more visible). We represent by E-+ and E-- the rooted branches in which the radial velocity x > 0 and < 0 at the origin, respectively.
b
a
Fig.
6.2. Orbits of families
E1+1 (left)
and
ET, (right)
in
rotating
axes, in the
vicinity
of the bifurcation orbit of type 3.
of branches. The number of second species branches e,, is tabulated Appendix C, Table C.6, as a function of n. Appendix C shows also that for 3.153... For all values of n, the number large n, e,, grows asymptotically as n of branches equals 4 or more, and therefore the orbit is indeed a bifurcation Number
in
.
orbit.
6.2.2 Partial Bifurcation
This is the
making up
case
where
one or more
the bifurcation orbit
deflection
belong
to
angles
more
than
do not vanish. The one
arcs
supporting keplerian
orbit. Each part of the bifurcation orbit from one non-vanishing angle to the a mavimal arc. One period of the generating orbit is thus
next will be called
made up from
one or
supporting keplerian
more
orbit.
maximal
arcs.
Each maximal
arc
has
a
unique
116
6.
Bifurcation Orbits
bifurcation, type 3: rooted branches for explanation of symbols).
Table 65.7. Total Table 6.6 for
M
+
1
+
I)
3T2 + +
E++ E+_
E_+ E__
+el +
-ele-
+
le
+ e
1
+
2i
+
illi
+
-lil-
+
-ii
+
i2
+
illi
+
-i2i-
+
iii,
+
2e
+
illll
+
-lel-
+
-il 11 li-
+
e2
+
1
+
-e2e-
+
Vill
1
e
+
+
+
+ +
-22-
+-13+31
+
13
+-2111-
+311
+-1121-
+-2111-
+21 +
-ill-
+
12
+211 + +
_1111112
+-1112+
113
+311 +
+
ell
+
-elle-
+
112
+
lle
+121
+
lel
+2111
+
ell,
+
-ell le-
+1112
+
We
1121
+
llel
+
+
+
131
+
311'
_11111-
+2111
+elll
+
_11111-
+
-el
+
1112
+
Vie
+
1211
+
lel'
+21111 +
_111111-
GlIll
+
-elllle-
+
Vile
1121'
+
llell
+ +
ie
+
ilill
+
-iei-
+
11111
+
ei
+
11,11
+
-eie-
+
11111
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
1131
le-
+
+-211111-111121-
I
+11112
-2111'-
+-11121-
11 i
_1111-
+-211+-112-
1
+
+211
+
+-31-
+31
ii-
1
4
+-12-
I
-el le-
+
+3
(see
4
+
3T4
3T3
:!
+
2
+
n
text and
Species and
6.2 Second
A maximal
the
made of different
a
a bifurcahng partial bifurcation orbit
or
Second
Species
in which the maximal
Since at least two different families pass at least one maximal arc is a bifurcating arc. arc, at least
one
117
if there exist two families
arc
arcs.
orbit, bifurcating
bifurcation In
will be called
arc
passing through
First
of the two families
is
made of
arc
through more
is a
than
It follows that the supporting keplerian orbit cannot be a hyperbola parabola, because it would then have only two collisions with M2, and contain only one arc (116non 1968, Fig. 3). It is therefore an ellipse. Combining this with what has been said in Sect. 6.2.1, we see that hyperbolic arcs are never involved in bifurcations. In a partial bifurcation, the bifurcation orbit may contain hyperbolic arcs; but they lie outside of the bifurcating arcs, and play no role in the bifurcation. one arc. or a
We state
now one
Restrictioia 6.2.1.
fundamental limitation of the present work:
Only pariial bifurcafions
with
a
single bifurcating
arc
w%ll be considered.
The
simply that this case is complex enough, as will be seen. Partial multiple bifurcating arcs will have to wait for some future
reason is
bifurcations with work.
Only the bifurcating
bifurcation; the other arcs continue on their respective arc families. It has a unique supporting ellipse, which we call bifurcation ellipse, and which is of type 1, 2, or 3. (It cannot be of type 4 because all arcs would then be of type 4, as shown by (4.8), and would therefore belong to a single supporting ellipse.) We shall correspondingly say that the partial bifurcation is of type 1, 2, or 3. Finally, the number n of arc can
be
implied
in the
bifurcating arc will be called the order of the bifurcation. use a designation of the form P; thus 1P7, for instance, represents the partial bifurcation of type I and order 7. In partial bifurcations, the relative phase of two branches is completely determined by the position of the bifurcating arc. Thus there is no need to basic
arcs
in the
We will
short
introduce free branches and rooted branches
(Sect. 6.2.1.1).
Type 1. In fixed axes, the bifurcation circle at two distinct points P and Q. 6.2.2.1
In addition to the collisions at the ends of the exist at least
one
intermediate collision. If this
ellipse
intersects the unit
bifurcating
were
arc, there must
not the case, there would
be only one possible decomposition of the bifurcating arc, as a single S- or T-arc; this contradicts the definition of a bifurcating arc. The collisions cannot be only at P or only at Q, because the families emanating from it would then consist of a succession of at least two identical T arcs, which is not allowed
(Proposition 4.3.2).
Therefore there
are
collisions
Q. Also, since there are at least three collisions, there exists at least two collisions in the same point. Consider the shortest interval between two such collisions on the bifurcating arc. It corresponds to an integral number I of revolutions of M2 and an integral number J of revolutions of M3. Therefore its duration is 27rl, and A must be of the form both at P and at
118
Bifurcation Orbits
6.
A
1
(6.32)
--
J
mutually prime. body M4 describing the bifurcation ellipse, the situation is then as portrayed on Fig. 6.1 for the case of total bifurcation. The bifurcating are corresponds to a finite interval on that figure, made of a sequence of n alternating basic arcs PQ and QP. The starting point can be either P or Q. The order n must be at least 2, but is otherwise arbitrary (it does not with I and J
For the fictitious
have to be
even).
Thus, the situation regarding the bifurcation ellipse is the same as in the case of a total bifurcation of type 1, and the computations of Sect. 6.2.1.2 are applicable. In particular, Z must be again be of the form L
z
J
where L is Arcs and arc
(6.33)
__
is the
integer.
an
arc
families.
The set of
arcs
which
can
belong
to the
bifurcating
in Sect. 6.2.1.2.
same as
Second species families. We must now find the second species families to a given bifurcation orbit belongs. By definition, the arcs outside of the
which
bifurcation
arc are
bifurcating
arc.
Symbols.
We
use
the
the
same
same
scribe the sequence of
arcs
in all these
symbolic representation as in Sect. 6.2.1.2 to making up the bifurcating arc in one family.
given sequence following rules:
Formation rules. A satisfies the
1) 2) 3)
families; they differ only inside the
Each number must be odd
of
de-
positive numbers is acceptable if
it
2.
or
There must not be two 2's in succession. The
sum
of the numbers must
Results. Table 6.8 lists all
equal
acceptable
n, the order of the bifurcation.
sequences for
again, same can only joined. belonging starting point of the bifurcating arc can be either P or Q. If this is specified, then a sequence from Table 6.8 determines a unique sequence of arcs (Sect. 6.2.1.2). Thus, Table 6.8 should properly be considered as representing two different tables in which the starting point is P and Q, respectively. The bifurcation orbits branches
to the
in the two tables Here
signs
are
each
be
n
--
2 to 7. Here
The
different.
family provides
two
branches, corresponding
to
opposite
of A C.
The are
again,
set
only
case n
=
I does not appear in Table 6.8, because in that case there 1. This corresponds to an ordinary generating orbit.
two branches
Table 6.8 is valid for all
partial
bifurcations of type 1.
6.2 Second
Table 6.8. Partial bifurcation, type explanation of symbols).
lP2
IP6
Species
and First
1: branches for
n
or
Second
:! 7
(see
133
lP7
1321
7
2
51
11
33
52
+
13111
321
511
+
1231
lP3 3
21 12 ill
+312 3111
+
331 3211
1312
1213 +
121 21 .
+231
313
213
3121
+2121
3112
12112 +
121111
115
2112
31111
+
1132
21111
25
+
11311
15
232
+
1123
31
132
2311
11212
211
1311
2131
112111
13
123
21211
+
121
1212
2113
+
111211
112
12111
21121
+
11113
1131
21112
+
111121
11211
211111
+111112
1113
151
1P4
lP5 5 32 311 23 212
2111 131 1211 113
1121 1112 ilill
+
11121 11112
11131
Species
119
Sect. 6.2.1.2 for
6. Bifurcation Orbits
120
of branches. The number of branches b,, is tabulated in Table CA as n. Appendix C shows that for large n, b-n grows asymptotically 1.839... For all values n !: 2, the number of branches equals 4 or more, n
Number a
function of
as
.
and therefore the orbit is indeed 6.2.2.2
Type
2. The bifurcation
P. There is
point
a
a
single
bifurcation orbit. to the unit circle at
ellipse is tangent extending from
basic arc,
one
one
collision in P to the
next.
Arcs and
arc
families. The
set of
acceptable
has been determined in
arcs
Sect. 6.2.1.3.
Symbols.
We
use
the
same
symbolic notation
Formation rules. A given sequence of following formation rules:
in Sect. 6.2.1.3.
as
symbols
is
acceptable
if it satisfies the
1) i and e can be present only if the sign is 2) There must not be two i or two e in succession. 3) Counting i and e as 1, the sum of the numbers must equal
n, the order
of the bifurcation.
acceptable
Results. Table 6.9 lists all
sequences for
partial bifurcations of type 2. The partial bifurcation of type 2 and order
n
=
1 to 4. This table is
valid for all it is
such
a
bifurcation orbit in which
case
no
deflection
n
I is remarkable in that
=
angle
among the bifurcations considered in the
vanishes. It is the
only
present Section 6.2.
of branches. The number of branches b,, is tabulated in Table C.2 as n. Appendix C shows that for large n, b,, grows asymptotically n'. For all values of n, the number of branches equals 4 or more, and
Number a
function of
as
therefore the orbit is indeed 6.2.2.3
Type
retrograde Arcs and
a
ellipse is single basic
3. The bifurcation
direction. There is arc
bifurcation orbit.
families. The
a
set of
the unit
circle, described
arc, of duration
acceptable
arcs
in the
7r.
has been determined
in
Sect. 6.2.1.4.
Symbols.
We
use
the
Formation rules. A
same
given
symbolic notation
sequence of
as in
symbols
is
Sect. 6.2.1.4.
acceptable
if it satisfies the
following 1) i, e, and odd numbers can be present only if the sign is 2) There must not be two i or two e in succession. 3) Counting i and e as 2, the sum of the numbers must equal formation rules:
n, the order
of the bifurcation. Results. For is
a
n
=
1,
we
find
simple continuation,
only
not
a
two branches
bifurcation
+
1 and
orbit, and it
1 1. Therefore this
+
is
table.
Table 6.10 lists all
acceptable
sequences for
n
=
2 to 4.
not listed
in
the
6.2 Second
Table 6.9. Partial
explanation
2P1
of
bifurcation, type
Species
and First
2: branches for
Second
< 4
(see
Species
2P3
2P4
1
2
3
4
i
11
21
31
e
ii
2i
3i
le
2e
3e
ii
12
22
ie
ill
211
el
iii
21i
ei
-
-
-
-
lie
-
lil
-
lie
-
lei
-
2le
2il 2ie
2el
-
lei
-
-
i2
13
-
ill
121
-
iii
-12i
ile
-l2e
iel
112
-
iei
lill
-
e2
iiii
ell
Me
eli
iiii
-
ele
llie
-
eil
Ilel
eie
-llei
-
-
-
-
-
2ei
-li2 lill iiii lile liel
liei le2 lell -
-
-
leli lele leil
-
leie
-
i3
-
i2l
-
i2i
-
i2e
-
i12
-
-
-
ille
-
iiii
-
ilie
-
ilel
-
ilei
-
ie2
-
iell
-
ieli
-
iele
-
ieil
-
ieie
-
e3
-
e2l
-
e2i
-
e2e
-
e12
-
ellt
-
elli
-
elle
-
elil
-
elie
-
elel
-
elei
-
ei2
-
eill
-
eili
-
eile
-
eiel
-
eiei
121
Sect. 6.2.1.3 for
symbols).
2P2
-
n
or
122
6.
Bifurcation Orbits
of branches. The number of branches b, is tabulated in Table C.3 as n. Appendix C shows that for large n, b,, grows asymptotically 3.153.... For all values n 2! 2, the number of branches equals 4 or more, n
Number a
function of
as
and therefore the orbit is indeed
Table 6.10. Partial
explanation
3P2
of
a
bifurcation orbit.
bifurcation, type
3: branches for
n
:! 4
(see
symbols). 3P3
+
3P4
+
*
2
+3
i
+
* e
31
4
+
+31
+
+21
+
311
+
+311
+
+3111
+
1112
22
+
illi
*
11
+211
*
ill
+
il
*
111
+
ill
*
J) J)
+
el
+
2i
+
We
+
ell
+
2e
+
11,11
+
12
+211
+
11,11,
+
ii
+2111
+
11,111
+
le
+2111
+
11,111,
+
111
+21111
+
113
+
ill,
+
i2
+
1131
+
lill
+
ie
+
1121
+
11,11
+
ill
+
112
+
ill
+
11 i
+
*
lle
+
*
1111
+
+
1111,
*
11111
*
J) J) J)
+
+
11211
+
ilil
illi
+
Vill
ii'll
+
llel
e2
+
llell
+
ei
+1112
+
ell
+
ell,
+
Vie
+elll
+
11111
I
illi
*
ellll
+
11111,
*
13
+
111111
*
131
+
111111,
*
121
+
11112
*
1211
+
ill'i
+
+
+ *
+
lel
+
*
lel
+
*
112
+
J)J)J)J)
Sect. 6.2.1.4 for
6.3 Third
6.3 Third
Species
One first species
Species
and First
family
and
or
and First
Second
or
Second
Species
123
Species
infinity of second species families include a species (Sect. 5.2). It corresponds to an end of the corresponding 3. As has been shown in Chap. 5, when an appropriate family, for C is used, this orbit is found to correspond also to the end system of axes of a family of v-generating orbits, for IF -oo. No other family is implied. Therefore there are only two branches, and we have a simple continuation, third
an
orbit --
-*
not
a
bifurcation.
6.4 Third In the
Species
and Third
Species
(x, y) system
of axes, there exists a single family of generating orbits species, and therefore no possibility of bifurcation. If we enlarge the vicinity of M2 by a [L-dependent change of coordinates (5-1), however, we find several families (in fact, an infinity) of v-generating orbits. We inquire now into possible bifurcations between these families. For 0 < v < 1/3, there is a single first species family and an infinity of second species families (see Sect. 5.3). The first species family cannot be implied in a bifurcation with a second species family since its orbits do not pass through M2 (except in the case of the orbit which reduces to M2; but this corresponds to the continuation of the family to larger values of V and will be covered below). Two second species families cannot have a common orbit either, because all deflection angles are non-zero (Sect. 5.3.2.3) and therefore an orbit has a unique decomposition into arcs. So there is no bifurcation in of the third
that
case.
For
1/3
there exist
only the two families of direct and retrograde obviously cannot have a common orbit. Finally we consider Hill's case, v 1/3, for which things become more interesting. Families of periodic orbits of Hill's problem can intersect. For < v,
circular orbits
(Sect. 5.6),
which
=
instance it
was
(1969) that gl (Fig. 5.1). This
found in H6non
orbit: the critical orbit branches emanate from
families g and g' have orbit lies inside both
a common
families;
it, known; but it seems exist between higher-order families of periodic orbits of Hill's problem. bifurcations can be found and studied only numerically. no
other bifurcation
6.5
4
and therefore it is
is
a bifurcation orbit. At present likely that other bifurcation orbits
These
Recapitulation
We have finished the
have been found:
inventory
of bifurcation orbits. Three distinct
categories
124
-
6.
Bifurcation Orbits
The direct circular orbits with
(I
1)/I,
mean
motion
n
=
(1+ 1) /1,
1
-_
1, 2,
and
2, 3,. (Sect. 6.1.2). They correspond species families, of the first and second kind, respectively. Total and partial bifurcation orbits of types 1, 2, 3 (Sect. 6.2). They correspond in most cases to bifurcations between two second species families, and in a few cases to a bifurcation between a first species family and a second species family. Bifurcation orbits in Hill's problem (Sect. 6.4), of which only one is known so far. They correspond to bifurcations between two families of V-generating n
--
-
I
=
..
to bifurcations
between two first
-
-
orbits. We call these three
categories first, second, ihird species bifurcation orbiis easily seen, they are mutually exclusive: first species bifurcation orbits stay at a finite distance from M2 (they do not belong to the second species; see Sect. 2.10), while second and third species bifurcation orbits pass through M2 (they belong to the second species); types 1, 2, 3, and 4 are mutually exclusive (Sect. 4.2). Therefore these three categories can be studied separately. By a natural extension we will speak of first, second, third species bifurcah'ons to designate the junctions of the branches in the vicinity of the bifurcation orbits. By far the richest and most complex case is the second species bifurcations, which will be the main object of study in what follows. respectively.
As is
7. Junctions:
We try
now
to determine how the various branches
joined for
are
listed
in
y > 0. We consider in turn the three
Species Bifurcations
These bifurcations and
common
A
were
found in Sect. 6.1.2. We consider first
to families
Idi and EI,I+1
wish to determine their
we
arriving at a bifurcation categories of bifurcations
Sect. 6.5.
7.1 First
orbit
Symmetry
(I
2
1).
a
bifurcation
Four branches arrive at
it,
junctions.
symmetric orbit intersects twice the
x axis at right angles. We call xO points, and we represent a symmetric periodic orbit by a point in a (xo, C) plane. A family of periodic orbits is then represented by a curve, called its characteristic. Guillaume (1969, Equ. (51)) showed that for small M, the characteristics of the families of periodic orbits in the vicinity of the bifurcation are described to first order by
and x, the abscissas of the intersection
I
(m
AC+
I)xOAxO
+
M2
I
AC
/I -_
sign (x 0) D (M) (- 1)
3(m
+
'
1)2,rn7r
(7-1)
where AC and AxO represent the displacements from the values corresponding to the bifurcation orbit for y -- 0; m is equal to I in the present notations for
family Idi; and D is a complicated integral which depends on m and easily verified, the two factors in the left-hand side represent
As is
tangents
EI,I+1 shape
to the
characteristics of the families of
seek
we
only
to establish the branch
determine which branch is of the
known
the
orbits Idi and
for y -- 0. Thus, (7.1) shows that the characteristics have locally the of a hyperbola which matches the characteristics asymptotically.
Here
sign
generating
xO.
right-hand
junctions qualitatively, i.e. to which, and therefore we need only the (7. 1). The sign of D for integral arguments is
joined
side in
to
(Guillaume 1969, Equ. (40)):
sign(V(m)) Substituting sign
--
into
AC+
-sign (m) [-sign (xo)]
(7.1),
we
(7.2)
obtain
(M + 1)XOAXO M2
M. Hénon: LNPm 52, pp. 125 - 135, 1997 © Springer-Verlag Berlin Heidelberg 1997
AC)
-[sigii(xo)]'+'
(7.3)
Junctions:
7.
126
For orbits of
Symmetry
therefore select the abscissa x0 with
We
We obtain the
simple result
sign
(I
the abscissas x0 and x, have
family EI,1+1,
(Sect. 3.3.1.1).
AC+
can
(M + I)XOAXO M2
I
AC
opposite signs positive sign.
a
(7.4)
1
This equation shows that the characteristics do not intersect the line Axo = 0. Therefore the junctions are established: the two branches with Axo > 0 are
joined, and so is a decreasing
are
the two branches with Axo < 0. The Jacobi constant C
function of x0 in
family Idi. Using
also
Figs.
3.3 and 3.4,
we
obtain
Proposition 7.1.1. The branch of family Idi corresponding to increasing C z.s Joined to the family segment EI',,+,. The branch of family Idi corresponding to decreasing C i's Joined to the family segment Ej,,+,The
same
results
can
be obtained
point, with abscissa x, < 0. The study of a bifurcation orbit
by considering the
common
to families
other intersection
Id, and EI,I_i
(I
!
2)
is similar and we omit details. Equs. (7. 1) to (7.4) are still valid; for family Ide, Guillaume's m is equal to -1 in the present notations. The Jacobi constant
C is
an
increasing
function of xo in
family Ide
-
We obtain
Proposition 7.1.2. The branch of family Ide corresponding to increasing C is Joined to the family segment Ee,,_,, The branch of family Ide corresponding the to is to decreasing C family segment EI',I-lJoined
7.2 Third We
dispose
Species Bifurcations
of this
simple
case
before
tackling
second species bifurcations.
One third species bifurcation is known, between families g and g' (Sect. 6.4). Here it seems that only numerical computations can establish the junctions between the four branches for small values of ft. What happens by Fig. 1 in Perko (1983), based on results of Broucke (1968)
shown
is
clearly
(see
also
Fig. 10.5):
Proposition
7.2.1. For M >
0, the branch g+ is Joined
'which goes to IF = +oo, while the branch goes to r -- -oo.
g'
7.3 Second
Species Bifurcations
We
to the most
come
now
many branches arrive at
a
is
Joined
of g of g which
to the branch
to the branch
complex case. As shown in Sect. 6.2, typically given bifurcation orbit, and the determination
7.3 Second
of the
junctions is much previous Sections. A first
more
Species Bifurcations
difficult than in the
simple
cases
127
of the two
strategy, which will be developed in this chapter and the next, properties of the orbit which remain invariant through a
is to search for
bifurcation. This will allow
us
to divide the set of branches into
disjoint
subsets, Only branches
each subset
a
corresponding to a particular choice of the invariants. which belong to the same subset can be joined. Therefore, if contains only two branches, they must be joined.
subset
Every subset must contain an even number of branches (Proposition 6. 0. 1); provides a check of the correctness of the derivations. In this chapter, we consider the property that the orbits are symmetric un-
this
der the fundamental symmetry E (Sect. 2.7). This property is invariant along a family: symmetric orbits cannot suddenly become asymmetric, because in
that
family would branch into the symmetrical branch of asymmetwell, and we would have the forbidden situation of Fig. 2.2a. Similarly, asymmetric orbits cannot suddenly become symmetric. (However, a family of asymmetric orbits can contain isolated symmetric orbits.) Thus, for each bifurcation orbit found in Chap. 6, the set of branches can be divided into two subsets, corresponding to branches of symmetric and asymmetric orbits, respectively. ric
case
the
orbits
as
7.3.1 Partial Bifurcation
We consider first the
case
of
partial bifurcation, which is simpler.
We call
furcating
complement the part of the orbit which remains when the biarc is removed. An orbit is symmetric iff (i) the bifurcating arc is
symmetric
and
(ii)
complement is symmetric. The two ends of the bifurexchanged in the symmetry. The two points of perpendicular intersection with the x axis are the middle points of the bifurcating arc and of the complement. If the complement is asymmetric, all orbits are asymmetric, and the symcating
the
arc are
metry criterion is useless. 7.3.1.1
can
Type
It also
arc.
be
1. The
exchanges
symmetry exchanges the
the
points
P and
Q.
bifurcating bifurcating arc
two ends of the
It follows that the
symmetric only if its order n is odd. work, we shall make the following
In the present
Restriction 7.3.1. In where the
complement
This
is the most
n
odd, only the
case
interesting because more junctions can be determined. occasionally point out the changes required when this restriction is
case
We will
partial bifurcations of type I with symmetric will be considered.
i's
not satisfied.
The symbols defined in Sect. 6.2.1.2 and used in unchanged in the symmetry, which therefore simply
the sequence.
Tables 6.4 and 6.8 reverses
are
their order in
Symmetry
Junctions:
7.
128
Table 7.1, derived from Table
6.8, shows the subsets obtained with the
symmetry invariant. They are separated by thick horizontal lines. For each odd value of n, the set of Table 6.8 is divided into two subsets, whose names
by appending
formed
are
previous
A
(for asymmetric)
or
S
(for symmetric)
to the
name.
satisfied, the two subsets are distinct. Only branches belonging to the same subset can be joined to each other. If Restriction 7.3.1 is not satisfied, i.e. if the complement is asymmetric, the two subsets should be joined into a single set; in effect we are back to Table 6.8. For even values of n, only asymmetric orbits exist and the previous set is If Restriction 7.3.1 is
not divided.
As
was
tables,
the
6.8, Table 7.1 represents actually starting point is P and Q, respectively.
for Table
case
in which the
Type 2. The symmetry reverses the order of the sequence and exthe symbols i and e. The other symbols are unchanged. The sign
7.3.1.2
changes
change. again
does not
We restrict
our
Restriction 7.3.2. In
complement
in
two different
Z's
field of
investigation:
partial bifurcations of type 2, only the
case
where the
symmetric will be considered.
For each value of n, Table 7.2 divides then the branches which were listed Table 6.9 into two subsets of symmetric and asymmetric orbits. For
n
1, consideration of the symmetry has split the former set of 4 branches, so that the junctions are established:
=
branches into two subsets of 2 +
lisjoinedto-
I
and
,
-
iisjoinedto-
e
.
again, the symmetry reverses the order of the sequence and exchanges the symbols i and e. The other symbols are unchanged. The sign does not change. We restrict again our field of investigation: 7.3.1.3
Type
3. Here
Restriction 7.3.3. In
complement
in
is
partial bifurcations of type 3, only the
case
where the
symmetric will be considered.
For each value of n, Table 7.3 divides then the branches which were listed Table 6.10 into two subsets of symmetric and asymmetric orbits.
7.3.2 Total Bifurcation
In this case, for
symmetric orbits,
an
additional invariant
can
be used: the
position in the sequence of the two perpendicular crossings of the symmetry axis. Each crossing lies either at the junction of two basic arcs, or in the
midpoint
of
a
Each basic
specify
the
basic arc
arc. is
position
by its midpoint into two basic arc halves. perpendicular crossing by the number h of basic
divided
of
a
halves between it and the origin. Since the total number of basic
arc
We arc
halves
7.3
Table 7.1. Partial
IP2 A
bifurcation, type
lP5 A
Second
Species
Bifurcations
1: branch subsets determined
21211
lP7 S
2113 2
32
7
21121
11
311
313
211111
23
232
lP3 S
+2111
3
ill lP3 A
+
133
21112
1321
1211
151
1312
113
12121
1121
11311
13111 +
1112
1231 1213 12112
21 12
lP6 A +51
lP7 A
511
321
331
31
312
3211
211
3111
3121
13
231
3112
121
213
31111
112
2121
25
2112
2311
21111
2131
lps S
+15
132 212 131
1311 123 1212
12111
1131 11211 1113 11121
11112
115
52
33
lP4 A
121111
1132 +
1123 11212 112111
11131
111211 11113 111121
111112
129
by symmetry.
130
7.
Junctions:
Table 7.2. Partial
2Pl S
2P1 A i
Symmetry
bifurcation, type
2P3 A
2: branch subsets determined
2P4 S
21
4
2i
22
2e
121
12
lill
iii
-
lie
-
liel leil
iii
-
i2e
lie
-
ille
2
lei
-
ieie
11
lei
-
e2i
ie
i2
-
ei
ill
-
2P2 A
iel
2P2 S
ii le
ii el 2P3 S
iei e2
eil
eiei
lile
liei
-
le2
-
lell
-
leli
-
lele
-
leie
-
i3
-
i2l
-
i2i
-
i12
-
-
-
ilie
-
ilel
-
ilei
-
ie2
31 -
3i
-
3e -
iell
-
ieli
-
iele
211 -
21i
-
2le
eie
-
-
2il
-
2ie
-
2el
3
ill
ile -
eli
e3
-
e2l
-
e2e
-
e12
-
elll
-
elle
-
elil
-
elie
13
12i l2e
112
iiii -
elel
-
elei
-
ei2
Me
iiii -
eill
-
eili
llel llei -
-
eile
-
eiel
li2 lill
ieil
-
2ei
llie
iiii
-
2P4 A
ell ele
elli
-
-
by symmetry.
7.3 Second
Table 7.3. Partial
bifurcation, type
3P2 S
3P4 S
3: branch subsets determined
+
1211
+
2
4
+
22
+
lel
+
ie
+
lel
+
ei
+
112
+
121
+
iii
+
lill
+
lle
+
11,111
+
+
+
3P2 A +
i
+ e
+11211
3P3 S +3 +
11111,
+
+
J) J) J) J)
+
+31
3'
+311
+
+311
+
+
+ + +
J) J) J)
3P3 A
+
3'1' 2i 2e
+211 +2111 +2111
+21 +211
+21111
+
+
+
+
i2
+
+
+
*
12
+
*
ii
+
e2
*
le
+
ell
ill,
+
ell
*
11,11
+
elll
*
V2
+
ellll
*
1
i
+
13
*
Ve
+
131
+ +
11 li
+
l1le
+
11,11
+
11,11,
+
11,111,
+
113
+
1131
+
1121
+
ilil
+
Vill
+
llel
+
llell
+
1112
+
illi
+
Vie
+
*
7
1112
+
+
1
+ e
+
+
3P4 A
Species Bifurcations
+ +
11112
+
1
+
Vile
+ + +
1i
131
by symmetry.
Symmetry
Junctions:
7.
132
is 2n, the values of h for the two crossings differ by n. We use the smallest value, which satisfies 0 :! h < n. h is an invariant in the bifurcation (because the position of the symmetry axis cannot suddenly jump from one position
to
another),
and
Subsets for other
by
a
we can use
given
a
simple
it to further divide the branches into subsets.
value of
n
and various values of h derive from each
shift of origin.
Type 1. The symmetry exchanges the points P and Q. It follows perpendicular crossing can lie only in the midpoint of a basic arc: h is 1. n 1, 3, always odd. We distinguish n/2 subsets corresponding to h 7.3.2.1
that
a
--
Table 7.4 divides the branches which The last or
symbol
in the
the letter A for
name
of
asymmetric
a
were
-
..
.,
listed in Table 6.4 into subsets.
subset is the value of h for
symmetric orbits, n 2, is
orbits. The first species orbit E, for
--
1. symmetric with h branches Only belonging to the same subset can be joined. For symmetric associated with a given free branch (see Table 6.4) branches rooted orbits, 6. different can now belong to subsets; this happens here for n --
--
Table 7.4. Total
1T2
I
bifurcation, type
IT6
1
1: branch subsets determined
1T6 A
E
-312-
321
11
15
-1212-
-231-
-1131-
1T4
1
-211-
13 1T4 3 31 -1121T4 A
+
1131
213
1212
132
1T6 3
312 +-1122-
114-
+231 -2121-
+33
123
3111 1113
211
+-2211-
-21111-
+-1311-
-11211-
21111
112
11112
1T6 5 -1111121
51
+
11211 -111111-
-213-
11121
-132-
12111
-111121311 2121
by symmetry.
7.3 Second
7.3.2.2
Type
distinguish
n
2.
Here, for symmetric orbits, h corresponding to h 0, 1,
subsets
=
Table 7.5 divides the branches which The first
species orbit E,
for
n
=
Table 7.5. Total
bifurcation, type
2Tl 0
2T3
2
were
Species
can ...'
Bifurcations
be odd n
-
1, is symmetric with h
--
0.
2: branch subsets determined
21i
2T4 1
-11il-12-
-211-
li2
1
21
13
i2l
iel
lile
eil
leli
2
2T3 A
2le
-llelle2
2T4 2
e2l
-11-
-
ie
-
2i
-31-
ei
-
-lil-
-13-
A
211
2il -lill-
M A
112
2e
-
-
2ie
-
ie2
-
2ei
-
ei2
e2 -
-
le
11i
el
lil ill
2T3 0
-
llie
-
iell
lle
-
lel
-
3
-ill-
ell
eli ile 2T3
-21-
-22-
ilel
-1111-
elil
lie
-
-liel-
lei
-
-
e2i
-
-
-leili2e
-
liel
-
elli
-
31 -112-
-
leil ille
12i
-
2el
-
-lell-
-
e12
-
l2e
-
111i
-
-
-
4
121
12
eill
i12
2T4 3
2T4 0
1
llei
2T4 A
-
-
-
3i
-
Me
-
llel
-
lell
-
elll
-
liei
-
ieil
-
eili
-
ilie
-2il-
leie
-
eiel
-li2-
-
A
-
3e
-
-2el-
-
-le2-
-
e3
even, and
we
1.
listed in Table 6.5 into subsets.
E
M 0
or
133
-
iele
-
elei
by symmetry.
134
Junctions:
7.
Symmetry
7.3.2.3 Type 3. Here again, corresponding to h 0, 1, =
.
.
.,
for n
-
symmetric orbits,
Table 7.6 divides the branches which The first
we
distinguish
n
subsets
1. were
listed in Table 6.7 into subsets.
species orbit I, for n 0. 1, is symmetric with h For n 2, in the first species family E1+1 (branches + E++ and + E+-), the pericenter and apocenter correspond to the junctions between basic arcs (Fig. 6.2); thus the branches are symmetric, with h 0. For family ET, (branches + E-+ and + E--), the pericenter and apocenter correspond to the midpoints of the basic arcs; the branches are symmetric with h 1. =
=
=
=
=
7.3.3 Conclusions
Symmetry one
case
is obviously a weak criterion: junctions have been established in only. However, the size of the subsets to be considered has been
generally reduced.
7.3
Table 7.6. Total
bifurcation, type
3T3 A
M 0
1
+
3T4 2
-iii-
+
+
E++
+
+
2
+
_11-
+
+
3T2
1
+
E_+
+
E__
1112
-13-
+
1211
211
+
112
11112
-iei-
el
+
-eie-
-Gle-
+
le
+
ell
+
-el le-
+
Ve
+
+
ill,
+ +
+
3T4 3 +
+31
+
+-112-
+
311
+
+
3T3 0
_11111-
+
+
+
+
+
+
3T4 0
+
+2111
+21111
+ +
Bifurcations
-31-
+
+
3T2 0
Species
3: branch subsets determined
+ +
Second
4
+-1112-
+
ii'll
-22-
+311
+
-ii'lliill'i
+
_1111-
+-11121-
+
+3
+
121
+3111
+
Vill
+31
+
_111111-
+-111121+
ell
+
-elle-
+
-Ill-
+
11211
+
-11
+
ie
+
1111
+
ei
1
1-
+
+
+
111,111
+
+
+
3T4 A
3T3 1 3T4 1 +-21-
+
2i
+
lle
+
lel
+
ell,
+
-ellle-
+
l1le
+
llel
-iii* *
12 112
+
13
+-2111-
+
1111
+
+
11,11
+-21111+
3T3 2 +-12-
+
1'31
+
11111
+
11,111, +
1
1
i2
+
-i2i-
+
-el
+
l1le
+
lel
+
2e
I
le-
I
+
e2
+
ellll
+
-e2e-
+
-el
+
Vile
+
Ilell
+2111 _11111-
11111
+21
+
131
+-211111-
+-1121-
+211
V3
1
+
+ e
+
1112
+
1121
I
lle-
by symmetry.
135
Principle
8. Junctions: Broucke's
another, more powerful invariant, based on what we call B,roucke's principle. It was enunciated by Broucke (1963, p. 75; 1968, p. 21). A proof has been given by Perko (1981a, 1981b). We
study
now
8.1 Definition
principle applies to the vicinity of second species bifurcation orbits (Sect. 6-2). Roughly speaking, it states that the side of passage of M3 with
Broucke's
respect
to
M2
Consider
is
invariant.
collision
a
point in
a
bifurcation orbit, at which the deflection
belongs to a total bifurcation orbit, or it lies angle (Therefore inside the bifurcating arc in a partial bifurcation orbit; see Sections 6.2, 6.2.2.) Consider the orbit Q in the vicinity of the collision point, in rotating axes (Fig. 8.1a). When we move away from the bifurcation orbit on a particular branch, two cases are possible: either it
is zero.
The collision In that case, not traverse
the
x
axis
point
was
M2
inside
as we
as soon
an
S-
or
anymore. We call this
either to the left
T-arc
(or
leave the bifurcation
or
to the
an
a
first
orbit,
species orbit).
the orbit Q does
antinode. The orbit intersects
right of M2 (Figs. 8.1b point, and we define
and
passage 0-
sign(xo
=
still
joined
and
e).
-
1)
point
in
was
a
o-
we can
=
junction
(Figs.
of two is
arcs.
generally
These two
arcs
(Figs
non-zero
are
8.1d
node.
8.1f and
g),
orbit for y > 0. It passes in the close either to the left or to the right, de-
of the deflection
=
and
at the
neighbouring periodic
of M2
vicinity
a
pending on the sign case of Figs. 8.1f, o0,
of
(8-1)
.
M2; the deflection angle
We call this
Consider
-+
We
as
The collision
M
c).
the side
call xO the abscissa of the intersection
+1 for the
then also define
+1 for Figs. 8.1e.
M. Hénon: LNPm 52, pp. 137 - 169, 1997 © Springer-Verlag Berlin Heidelberg 1997
case a
angle. We define: oof Figs. 8.1g. By going
side of passage
-I for the
=
o-
=
to the limit
-1 for
Figs. 8.1d
138
8.
Junctions: Broucke's
m
Principle
x
2
0
N
\
-
Fig.
8.1. Side of passage.
a
8.2 Side of
Passage
for
a
Node
139
principle states that o- is invariant in a bifurcation. For [L > 0, a branch showing the behaviour of Fig. 8.1c, for instance, cannot be joined 0, we to a branch showing the behaviour of Fig. 8.1f. Going to the limit Y deduce that branches corresponding to Figs. 8.1b and d belong to one subset (o- -1), while branches corresponding to Figs. 8.1c and e belong to another Broucke's
-+
-_
subset
(o-
+1).
=
vanishing angles, Thus, in a total principle applies separately and in principle we can distinbifurcation of order n, we have n invariants, guish 2' subsets. (In practice many of these subsets are empty.) For a partial If the bifurcation orbit involves several collisions with
to each of them.
then Broucke's
bifurcation of order n, We must
We call of
Passage
Cos
-Y
I +
=
V Cos
The Jacobi constant is
C
Fig.
=
3
-
I invariants. various cases.
and for antinodes
for
a
This
computation
will be
(Sect. 8.3).
Node
0
,
Vsin7
=
simply related
Vsino
to
(8.2)
.
v:
(8.3)
2
v
Similarly,
T
-
8.2. Definition of the v, 0 coordinates.
the end of For
n
in the
polar coordinates of the velocity of M3 at the beginning in roiaiing coordinates, with the y axis as origin for 0. They are the polar coordinates V, 7 in fixed axes (Sect. 4.2) by (Fig. 8.2)
related to V
o-
and 0 the
v
arc,
an
have
(Sect. 8.2)
different for nodes
8.2 Side of
we
compute
now
a
arc
we
an
call
arc.
v
(v
and 0' the has the
polar coordinates
same
velocity of A113 at by virtue of (8-3).)
of the
value at both ends
S arc, we have 0' = -0 as a consequence of the symmetry, while for a 0 since both ends correspond to the same point on the we have 0' =
supporting ellipse.
140
Principle
Junctions: Broucke's
8.
designate by the subscripts b and a (before and after) the quantities relating to the two arcs which are joined at M2. The deflection angle is thus We
60
0,1
=
angle vicinity of
This
(8.4)
*
is small in the
a
quantity
considering here, which is symbol generally represent of the first arc b to the beginning of
which
case
bifurcation orbit. The
a
variation of second
Obb
-
from the end
we
are
6 will
crossing of the x axis IS towards increasing y, i.e. if cos 0 > 0, angle 60 is positive, the orbit passes on the right of M2: o(This is the case represented in Fig. 8.1g.) Examining the three other we find that a is generally given by the deflection
This
sign(60 cos 0)
=
sign[b(sin 0)]
=
Type
8.2.1
0
(8.3)
and if =
+1.
cases,
(8.5)
.
=
sign(sin 0,,
-
sin
0') b
(8.6)
.
I
(Fig. 8.2)
We have
is
the
also be written
can
o-
the
arc a.
If the
a
the
=
V2
+ 2v
shows that
cos
0 + I
(8.7)
.
is invariant
v
the collision. Therefore the variation of V
in
given by 2V 6V
V is
an
-2v sin 0 60
--
increasing
(8-8)
.
function of A
(see (4.12b)). Combining
with
(8.5),
we
obtain o-
--
-sign(6A cos 0 sin 0)
Consider
(8-9)
.
particular arc on a second species orbit in the vicinity of the represented by a point in the (A, Z) plane. We call AA, AZ its displacement with respect to the point representing the bifurcation orbit. (The symbol A will generally represent the variation of a quantity as we move away from the bifurcation. It should not be confused with 6, defined a
bifurcation. It is
above.) (i) For a
straight 13
=
a
S arc, the characteristic of the
line of
/3o
family in the (A, Z) plane
is
slope
k +
arc
1
(8-10)
2
(equation (6.21)),
where k is
an
odd
integer characterizing
the
arc
family.
Therefore AZ
--
(00+
k 2
1J)
AA.
(8.11)
8.2 Side of
On the other
hand, the second species orbit
value of
which is the
AZ
AC,
((9Z) jc-
=
k
)30 We define
I +
=
Node
We
corresponds to a given have, to first order
(8-12)
.
C
(8.12):
AC
(8-13)
("Z)c
-
M
[(i9Z) 00]
(8.14)
_
-
aA
J
141
K, characterizing the bifurcation, by
number
a
and
a
whole
as a
arcs.
for
A
j
2
2
K
I
-
+
AA
aA
(8.11)
(jC az)
for all its
(az)
AC + A
We obtain AA from
AA
same
Passage
C
play a fundamental role in the following classifications. It has a simple geometrical interpretation: (8. 10) and (8.14) show that the slope P of a characteristic in the (A, Z) plane is smaller than the slope (aZlaA)c Const. passing through the bifurcation (see Sect. 4.6) if of the curve C k < K, and larger if k > K. We obtain finally, for a S-arc This number will
--
-2 (aC)A
AA
(ii)
(8.15)
k-K
J
things are much simpler: vertical line, and therefore
For
plane is
AC
a
a
T arc,
the characteristic in the
(8-16)
AA=O. We compute 6A Four
=
(AA),,
If both
6A
2 =
=
the variation of A
(AA)b
-
through
the
collision, which
is
(8-17)
-
arcs are
S arcs,
WA AQkb .(aC (kb
J
Combining o-
now
must be considered.
cases
(i)
(A, Z)
-
K)(k,,
this with
sign
az) 1(aC
-
-
(8.9),
we
have
k,,)
(8-18)
K) we
A C(k,,
obtain
-
A
kb) (kb
-
K) (k,,
-
K) cos 0 sin 0
I
-
(8-19)
P, we have sin 0 > 0, kb < 0, k,, > 0 (see Sect. 6.2.1.2). If the collision is in Q, the signs are reversed. Therefore there is always If the collision is in
kb sin 0 and
<
0
k,,
sin
0 > 0
(8.20)
142
Junctions: Broucke's
8.
(k,,
kb) Sin 0
-
0
>
(8.21)
-
Therefore these two factors 8
Principle
be taken out of
can
(8.19).
We define
2 Sign(,61C)
::--:
(8.22)
with
A2 s
__
is the
(aZ)
Sign
OC
sign appearing
01
Cos
A
(8.23)
-
in Tables 6.4 and 6.8. The
value 1 for each bifurcation orbit of type 1. value in P and in Q.) We have now o-
s
_-
(11)
[- (kb
If the first
6A
o-
sign
=
-(AA)b
sign
_-
-
K) (k,,
arc is
-
quantity A2
(In particular
has
a
d efinite
it has the
K)]
same
(8.24)
S and the second
arc
is
T,
we
have
(8.25)
;
I (OC ) OZ
AC(kb
-
K) cos 0 sin
A
01
(8.26)
-
So a
s
--
sign [-(Icb
(iii) Similarly, 6A
=
-
K) sin 0]
if the first
arc
(8.27) is T and the second
arc
is
S,
we
have
(AA),,
(8.28)
and o-
s
--
(1v)
sign [(k,,
If both
-
arcs
K) sin 0] T-arcs,
were
indeterminate. However
(8.29)
(and
this
we is
would have 6A
probably
not
a
forbidden
by the formation rules (Sect. 6.2.1.2 and Equations (8.24), (8-27), (8.29) allow us to compute
fied if
with
introduce the concept of given value of K.
we
a
Definition 8.2.1. A normal same
k
-
sign
-
k,
or a
K and k have An
-
as
T
arc.
opposite
equivalent definition
arc
a
normal
is either
Conversely,
an
arc.
a
S
-
o-.
Consider
arc
0 and
would be
o-
coincidence), this case Proposition 4.3.2). Their a
arc
is
is
simpligiven bifurcation,
such that k
abnormal
is
a
S
use
-
K has the
arc
such that
signs.
is
1, all arcs are normal. 1, ingoing S arcs with 0 < k
If -I < K < If K >
< K
are
abnormal;
other
abnormal;
other
arcs
are
arcs
are
normal. -
If K <
normal.
-1, outgoing S
arcs
with K < k < 0
are
8.2 Side of
Values of K
can
be
Passage
for
computed from (8.14) and (4.79). They
in Tables 6.2 and 6.3. As far
a
Node
are
143
tabulated
results, number, which never takes integral values. Therefore the sign of k K is always well defined. Using the properties (8.20), we can easily verify that the following rules hold at a junction between two arcs: K is
arbitrary
an
judged
be
as can
from these numerical
real
-
between two normal arcs, a normal arc and
+8
between
-S
A
junction between
(k
arc
>
0)
abnormal
can
only
two abnormal
be abnormal
only
an
is not
arcs
abnormal
(8-30)
arc.
possible, because an ingoing S an outgoing S arc can be
if K > 0, while
if K < 0.
majority
We remark from the tables that in the
of
cases
there is -1 <
nearly horizontal in a large part of the (A, Z) plane, and in particular for large A (Fig. 4.15). For (OZIOA)c -_ 0, (6.14) and (8.14) show that -1 < K :!! I approximately. K < 1. This is because the
Hence
arcs are
normal in most
same as
sign Looking D, and
keeping
-"Z ) (OC
-_
A
( 07 )
cos
=
0
can
(8.31)
.
A
be obtained from
sign(V cos 7
negative
in most
-
1)
is
negative
in
domains
(8.2):
(8-32)
.
cos
0 is
always negative. In domain D1,
(in particular it is negative for all 6.2), but it can also be positive. The
cases
ellipses tabulated in Table ellipse with the smallest I for --
we
OZ
In domain D2 , there is V < I and
L
(4.6)
constant is
have then
Figs. 4.6 and 4.7, we find that (19ZIOC)A D2+ and positive in domain D 2
sign(cos 0) cos
are
at
The sign of
0 is
Const.
cases.
V constant. From
-sign
--
A2- (4.12b) shows that keeping A
We need also the value Of
the
C
curves
which this
happens corresponds
to I
bifurcation
bifurcation =
8, J
=
5,
-4.
A2 is tabulated in Tables 6.2 and 6.3. 8.2.2
Type
2
Equation (8-9) We go back to o-
=
can no
be
used, because
sinO
--
0 at the bifurcation.
(8-6):
sign(sin 0,,
signs of sin 0 separately. The
longer
-
sin
0') b
and sin 0'
(8.33)
.
are
given by
Table 8. 1. We consider various
cases
(i) Two S-arcs. 0 is positive for an ingoing arc, negative for an outgoing arc (see Sect. 4.3.1; (6.23) and (6.24) show that 3 is never zero). Therefore, using (6.26), we have
144
Junctions: Broucke's
8.
Table 8. 1.
Signs
Principle
of sin 0 and sin 0'
sign (sin 0')
sign (sin 0) ingoing S-arc outgoing S-arc ingoing T-arc (T') outgoing T-arc (T')
sign(sin 0,,)
+ + + -
sign(#,,)
=
+ -
-sign[c'AC(A
=
-
1)]
(8.34)
.
Similarly,
sign(- sin 0') b
sign(sin Ob)
=
sign(,3b)
--
=
-sign[cAC(A
-
1)]
(8.35)
Therefore o-
=
-sign[c'AC(A
sign(c'AC) (ii) Two
is the
=
(iii) sin
sign appearing
in Tables 6.5 and 6.9.
should not be identical
arcs
From Table 8.1
+1
for
-1
for
A S-arc and
0,, and
(8.36)
.
T-arcs. These two
Proposition 4.3.2). 0-
1)]
-
we
junction T'Ti a junction T'T' a
a
(in
T-arc
any
0'b have the
(Sect.
6.2.1.3 and
find that
,
(8-37)
.
order).
In that
it may happen that must determine which arc has case
same sign, we largest absolute value I sin 0 1. This is also the arc which has the largest 2 sin 0; or the smallest COS2 0; or, using (8.7) (in which v is invariant), the sin
and
the
smallest
Vsign(cos 0).
For the
S-arc,
we
T-arc, V has the
have from
sign(AV) Since
a
T
arc
=
sign (AV)
sign[AC(c'- V)]
Vy, v.
=
-
(8-38)
1)
c'AC
<
0
(Sect. 6.2.1.3),
and
(8-39)
.
vertical component of
V; E'V
-
I
--
v
Y
-
v cos
0,
vertical
Therefore sign (cos
sign[AVslgn(cos 0)] Thus, the term Using Table 8.1, we 0'
in the bifurcation orbit. For the
.
must have
we
sign(C'V
We have e'V
component of
as
(6.25)
is present,
sign(AV)
value
same
0)
=
(8.40)
,
+1
(8.41)
.
associated with the T-arc
always
dominates in
(8.33).
obtain
+1
for
a
-1
for
a
junction ST' junction ST'
or
T'S
or
T'S
(8.42) .
8.2 Side of
In summary: if o-
If
-sign(A
=
sign(cAC)
--
-
sign(c'AC) 1)
=
a
type
2
second i
first
arc
e
1, 2,
Type
8.2.3
...
Node
145
present, and
o-
given by
is
Table 8.2.
node, for sign(c'AC)
arc
1, 2,
e
+
+
+
sign(A
-
1)
3
again (8.33) to determine o-. 0, the only arcs are S-arcs given by (6.30), (i) < 0). So we always have in this case
We
a
(8.43)
for
a
are
for
.
-1, the value of
Table 8.2. Value of
+1, only S-arcs
Passage
use
If A C <
0-
-1
-_
which
outgoing
are
(8.44)
.
are given by (6.28), (6.29), (6.31). TIl and Tl',. The corresponding families lie in domains Di, D2 and D3, and they all end in the point (A 1, Z 0) (see Sect. 4.5, points (iii), (v), (vii). The vicinity of this point is represented on Fig. 8.3. The symbols for the branches are those which have been defined in
(ii)
If AC >
In addition
we
0, the possible S-arcs
have the
arcs
-_
,
=
Sect. 6.2.1.4. At any junction, the value of o- is determined in (8-33) has the largest I sin 0 1. This is also the arc with the smallest
by
the
arc
which
Vsign(cos 0) (see
1 here, the arc with the largest A. previous Section); or, since cos 0 Const. in the Figure 8.3 shows schematically the shape of a curve C =
-
--
(dotted line).
vicinity of the bifurcation and
D2 respectively.
From this
,
the order of
...
decreasing
<
sin 0'
are
...
read the dominance
given by
i,
>
order,
i.e.
e
>
...
> 6 > 4 > 2
-
(8.45)
Table 8. 1; the S-arcs 1, 3, 5, are and 2, 4, 6, are outgoing .
the S-arcs 11, 31, 5 1,
.
..
.
..
..
0).
From these basic facts
ble
one can
parts, lying in D,
A:
> 5 > 3 > 1 > 11 > 31 > 51 >
The signs of sin 0 and ingoing (0 0) while
(0
figure
It consists of two
8.3), excepting (Sect. 6.2.1.4
bidden
the
we
cases
and
derive the value of of two i
or
two
Proposition 4.3.2).
e
o- in
in
the
various cases
succession,
which
are
(Tafor-
146
8.
Junctions: Broucke's
z
Principle
4
5
2
3
i,e
3'
A
Fig.
8.3.
Type
3 node:
Table 8.3. Value of
o-
in the
arcs
for
a
type
(A, Z) plane.
3 node.
second
1, 3, 1, 3, 11, 31, first
+
...
11, 31, +
arc
i
e
+
+
2,4, +
+
+
arc
2, 4,
+
+
+
+
+
8.3 Side of
8.3 Side of
Passage
for
an
Passage
for
an
Antinode
147
Antinode
We need to find the side of passage for points which are inside a S- or T-arc. In the case of a total bifurcation, we must also establish the sides of passage for the first
8.3.1
species
Type
orbits.
1
analysis of Sect. 6.2.1.2 to the vicinity of a bifurcation orbit. origin the time of ingoing crossing of the unit circle by M4. We consider a perturbed supporting ellipse, in which A and Z are replaced by A+ AA and Z+AZ, and the crossing of the unit circle takes place in rotating axes in a point with coordinates (cos AO, sin AO) instead of (1, 0) (Fig. 8.4). We extend the
We take
as
Fig.
Antinode, type
8.4.
1.
Q the points
We continue to call P and
of intersection of the
perturbed
supporting ellipse We consider only
with the unit circle.
the passages through P and Q which correspond to unperturbed bifurcation orbit. With this convention, M2
collisions in the passes
now
t2P and
M4 t4P
--
AO
+
passes --
-
(8.46)
through
we
t4P
P at times
27rpl
27rp(A
Using (6.3), 12P
through
--
+
P at times
AA)J
(8.47)
.
have
AO
-
27rpJAA.
(8.48)
Suppose for instance that t2P 14P > 0, and that the crossing of the x axis is in the positive direction (cos 0 > 0, with 0 defined in Sect. 8.2). We have then the situation shown by Fig. 8.4, and the side of passage is: o+1. More generally, it is easily seen that -
=
148
Junctions: Broucke's
8.
0'
Sigll[(t2P
::::--
Next
t4P)
-
COS
Principle
01
-_
sign[(Ao
27rpJAA) COS 0]
-
consider the former collisions in
we
Q.
(8.49)
.
In the bifurcation
orbit, M2
through Q at times 20 + 27r(ao + pI) (see (6.5) and (6-6)). perturbed supporting ellipse, M2 passes through Q at times
In the
passes
t2Q
AO
-
(t20
+
At20)
+
+
27r(ao
+
pl)
(8-50)
.
In the bifurcation
orbit, M4 passes through Q at times 140 + 27rA(flo perturbed supporting ellipse, M4 passes through Q at times
In the
t4Q
(t40
-
Using (6.5) t2Q
-
t4Q
A0
=
an
+
outgoing -
-
+
AA) (go
+
pJ)
and the definition
+ 27rAZ
-sign[(t2Q sign[(-Ao
-
-
27r(A
(6.6) again,
and
Since this is 0-
A140)
+
-
27rAA(po
passage,
we
+
have
+
(8-51)
.
(4.28)
pJ)
pJ).
of
Z,
we
obtain
(8-52)
.
now
t4Q) COS 01 27rAZ +
27rpAA(flo
+
pJ))
Cos
0]
(8-53)
Up to now the perturbed supporting orbit was quite general, with arbitrary values of AA, AZ, A0. We determine now the sides of passage for perturbed arcs, by considering more specialized cases. 0. It has (i) T' arc. The perturbed supporting ellipse satisfies AA --
collisions
A0
=
P. Therefore
in
0
we
have from
(8-54)
.
The side of passage at the intermediate o-
with
=
s
sign(-AZ COS 0)
defined
in
T'
There is
(ii)
=
=
point Q
sign [- (OZ160A
=
The
=
(iii)
AC COS
01
-
(8-55)
's
perturbed supporting ellipse has (8.56)
sign(-AZ Cos 0)
--
s
point
P is
given by (8.49):
(8-57)
.
First
0. This orbit is symmetrispecies orbit E. There is again AA x axis; therefore the time lags 12P t4p and 12Q 14Q; by (8.48) and (8.52), should be opposite: -
cal with respect to the given
given by (8-53):
-27rAZ
and the side of passage at the intermediate o-
is then
Sect. 8.2. 1.
0. again AA Q. Therefore, from (8.52)
arc.
collisions in
A0
(8.48)
A0
=
-(Ao
from which t2P
-
we
14P
+
-
-
27rAZ)
(8-58)
deduce
-
AO
=
_7rAZ
and the sides of passage
near
t2Q P and
Q
-
14Q
are
-
both
7rAZ
given
(8-59)
,
once more
by
Passage
8.3 Side of
o-
=
sign(-AZ cos 0)
(iv) Ingoing (8.48),
s
--
for
an
Antinode
149
(8.60)
-
S-arc. We have
an
initial collision in
P, for
p
_=
Therefore,
0.
from
A0
0
=
(8-61)
.
For later passages o-
--
P, with
near
sign(-AA cos 0)
There is also
a
p >
0,
(8.49)
have from
we
(8-62)
.
final collision in
Q,
for
value p
some
--
po, and from
(8.52)
we
have +
poJ)
For earlier passages
near
AZ
t2Q
-
and from o-
=
AA(Po
-
14Q
=
0
=
(8-63)
.
Q, with
27rAA(po
p < po,
we
have
(8-64)
p)J
-
(8.53)
sign(-AA cos 0)
(8-65)
.
So this formula is valid for all passages. We substitute the exp ression and obtain o-
--
1
-sign (k
(v) Outgoing (8.52),
-
K) cos 0
(OZ) ac
S-arc. There is
an
AC A
]
-_
s
sign (k
initial collision in
-
Q,
K) for p
(8.15) (8-66)
=:
0.
Therefore,
from
A0
+ 21rAZ
-
For later passages
t2Q
-
and from o-
=
t4Q
=
2-x o AA
0
--
Q, with
near
(8-67)
.
p >
0,
have
we
(8-68)
-2rpJAA
(8.53)
sign(AA cos 0)
There is also
a
(8.69)
.
final collision in
P, for
some
value p
=
po, and from
(8.48)
we
have
A0
-
2vpoJAA
For earlier passages o-
--
--
(8.70)
0.
near
sign(AA cos 0)
P, with
p < po,
we
have from
(8.49) (8.71)
.
We obtain for all passages o-
=
-s
Since k
sign(k >
group the two
S
arcs:
0 for
-
K)
an
(8.72)
.
ingoing
S-arc and k
equations (8-66) and (8.72)
<
an outgoing S-arc, we can single equatio n, valid for all
0 for
into
a
150
Junctions: Broucke's
8.
o-
s
--
sign [k (k
-
K)]
(vi) Recapitulation: for
a
-8
for
an
Type
8.3.2
all
covered
arcs are
by
the
simple
result
(8.74)
2
The method for
(i)
(8.73)
.
normal arc, abnormal arc.
+S 0-
Principle
determining
the side of passage is different from type 1. case A < 1. The bifurcation ellipse is
S-arc. We consider first the
tangent to the unit circle and lies in its interior. In rotating axes, the orbit in the vicinity of M2 is as shown on Fig. 8-5a. When we move away from the bifurcation, the supporting ellipse intersects the unit circle. Successive intersections in the vicinity of M2 are separated by an angle -21rJAA. For an ingoing S-arc, the situation is as shown on Fig. 8.5b. (The figure corresponds to the
case
rn
3: the S-arc is made of 3 basic
=
the vicinity of M2
are
Successive passages in
arcs.
numbered.)
M2
M2
0 1 2
M2
3 2 1 0
b
Fig.
Antinode, type 2, S-arc.
8.5.
Thus, all intermediate
outgoing
arc
(Fig. 8.5c),
on
the
right
and the above
figures
are
of M2:
o-
--
+1. For
an
o-
reversed
to the unit circle
right-to-left;
on
its
all sides of Passage
the left of M2.
are on
Thus, o-
passages are have again
we
1, the bifurcation ellipse is tangent
For A >
outside,
C
--
(ii)
o-
is
-sign(A
T-arc. In
therefore has
(ill)
always
First
Therefore,
no
-
given
by
(8.75)
1)
type 2,
a
T-arc
corresponds
to
only
one
basic arc, and
apocenter
at the collision.
antinodes.
species orbit
as we move
E. If A <
1, M3 is
at its
away from the bifurcation
along
the first species
family,
8.3 Side of
Passage
for
an
Antinode
151
the side of passage is on the right if the eccentricity increases, on the left if it decreases (the semi-major axis is invariant). For A > 1, this is reversed. Thus o-
-sign[(A
--
Using
the relation
o-
sign[(A
=
Type
8.3.3
(1) S-arc,
I)Ae]
-
-
(3.11)
(8.76)
.
between
e
and
C,
we
obtain
I)cAC]
(8-77)
3
odd. As
approach the bifurcation orbit, the end points P and Q diametrically opposite points of the unit circle. The computations of Sect. 8.3.1 are applicable. There is cos 0 1, and AA is always positive (see Fig. 8.3). Therefore, for an ingoing S-arc (symbols 1, 3, 5, ...), we have rn
we
tend to
--
for all passages, from
0-
(8.78)
outgoing
an
-1
_-
Q
11,
3
1,
from
5
(8.7 1), (8-79)
Points of P
As
approach the bifurcation orbit, the end points point of the unit circle, as for type 2. even rank along the arc (counting from its origin) are in the and Q. The considerations of Sect. 8.3.2 are applicable and we rn even.
tend to the
vicinity of
(symbols
S-arc
.
(ii) S-arc, P and
(8.65),
+1
0--_
and for
-
we
same
have o-
=
-sign(A
-
1)
Points of odd rank
Therefore o-
we
=
(since
+sign(A
The sign of A have 0-
=
(iii) O-_-
-
-
T-arc. The
-1
orbit,
are
at the
diametrically opposed point
the orbit must intersect the unit
1)
of the orbit.
circle) (8-81)
.
I is the
I +sign(AC) -sign(AC)
bifurcation
(8.80)
.
opposite of the sign
for for
points points
of
even
(see
Sect.
6.2.1.4).
rank,
as
So
(8.82)
of odd rank.
Til family lies in D3, and
there is AZ < 0
of AC
we
move
(Fig. 4.8). Therefore,
from
away from the
(8.55) (8.83)
.
species orbit El- ,. The intersections with the x axis are alternatively right and to the left of M2 (Fig. 6.2a). Thus, o- equals alterand -1. According to the definitions laid down in Sect. 6.2.1.4, natively +1 o+1 at the origin of the rooted branch E++, and o-I at the origin for
(iv)
First
to the
=
E+-.
--
152
Junctions: Broucke's
8.
Principle
species orbit E 1. Both intersections with the x axis are at the left of M2 (Fig. 6.2b). Thus, o- equals -1 in both cases. (vi) First species orbit 1,. This is a circular orbit. In the vicinity of the bifurcation orbit, there is da/dC < 0 (see Fig. 3.1). Therefore
(v)
o-
First
-sign(AC)
--
(8-84)
.
8.4 Results: Partial Bifurcation Two branches n in
can
joined only if the
be
I intermediate collisions of the
-
Sect. 7.3.1
sides of passage
bifurcating
arc.
are
the
same
at all
Each of the subsets found
thus be further subdivided -into 2n-1 smaller subsets.
can
In older versions of this work, each subset was identified by a semigraphical representation (the "sch6mas de H6non" mentioned in Guillaume's thesis (1971), pp. 108-109, 142, 151-156, etc.). Here we use a more compact and more convenient notation, in which the sequence of the o- signs is simply listed in the name of the subset, before the symbol which describes the symmetry. Thus, the name of a subset is of the form
or
T><szgn sequence>< symmetry svmbol>
for instance: lP4++-A.
Type
8.4.1
I
help
With the
of the rules established in Sects. 8.2.1 and 8.3.1
(specifically
(8.30) (8.74)), the sequence of signs can be computed for each entry of Table 7.1. We recall that the sign appearing in this table is the value and
of
s
(see
tables, (Sect. 6.2.2.1). Thus,
which must be For
given
a
(8.22)),
actually starting point is P and Q, respectively each line in Table 7.1 corresponds to 4 different entries, considered separately. entry, the procedure is as follows:
Sect. 6.2.1.2 and
two different
and that the table represents
in which the
1. Associate successive collisions in the bifurcation orbit
P and
with
Q.
2. Determine the value of k for each S
IkI
alternatively
arc
(odd symbol):
its absolute value
m equal symbol, sign is + for an ingoing for an outgoing arc QP (Sect. 6.2.1.2)arc PQ, Identify abnormal arcs: they are the S arcs with 0 < k < K if K > 0,
to the value
is
of the
and its
-
3.
K < k < 0 if K < 0. Abnormal 4. Set o-
=
o-
--
-s
arcs are
for all collisions inside
+s elsewhere.
or
underlined in Tables 8 -4 to 8.11. at the ends of
an
abnormal arc;
8.4 Results: Partial Bifurcation
The number of
153
to be considered can be reduced by taking advansymmetries. We consider first the following operation: exchange of P and Q, and change of sign of K. Each k has its sign changed; therefore normal arcs remain normal, abnormal arcs remain abnormal, and
tage of
cases
two internal
the sequence of signs is not changed. Thus a single table serves for both cases. (The captions of Tables 8.4 to 8.11 explicitly describe the two cases to
which each table
starting point
applies.)
For
simplicity,
we
describe
only the
case
where the
is P.
A second reduction
signs (i.e.
can be achieved. Suppose that we change all the s replace each branch by the opposite branch of the same family).
we
Then all sides of passage sign sequence is replaced
o- change sign. A subset corresponding to a given by another subset in which all signs are changed. These two subsets are isomorphic and it is sufficient to study one of them. We will therefore consider only the sign sequences which begin with a + sign. The sides of passage depend on the value of K, which is fixed for a given bifurcation orbit. K is used in comparisons with odd integers k. Therefore the following cases must be considered separately:
3 < K < -1
-1 < K < I
,
I < K < 3
,
3 < K < 5
,
....
(8-85) K does not take
integral
For
m
< n
starting point
for all
is P. It
odd values of K
lying only
values; therefore there are only the K < -1 n
cases
only a finite number of cases have to be distinguished, -n is excluded since the Moreover, the value k follows that changes in the subsets happen only at the =
n
three
are
+ I
cases
1)
:5' K
<
n.
There
are n
such
have to be considered.
Thus,
for
=
-
n
2,
cases
the four
-1 < K < 1
,
-(n
in the interval
,
(8.86)
1 < K
cases
,
I < K < 3
,
(8-87)
3 < K
so on.
Table 7.1 lists the branches for
n
<
7. Therefore
consider. Tables 8.5 to 8.10 show the results for the 6 -5 < K < 7. For K <
except for the
sign
boundary
and therefore
arcs.
-1 < K < I
,
3, there
--
K < -1
and
(Sect. 8.2.1)
given order n,
a
because
for
values
ignored.
be
can
as
shown
by
identical to those of Table
sign
is
are
have 8
to
cases
in the interval
identical to those of Table
8.5,
two subsets 1P6 ..... A and 1P7 ...... S, in each of which
changed
is
-5, the results
we
cases
changed
Table 8.4.
Finally,
for 7 <
8.10, except for the subset
K,
the results
a
are
1P7 ...... S, in which
by Table 8. 11. simple. All arcs are normal; all sides of passage have the same sign os. Thus, the sequence of signs is either +++. or Broucke's principle is not very helpful in this case; it simply separates each set of branches in Table 7.1 in two subsets, comprising the branches with a
The
case
as
shown
-1 < K < 1 is --
.
---.
.
.
.
154
Junctions: Broucke's
8.
Table 8.4. Partial for
Principle
bifurcation, type
1: branch subsets lP6+++++A and lP7+
starting point
subsets
are as
in P and K < -5, in Table 8.5.
1311
+232
1113
-
151
-
iiiiiii
11111.1
bifurcation, type starting point in P and
or
starting point
in
Q and
++S,
i
5 < K. Other
Table 8.5. Partial
1: branch subsets determined
Broucke's
principle,
-5 < K <
in
for
-3,
or
by starting point
3 < K < 5.
lP2+k + -
lP5++--A
IP6++---A
lP7 ...... S
2
+311
+33
+313
11
+2111
+3111
+21112
lP3++S
+
1211
+213
+
12121
-
113
+21111
-
11311
-
1121
+
123
1112
+
12111
3 -
ill lP3++A
lP6+++++A
+21
15
+
lill lP4++-A
1311
+232
+21121
1113
+151
+1213
-
iiiiii
-
+
51
+211
-
132
-
11112
121
-
112
+
5
+2112
+212 -
131
lP5 +32 +23
... A
-
13111
-
11131
lP7 ......A +511
+3211 +312
+
1212
12112
-
112111
-
111211
lP7 ------ A
25
-
... S
+
+331 +31111
+2131
IP6++--+A lP5
lP7 ......A
+
+231 +2121
iiiiiii
+52
+321
+31
+
3112
+2113
-
lP6 ..... A
13
7
-
12
lP4+++A
+3121 +
+
+
IP7 ......S
lP7 ...... A
+
23T11
+211111 +
-
115
-
1132
+21211 -
-
1123
-
11212
133
-
1131
-
1321
-
11211
-
1312
-
11121
-
11113
-
111121
-111112
1231
+121111
Q
and
8.4 Results: Partial Bifurcation
Table 8.6. Partial
principle,
bifurcation, type
for starting
point in P and
1: branch subsets determined
Broucke's
-3 < K <
in
-1,
or
by starting point
1 < K < 3.
lP2+A + -
lP5 .... A
lP6++--+A
1P7 ...... A
2
+32
+312
+511
11
+23
+2112
+3211
+1212
+2311
lP3++S + -
3 ill
lP5++--A
+
13 lill
lP4++-A +31
+
13111
-
11121
-
11113
-
1211
111121
-
111112
lP6++---A
113
-
1121
+3111
-
1112
+21111
12
lP4+++A
+21211
11211
-
+21 +
1131
-
+2111 -
lP3++A
-
+311
+
+
lP6+++++A +
33
-
121
-
112
lP5 .... S
+313
1113
+21112
lP7 ......S 7
+
15
+
+
132
+232
+
123
+
-
iiiiii
151 1111111
lP6 .....A +51
+321
T +231
1P7 ......A +52
lP7 ...... A +3121 +3112
+21121 +
1213
+
12112
-
112111
-
111211
+25
+
1311
+
133
-
11112
+
1321
11111
11311
+331
+2121
-
12121
-
+2113
+5
131
+
+213
+212 +
lP7 ...... S
12111
+211 +
155
+2131
+1312 +
1231
lP7 ------ A
+31111 +211111 +121111 -115 -
1132
-
1123
-11212 -11131
Q
and
156
8.
Junctions: Broucke's
Principle
Table 8.7. Partial
1: branch subsets determined
principle,
and -I < K < 1.
1P2+A
for
bifurcation, type starting point in P or Q
lP6+++++A
1P7 ......A
*
2
+51
+52
*
11
+33
+511
+321
+331
+312
+3211
+3111
+3121
+231
+3112
+213
+31111
+2121
+25
+2112
+2311
+21111
+2131
1P3++S 3
ill IP3++A +21 +
12
lP4+++A
+31 +211 +
13
+
121
+
112
+
1111
1P5 .... S +
+
15
+21211
+
132
+2113
+
1311
+21121
+
123
+211111
+
1212
+
133
+
12111
+
1321
+
1131
+
1312
+
13111
+11211 +
1113
+
1231
+
11121
+
1213
+
11112
+
12112
+
111111
+
121111
+
115
+
1132
+
1123
+
11212
5
+212 +
131
+
11111
1P5 ....A
1P7 ......S +
7
+313 +232 +21112
+32
+
+311
+
+23
+
+2111
+
*
1211
*
113
*
1121
*
1112
151
12121 11311 1111111
+112111 +
11131
+
111211
+
11113
+
111121
+
111112
by
Broucke's
8.4 Results: Partial Bifurcation
157
Table 8.8. Partial bifurcation, type 1: branch subsets determined by Broucke's principle, for starting point in P and 1 < K < 3, or starting point in Q and -3 < K < -1.
lP2+A +
2
IP5+--+S
lP6+----A
lP7 ------ A
+212
+21111
+31111
131
lP3++S +
3
lP5+--+A
-
121 1
-
1121
-
15
-
132
-
123
-
1212
-
1131
.
lP3+-A
+21
lP5+---A
+2111
-
-
-
12
-
113
lP4+++A
lP6+++++A
+31
+51
11113
-
IP7 ------ A +2113 13111
-
lP7 ...... S
12111.1
-
112111
-
7 lP7+--++-A
+313 +232
+2131 +21211
lP7 ...... A
+21121 1312
-
-
lill
+33 +321
lP4+--A +211 -
13
-
121
-
+312 +231
lP6+++--A +3111
+
... S
-
11121
-
11112
+23
+25
+311 -
1112
12112
-
11212
lP7 ------ S
+511
+21112
-
lP6+--++A +213
+3211 +3121
-
12111
-
11211
111112
1P7 ...... A
1 51
-
12121
-
11311
ip
------ A
-
1321
-
1231
+3112
IP7 ------ A
-
11131
-
11121 1
+2112
1311
-
-
+2311
+2121
-
-
+33 1
.
-
lP5+++-A
-
lP7 ...... A
-
iiiii
+32
1113
-
5
lP5 .... A
52
111111
112
IP5
+
11112.1
+211111 -
133
-
1213
-
115
-
1132
-
1123
158
8.
Junctions: Broucke's
Principle
bifurcation, type 1: branch starting point in P and 3 < K
Table 8.9. Partial
principle, -5 < K <
lP2+A + -
2 11
lP3-S
-
-
3 ill
for -
+212
+231
-
133
+21111
-
13111
-
131
lP5+--+A
-
-
-
15
-
1213
-
132
-
121111
-
123
-
1123
-
1212
-
112111
1211 1121
+23
lP7 ...... S +
-
-
31
lill
IP4+--k +211
+2111
lP6+++++A +51 -
3111
-
1131
-
Mill
lP6+++--A
-
13
-
121
-33
-
112
-321
-312
+
-
1113
-
11121
5 -
iiiii lP5 .... A
-
11311
-
iiiiiii
IP7 ...... A +
-
311
-
113
lP5+++-A
-
-
32 1112
+213
+ -
12111 11211
+21121 -
1312
-
12112
-
11212
lP7 ------ S +232
+21112
31T12
-
151
-
12121
lP7 ------ A
-
1132
-
1321
-
111112
-
1231
lP7 ...... A
lP7 ------ A
-331
+2311
-
3211
+2113
-
3121
+211111
-
11131
-
111211
-
111121
1311
-
+21211
11113
+511
+2112
-
+2131
25
1P7 ...... A
+2121
-
52
lP7+--++-A
-31111
11112
lP6+--++A
7
-313
-
lP5 .... S
Broucke's in
lP7 ------ k
lP6+----A
+21
lP4+++A
or
lps+--+s
1PS+---A
12
5,
3.
lP3+-A
-
by starting point
subsets determined <
-
115
Q
and
8.4 Results: Partial Bifurcation
159
Table 8.10. Partial bifurcation, type 1: branch subsets determined by Broucke's principle, for starting point in P and 5 < K < 7, or starting point in Q and -7 < K < -5.
lP2+A + -
2
+212
11
-
lP3++S
-
lP5+--+S
3
131
lP5+--+A -
ill
-
+231
-
-
13111
1PS+---A
+21
+23
15
-
1213
-
132
-
121111
-
123
-
1123
-
1212
-
11211 1 .
lP7 ...... S +
-
12
+2111
-
-
.. A
31
-51
lill
-3111
lP4+--A +211 -
lP6 ..... A
-
1131
-
iiiiii
IP6+++--A
-
121
-33
-
112
-321 -312
lP5
... S
-
1113
-
11121
5 -
iiiii
lP5 .... A
+2131
1131.1 iiiiiii
lP7 ...... A
-511
311
-
113
lP5+++-A -32 1112
+213 +2121
+2112 -
1311
-
12111
-
Y112T11
-
1312
-
12112
-
11212
lP7 ------ S
-
115
+232
-
11113
+21112
lP7 ...... A -52
-
151
-
12121
IP7 ------ A
-3112 -
1132
-
1321
-
111112
-
1231
lP7 ...... A
-
-
+21121
-31111
11112
lP6+--++A
lP7+--++-A
+21211
13
-
-
7
-313 -
lP4
133
-
112 1
lP3+-A
IP7 ------ A
+21111
1211 .
-
lP6+----A
331
lP7 ------ A +25
-3211
+2311
-3121
+2113
-
11131
-
111211
+211111 .
-
11112 1
160
Junctions: Broucke's
8.
Principle
bifurcation, type 1: branch subset 1P7 ...... S, for starting K, or starting point in Q and K < -7. Other subsets are as in
Table 8.11. Partial
point in P and 7
<
Table 8.10.
1P7
..... S
-7 313
-
-
11311
-
1111111
s
+I and
=
if
1, respectively. In other words,
s
they have the
same
sign
In other cases, the situation is
distinguished.
two branches
can
be
joined
of AC. more
complex
and many subsets are left in one
can
be
subset; other instances, however,
two branches
instances, only joined. In many the number of branches is still larger than 2. It can be noticed in the tables that the signs occur in pairs in the sign sequences. More precisely, if K > 1, the sign is always the same for two successive collisions of ranks 2j and 2j + 1; if K < -1, the sign is always the I and 2j. This property is same for two successive collisions of ranks 2j easily proved from the above rules. As a consequence, the number of distinct sign sequences for a given n cannot exceed Dn-l)/2]. (This upper limit is In several
these two branches must then be
-
2 to 7: all allowed sequences appear.) actually reached in Table 8.8 for n Taking the possible symmetry or asymmetry into account, we have an upper limit of 2 x 2L(n-l)/2j for the number of subsets. Thus the number of subsets grows at most as 2n/2 (1.414 )n. on =
=
...
the number of branches grows as (1.839...)' (Sect. C.1.1), i.e. faster than the number of subsets. It follows that the average number of the other
hand,
branches per subset increases at least as (1.301 )n, and Broucke's branch be sufficient establish the cannot to junctions generally. ...
8.4.2
Type
With the
principle
2
help
8.3.2, the sequence They depend on the 1, so that two cases should in principle be considered. We notice, sign of A however, that all sides of passage change their sign if we change the sign of A I and we exchange i and e. Thus, there is an isomorphism between the two cases, and for the sake of brevity we consider only the case A > 1. Table 8.12 shows the resulting division into subsets. The maximal number of distinct sign sequences is 2". (All sequences actually appear up to n 4.) Taking the possible symmetry or asymmetry of
signs
can
of the rules established in Sects. 8.2.2 and
be
computed
for each entry of Table 7.2.
-
-
--
into
account,
we
obtain
an
upper limit of 2
x
2'-' for the number of subsets.
8.4 Results: Partial Bifurcation
Table 8.12. Partial
principle,
2PlS
bifurcation, type
2: branch subsets determined
for A > 1.
2P3-+A
2P4+-+S
2P4-+-S
+
1
-
21
-
121
-22
-
1
-
2i
-
liel
-
leil
lei
-
e2i
-
ille
-
lei
-
eiei
-
ieie
-
ill
-
2PlA
-
i
iii
2P4+-+A
iel
-
-
11
ei
2P2+A
2P3--S +3 -
3
2il
-
llei
-
2ie
-
lill
-
le2
-
iiii
-
lele
-
-
leie
-
i12
-
ile
-
elel
-
iiii
-
elei
-
ilie
-
eill
-
ie2
-
eili
-
iele
-
eiel
-
ieil
21
2e
+2
-
-2
+
12
-
i2
2P4 ... S 2P2-A
2P4+--A 13
-
l2e
-
-
li2
-
2el
-
lile
-
2ei
elli
le
2P4 ... A
-
-
iiii
-31
3i
-
e3
-
i2l
-
e2e
-
i2i
-
ei2
-
ilel
-
eile
-
ilei
ii 2P3++S
2P4--+A
-
lill
ill
liei e2l
+
ie
2le
-
-
2P2-S
-
-
llel
ill
2P3--A
11
12i
-
+
el
+
-
iei
2P2+S
2P4-+-A
elll
eli
2P4-++k
2P4
-211
+4
---
S
2P4++-A 2P3++A
ell 2P3+-A
-
12
-
lie
iii lie
21i
-4
-
lell
+22
iiii
-
leli
+
121
llie
-
+
lill
-
112
-
Me
-
-
-
e12
-
elle
-
elil
-
elie
-
i2e
-
-
iell
-
ieli
2P4
---
+31 -
3e
e2
+211
ele
+
eil
+
eie
-
13 112 i3
A
161
by Broucke's
162
8.
Junctions: Broucke's
Principle
On the other hand, the number of branches grows as 3' (Sect. C.1.2). Thus the average number of branches per subset increases at least as (1-5). One particular case can be resolved for all values of n: a sign sequence
composed of + signs only. In that case, it can be deduced from the rules that 11... Ili, and 11... 11, ell... Ili, only possible branches are 11. Furthermore, the first two branches are symmetric and the last two e 11. are asymmetric. We have thus two subsets of two branches, and the junctions are established (we use Restriction 7.3.2: the complement is assumed to be the
-
.
-
-
-
.
symmetric). 8.4.3
Type
3
With the
help of the rules established in signs can be computed for each entry resulting division into subsets.
of
As in the
case
of type
2,.
we
have
hand,
an
upper limit of 2
(All sequences actually the number of branches grows as
number of subsets. other
Sects. 8.2.3 and 8.3.3, the sequence of Table 7.3. Table 8.13 shows the
average number of branches per subset
2'-' for the
x
4.) On the (3.153.. .)n (Sect. C-1.3). The increases at least as (1-576 appear up to
n
--
...
8.5 Results: Total Bifurcation Two branches n
can be joined only if the sides of passage are the same at all collisions of the bifurcation orbit. Each of the subsets found in Sect. 7.3.2
can
thus be further subdivided into 2n smaller subsets. Each subset is identified
origin. For the sake of
by
the sequence of the o- signs, we repeat the sign of the
symmetry,
starting from the origin at the end
of the sequence. Thus, a sign sequence contains n + I signs, with the first and last signs identical. This sequence is listed in the name of the subset.
8.5.1
Type
1
The sequence of signs can be computed for each entry of Table 7.4, using the same procedure as in Sect. 8.4.1. Here the origin is in P (Sect. 6.2.1.2). Each line
Table 7.4
corresponds to two different entries for s 1, which must be considered separately. As in Sect. 8.4.1, the number of cases to be considered can be reduced by taking advantage of the isomorphism in which P and Q are exchanged, and the sign of K is inverted. (The origin can be brought back in P by a shift of one basic arc.) We will therefore consider only the case K > 0. in
--
As in Sect. 8.4.1, the number of cases to be considered can be again halved by consideration of the isornorphism in which all s signs are changed. We will therefore consider only the sign sequences for which the origin has a + sign.
8.5
Table 8.13. Partial
bifurcation, type
Results: Total Bifurcation
3: branch subsets determined
principle.
3P2+S -
+
3P3-+A
3P4++-A
2
+21
+
112
+
4
11
+
ii
+
iii
+
ei
+
el
+
lle
+
11111,
+
1112
+
illi
3P2+A +
+
3P3--S
+
+31
3P2-S +2 +
+
I) I) I)
3P3--A
I) 1)
+211 3P2-A
i
3P3++S
+
ill
+
ell
+
112
+
11i
+
Ile
+
l1le
+
111111,
3P4+-+S
3P4-+-S
3P4-+-A +
2i
+
e2
3P4--+A +311 +2111
-4
+
-22
illi
+elll
+
121
+
11,111
3P4+-+A
+1121 +
llil
+
llel
+
111,111
+
+3
3P4 ... S +
1111
+
11111,
+
+
3P4+--A
+
ill,
+
1111
3P3+-A
3P4 ... A +31 +311 +
13
+
ilill
+
11111
131
ie
+11211
+1211
+
I' 1' 1)1)
iii +
lell
+
1112 111i
+
lil
e
11,111,
2e
+21111
i2
+
ellll
+
1131
+
Vill
ill
+
llel,
ill,
+
11112
ell
+
1
ell
+
Vile
11,11
+
ii
+
11,11,
+
le
+
113
+
11111
+211
+
111111
+2111
3P4-++A
4-
3'11
+
ii 11,
+
+
+
+
12
+
3P4---A
+
+
+
+22 +
+
3P3++A
3P4---S
lel
1
11 i
163
by Broucke's
164
Junctions: Broucke's
8.
Principle
again, only a finite number of cases have to be distinguished; changes happen only at the odd values of K lying in the interval n. Thus, for n 2, there are only the two cases
Here
the subsets
in
0 :! K :!
=
0 < K < I
for
n
--
4, there
0 < K < I and
(8-88)
I < K ;
,
the three
are
1 < K < 3
,
cases
,
(8.89)
3 < K
so on.
Table 7.4 lists the branches for
n
<
6. Therefore
consider. Tables 8.14 to 8.16 show the results for the 3 0 < K < 5. For K >
5, the results
are
we
have 4
cases
cases
to
in the interval
identical to those of Table 8.16, except
for the three subsets 1T6 ....... 1, 1T6 ....... 3, IT6 ....... S,in each of which
sign
a
is
changed
Table 8.14. Total
principle,
as
shown
1T6 .......1
IT6 .......A +321
E
+-312-
*
11
+
IT4+++++l
+-1131-
+1131
+-2211-
+-21111-
+-211-
+213
+1212
13
+
IT6 .......3
1T4 ..... 3
+-112-
+231
+
+3111
lT4+++++A
+
121
123
+-1311-
1113
-11211-
+211
+
+312
+-2121-
+33
-1111-
132
-1122-
+-114-
+31
+
by Broucke's
+-1212-
15
+-231-
112
1: branch subsets determined
bifurcation, type
*
+
Table 8.17.
for 0 < K < 1.
lT2+++l
+
by
+21111
+11112 1T6 .......5
+11211 -111111-
+51
+
+-213-
+11121
+-132-
+
12111
+-11112+
1311
+2121
The
case
have the
0 < K < 1 is
same
sign
o-
=
s.
simple: all
arcs are
The sequence of
normal and all sides of passage
signs
is either
+++.
..
or
8.5
Table 8.15. Total
principle,
bifurcation, type
Results: Total Bifurcation
1: branch subsets determined
for I < K < 3.
-.111--
+ k'
. 11
........
+ bi
le 1. J
+-132-
+-11112-
lT4+++++3
1T6 ....... A
+2121
+31
+321
IT6 ....... A
+-112-
+-1212-
-
11
-
lT4++--+l +-211-
13
lT4++--+A +211 -
112
+-1131-
+213
+312
+-2121-
+-1122-
-
IMI
+231
-
12111
1T6 ....... 3
121 1T6 ....... 1 +-312-
+-2311T6 ....... 3 +-114-
+33
1T6 ------- 1
+-411-
-
15
+3111
-
1131
-
-1111-
1311
1113
+-112111T6 ....... A
+-21111-
1212
lT6++----+A
+-2211-
-
132
+-1311-
-
123
-
11112
+21111
-
11121
+
-111111-
165
by Broucke's
166
Junctions: Broucke's
8.
Table 8.16. Total
principle,
Principle
bifurcation, type
1: branch subsets determined
by Broucke's
for 3 < K < 5.
A
bI
11
-11112-
+213 --1122-
lT4+++++3
-
31
1T6 ....... 3
+-2121-
11211
-
12111
+-411-
--112-
-
-
IT4++--+l
+
33 1113
1T6 ------- 1
-11211-
+-211-
1T6 ....... A
13
lT4++--+A
-
+-2211-
+211
+
112
11112
-
11121
121 1T6
...... 1
+-312-
Table 8.17. Total
1T6 ------- A +-1131-
1T6 ....... 5 +
-213-
-
-132-
-
1311
+
2121
bifurcation, type
+
I
lft7
3111
-111111-
1: branch subsets 1T6 ....... 1,1T6 ....... 3, are as
in Table 8.16.
..........
-
123
-
+21111
1T6 ....... 5, for 5 < K. Other subsets
1131
132
+231
1131
1T6 ....... 3
312
1212
-1311-
-
-1111-
15 -231-
+-21111-
321
-
-
+
........
-01 -
-11112-
8.6
Recapitulation
167
simply separated in two subsets, com+1 and s -1, respectively. In other cases,
Each set of branches in Table 7.4 is
the branches with
prising
for
a
--
=
more
the same for two successive collisions of again, the As 1. a + 2j 2j consequence, the number of distinct sign sequences 2' /2-1 (This upper limit is actually reached in exceed cannot n given
Here ranks
s
complex. sign is always
the situation is and
.
6.)
The number of symmetry cases is at limit most n/2 + 1. So we an upper (n/2 + 1)2 n/2-1 on the number of subsets. On the other hand, the number of branches grows as (1.839.. .)n (Sect. C.2.1), so that the average number of branches per subset increases at
Tables 8.15 and 8.16 for
n
2 to
=
have
least
as
(1.301
.)n /n.
..
The bifurcation for
n
-_
2 has been studied
quantitatively by
Guillaume
112-119).
(1971,
pp.
8.5.2
Type
2
The sequence of
for each entry of Table 7.5, using the in Sect. 8.4.2. Table 8.18 shows the resulting division into
signs
can
be
computed
procedure as subsets, for A > 1. Most subsets (but thus establishing a junction. same
not
all)
contain
only
two
branches,
An upper limit on the number of distinct sign sequences is 2 n. An upper on the number of symmetry cases is n + 1. So we have an upper limit
limit
(n + 1)2 on the number of subsets. The (Sect. C.2.2), and the average number of least as (1 .5 )n /n. n
8.5.3
Type
3
The sequence of same
number of branches grows as 3' branches per subset increases at
procedure
signs as
can
be
computed
7.6, using the resulting division into
for each entry of Table
in Sect. 8.4.3. Table 8.19 shows the
subsets.
(n + 1) 2n on the number of The number of branches grows as (3.153 )n (Sect. C.2-3), and the number of branches per subset increases at least as (1.576 )'/n.
As in the subsets. average
8.6
case
of type
2, there is
an
upper limit
...
...
Recapitulation
powerful: branch junctions have been established in sufficient, however, to establish junctions in general; the many number of branches always grows more rapidly than the number of subsets. The number of branches in a subset is always even, as predicted by Proposition 6.0.1. This provides a good verification of the method and of the computations. Broucke's
principle
cases.
It is not
is
168
8.
Junctions: Broucke's
Table 8.18. Total
principle,
2Tl++O
bifurcation, type
2: branch subsets determined
2T3+--+O
+
E
-
1
-
3
ile
2T3+--+A
-
E
-
i2
+
1
-
2e
2T4+++-+2
2T4+-+-+k
-
112
-
2le
-
llie
-
le2
-
2il
2T4+++-+A
-
ilil
-
Me
-
2T3-++-O
2T4++-++O
-
2
-
-ill-
-
121
-
ie
-
eli
-
liel
2T2+-+A -
-
il le
2T2-+-O
2T3-++-A
-
-
11i
-11-
--12-
ei
-
-
-
li el
-
2T3-+--2
-
2T2-+-A
-
ell
-
eil
2T3-+--A
-
-lil-
-
e2
2T4++-++A
lill
-
13
-
lile
2T4++--+A
-
-
li2
2T3--+-l
2T4+-+++2
+2
--21-
-211
-11-
2T3++-+l
-
lei
-
2T3--+-A
-
12
-
2i
-
lie
-
-lel-
iele
2T4+--++3
-
illi lell
2T3++-+A
2T3
-
lil
+3
-
lle
+
2T3+-++2 -21 -
iel
2T3+-++A
0
-ill-
2T3
----
1
+-21+
12
2T3
----
ill
+-12-
lel
+21
2
2T4+-+-+O -
leil
-
ille
2T4+-+-+2 -
2ie
-
ie2
2T4-+-+-2
2T4---+-A
llei
-
3i
-
eill
-
-2el-
2T4-+-+-k
2T4-----O
-
-llel-
+
-
e2l
+-22-
-lill-
+
12i
+121
-
liei
-
eili
i2l
-
eiel
-
2el
-
elei
2T4-+---2
4
-1111-
2T4
-----
1
+-211+
2T4+---+O
2ei
-
-
13
2T4
-----
2
-4 -
i2e
2T4+---+A
-
ei2
2T4-+---A
+-13-
+211 +
D
-li2-
3e
e3
2T4-+++-O
2T4--++-l
112
2T4
-----
+31 +-112-
-1111elli
2T4-+++-A -
----
-
ilel
2T4+--++A
2T4+-+++A
-
--31-
e2i
-
iell
-
-liel-
-
-31
l2e
2T2---O
+
leie
-
2T4---+-2
-
ilie
-
llel
2T4++--+l
i12
2T4-+-+-O
ieil -
2T2+-+O
by Broucke's
for A > 1.
-
2Tl--O
Principle
-
111i
elll
2T4-++--3 --112-
elil
2T4-++--A
--211-
2T4--++-k
-
21i
-
-lell-
2T4--+--O --22-
e12
-leil-
2T4--+--A
-11ii-
leli
-2il-
-le2-
3
Recapitulation
8.6
Table 8.19. Total
bifurcation, type
3: branch subsets determined
169
by Broucke's
principle.
3T3++-+A
3Tl++O
3T4 ..... 1
3T4+-+++2
3T4-++--A
+211
+
-
I
+
ii
+-211-
+
1
+
le
+
13
+
111,11
+-2111-
3T3+-++2
3Tl--O +
+21
1
+
3T2+++l
3T3+-++A
+
+
3T2+-+O + -
11111
+
11111
3T4 ..... 3
+
3T4+-+++A +
ill
+
ell
3T4+ -+-+2
+-112-
el
+311
3T3-++-O
E++
+
-ill-
_11-
+
1111,
+-1112+
1111,
+
11,11
--13-
3T4+--++3 +311 +
111,111
+ -
3T4+--++A
3T3-++-A
E+_
+
-ili-
2
+
-ele-
+
112
+
iii'll
3T2
---
0
3T3
----
0 +
+
+
+31
2
iii lle
3T4++-++O 3T2
---
1
3T3
----
1 121
E_+ E__
3T3
----
+
3 1111
+-1121-
+
3T3 .... 2 +-12-
----
ill
+
-il)i-
+
11 i
+
ell
+
-el lelle
+
lel
+
131
+
11,11
3T4++--+A +
1112
+
1211
+
12
+
11,11
3T4-+-+-O
3T4
+
3T4-+---2 +-31-
3T4-+---A -lil-e2e-
+
-211 l-
+elll
+
111,11,
3T4+---+O
3T4--++-A
+-22-
+2111
ie
3T4+---+A +
i2
+
2e
3T4-+++-O
3T4
+
l1le
+
lell
+-11121+
111111,
-----
2
+
11112
3T4
-----
3
+3111 +
-1'1 12'-
3T4
-----
A
+ +
+
Vill
+ellll
_11111-
+
-el 11 le-
ill,
+
Vile
+
-il
+
llell
+
ell'
+
-el
I
I
li-
le-
3T4--+--O +
4
+
ei
3T4--+--A +
2i
+
e2
-elle-
3T4-++--3
1131
+
+
+
1
+
+
3T4-+++-A
-----
+21111
+ -ele-
+
-111111-
-2'11 l'-
--22-
+ +
3T3++-+l
+
+11211
A
+
+
+
Vle
3T4--++-l
+211
3T3
+
llel
+
+
3T4++--+l 3T3 .... 1
3T4-----O
illi -ell le-
+
2
-
+
+
112
3T4++-++A 3T3 .... 0
+1121
+ +
-lel-
+ +
+
-i2i-
+
i-
+
+2111
3T4+++-+A
I
+
4
3T4+++-+2 3T2-+-O
-ill
+
+31
il
ill
113
11111
+
it,
+ +
_11111-
+1112
3T4---+-A
3T4---+-2
+
-iei-
9
a.
Fragmi ents
9.1 Introduction In
Chaps.
3 to
5, families of generating orbits have been assembled in
a
natural
with Definition 2.9.2. In way from individual generating orbits, in accordance other words, a family of generating orbits is simply a natural set of generating orbits. No attention has been
paid
to the existence of bifurcation orbits: the
right through bifurcations. For instance, of in a simple bifurcation involving only two families (Fig. 9.1), the family generating orbits coming in along branch I (solid line) continues along branch 3, and the family coming along 2 continues along 4.
families
as we
have defined them go
4
2
9.1. Families of
Fig. for p
=
generating
vs. generating periodic orbits for
orbits
0. Dashed lines: families of
families. Solid lines: families tt > 0.
0, however, it is preferable to change now generahng families, which we have our 0 of families of periodic orbits (Definition 2.9.3). defined as the limits for /-t This agrees also with the definition used by Bruno (1993a). In a bifurcation, the characteristic of a generating family changes direction abruptly in general. For instance, if the characteristics for M > 0 are as shown by the dashed lines in Fig. 9. 1, then one generating family comprises branches 1 and 4 (solid lines), and another comprises branches 2 and 3. Families of generating orbits belong to the first, second, or third species (Sect. 2.10). On the contrary, generating famffiies do not belong to a definite
comparison with point of view, and to
For
a
the
case
/-t >
consider
--+
species; they frequently bifurcate from
M. Hénon: LNPm 52, pp. 171 - 202, 1997 © Springer-Verlag Berlin Heidelberg 1997
one
species
to another.
172
9.
In
Fragments
principle,
all that is necessary for the construction of generating famipreceding chapters, and this construction could
lies has been laid out in the
as an exercise for the reader. In practice, however, the trivial; the necessary ingredients are numerous and scattered throughout the preceding pages. Therefore we will show in some detail how this construction proceeds, provide detailed recipes, and give several examples. In Chaps. 3 to 5, families of generating orbits were decomposed into family segments, separated by extremums in C. In this chapter, we consider in turn each family segment and we decompose it into smaller family fragmenis, separated by bifurcation orbits. We use the systematic enumeration of bifurcation orbits made in Chap. 6. In the next chapter, these fragments will be reassembled in a different way to form the generating families.
theoretically
be left
is far from
exercise
9.1.1 Accidents
family fragment. As a preparation for now an inventory of the possible kinds of accidents. follow, we An accident corresponds either to an end of a family segment, or to a bifurcation orbit inside a family segment. By reviewing all ends of family segments found in Chapters 3 to 5, and all bifurcation orbits found in Chap. 6, we find the following list of possibilities (see Table 9.1). First, an accident may correspond to a natural termination (Sect. 2.5). The generating orbit tends toward a limit without ever reaching it. In that An accident
defined
is
what will
case
as
the end of
a
make
there is of
course no
continuation.
In all other cases, the accident
corresponds to a definite generating orbit, actually reached, and the important question is then how many branches emanate from that orbit. If the number of branches is only two, we have a simple continuation of one family segment into another. This happens in the following cases. which is
-
An extremum in A
particular
itself
C,
case
is
in which two segments of the same family are joined. the reflection, where the family continues back into
(Sect. 2.5).
frequently happens that an extremum in C along a family is at the same a bifurcation orbit, because other families pass through the generating orbit. In that case, the bifurcation property supersedes the extremum property: the junctions between branches are no more trivial and must be determined. In fact, since there are more than two branches, such a case does not belong here, but in the next paragraph (bifurcation orbits)A second species family including one or more arcs S O, and where C de-0-720283. At that point, each arc creases and reaches the value Cp S O smoothly continues into an arc Sh (Sect. 4.5, point (x)). The family as a whole continues into another family, with Sh substituted for S O It
time
-
=
everywhere,
and C continues to decrease.
Introduction
9.1
173
reciprocal case of a second species family including one Sh, where C increases and reaches the value Cp. It continues then into another family with S O substituted for Sh everywhere, and C
There is also the or more arcs
continues to increase.
ellipitc-hyperbolic continuation. It is not a bifurcation, whatarcs implied, because C ever Cp is not an extremum. If not be C > Cp, only S O arcs can Sh arcs; and conversely if present, C < Cp, only Sh arcs can be present. Thus there are only two branches. The continuation of a second species family into a third species family, or conversely (Chap. 5; Sect. 6.3). We call this a Hill coniinuaiion. We obtain We call this
an
the number of
-
the
of the next segment from Table 5.2. also, there are only two branches, whatever the number of
name
Here
C
--
<
3,
species
all
arcs
belong
to the second
species;
if C
--
3, all
arcs
arcs:
are
if
third
v-arcs.
generating orbit corresponding to an accident, it is a bifurcation orbit. This can happen either inside a family segment, or at one end. It will be useful to distinguish between first, second, and third species bifurcations (Sect. 6.5), and further to divide second species bifurcations into type 1, type2, and type 3 (Sect. 6.2). We distinguish thus 10 classes of accidents, which are numbered and listed in Table 9.1. Class I is a natural termination; classes 2 to 5 are simple continuations; classes 6 to 10 are bifurcations. (The right part of Table 9.1 will be explained in the next Section.) If
more
than two branches emanate from the
Table 9.1. The ten classes of accidents. Class
Accident
Description
1
natural termination
natural end
2
extremum in C
3
reflection
minlmax min1max
4
elliptic- hyperbolic
5
Hill continuation
6
first
7
second
8 9
10
9.1.2
continuation
species bifurcation species bifurcation, type second species bifurcation, type second species bifurcation, type third species bifurcation
Explanation
of the
species
I
J
Ist
I
I
J
L
2
1
J
2..
1
1
3..
3rd
species
3
1..
Fragment Tables
Each of the tables 9.2, 9.3, 9.7 to 9. 11 represents the decomposition of family segments into family fragments, in a standard format. Each subtable (between two thick lines) corresponds to one family segment, whose name appears in
Fragments
174
9.
the
heading.
In the
body of a subtable, each line corresponds to one accident, corresponds to one fragment.
and each interval between two consecutive lines
The first column indicates the class of the accident. The second column indicates the value of the Jacobi constant C. This value
always
the bottom. The next columns describe the accident
increases towards
precisely; the naspecified in Table 9. 1 for each class. In classes 2 and 3, 'min' or 'max' is selected according to the nature of the extremum in C. In classes 2, 4, 5, the arrow is followed by the name of the continuing segment. In class 3, the symbol --) indicates the reflection fo the family over itself. In classes 6 to 9, columns 3 and 4 give the values of I and J. In class 7 (type 1), column 5 gives the value of L (a superscript + or indicates that the bifurcation orbit is in D+ or D-). In class 8 (type 2), column 5 gives 2 2 the value of c'. In classes 7 to 9, column 6 identifies the bifurcation in the ture of the information
given
more
is
-
shorthand notation defined in Sects. 6.2.1 and 6.2.2 first character is the type, the second character is T bifurcation, and the third character is the order.
(for
or
instance
P for
a
total
3TI): or
the
partial
symbol printed between two lines, in the rightmost column of the table, name of the generating family to which the fragment belongs gives (see A
the
next
chapter).
In Tables 9.8 to
9.10, corresponding to composite family segments of the species, an additional column is present (second from left); it will be explained in Sect. 9.3.3-2. second
9.2 First
Species Family Segments
We examine here the
9.2.1 First Kind-.
In
family segments found
in
Chap.
Retrograde Orbits
family I, of retrograde circular orbits, which
C increases from
3.
-oo
to +o
as
motion decreases from 0 to
is
also
a
the radius decreases from
family segment, oo
to 0 and the
(Sect. 3.2.2). family includes one bifurcation orbit of type 3 (Sect. 6.2.1.4) for n -- -1, C -- -1, which belongs to the set M. Thus the family is decomposed into two fragments (top part mean
of Table
-oo
This
9.2).
9.2.2 First Kind- Direct Orbits
In Sect. 3.2.2
we
family segments: orbits Ide
found two families of direct circular
are
the
exterior
also
-
We consider first increases
orbits, which family of interior orbits Idi and the family of
family Idi. As the radius a decreases from 1 to 0, C Family Idi includes an infinite sequence of bifurcation
from 3 to +oo.
First
9.2
Fragments
Table 9.2.
I
C
1
-
00
9
1.000000
1
+00
of
family segments
natural end M
M
1
h
natural end
Idi 1
3.000000
natural end
6
3.008062
6
7
Ist
6
3.011315
5
6
1st
6
3.017033
4
5
1st
6
3.028534
3
4
1st
6
3.057532
2
3
Ist
6
3.174802
1
2
1st
1
+00
I)II,
I
(first kind).
species species species species species species
1
3.000000
na tural
6
3.010866
6
5
1st
6
3.016209
5
4
Ist
6
3.026767
4
3
1st
6
3.052571
3
2
1st
6
3.149803
2
1
Ist
1
+00
family Idi
is
species species species species species
natural end
i
the direct circular orbits with
a mean
thus divided into
an
infinite number
motion
(I
n
+
given by (3.8). The of fragments; the first few
1, 2,.... The corresponding values of C
--
end
natural end
(Sect. 6.1.2):
orbits
175
Ide
i
1
I
Species Family Segments
are
shown in Table 9.2.
are
The
case
of
is similar. As the radius
family Id,
a
increases from 1 to
co, C increases from 3 to +oo. Family Id, includes an infinite sequence of bifurcation orbits:'the direct circular orbits with a mean motion n = (I- 1)/I,
I
=
2,3,.... The family Id, is thus also divided
into
an
infinite number of
fragments. 9.2.3 Second Kind
Families of cations
asymmetric orbits
(see Chap. 6).
orbits. For
given
family segments
E+
and
So
we
I and J
Ej'j
and
ETj (Sect. 3.3).
of the second kind do not have proper bifuronly to examine the families of symmetric
have
(mutually prime),
if I + J is
odd, there
are
two
Ej'j; if I+ J is even, there are two family segments (3-16) shows that on a given segment, C varies in
the interval
2V/a
C
with
a
+
2VIra
a
a
(1/j)2/3.
These segments exhibit several bifurcations, enumerated in consider them in turn. The results for all families I :! 4, J <
other families which will be
needed,
are
Chap. 6. We 4, and a few
collected in Table 9.3. The values of
176
Fragments
9.
I and J
redundant
are
family; they
kept
are
here, since they already appear in the compatibility with the other tables.
name
of the
for
9.2.3.1 Bifurcation with First Kind. In the
particular
case
JJ
-
11
=
1,
the segments Ej'j and Ejj have a bifurcation with the family Id (Sects. 6.1.2 and 7.1). This direct circular orbit corresponds to 0 = 0 and to the upper
boundary
of each segment
(Fig.
3.2 and
Equ. (3-16)).
9.2.3.2 Bifurcation with Second a
first
species
to
second
a
Species, Type 1. In a bifurcation from species family, all deflection angles vanish in the
second species family: it must be a total bifurcation. We consider first total bifurcations of type I (Sect.
ellipse
tion
J I
the
are
i4
ues
same
J. For
(see
characterized
is
given
three numbers
values of I and
J,
L has
Each of these values
El'j
family segments
and
6.2.1.2).
1, J, family segment.
those of the second kind
6.1).
Table
of the two
as
by
finite number of
a
corresponds
The bifurca-
L. The values of I and
to
a
There must be
permitted val-
bifurcation
on one
odd, or Ej+j and Ejj the appropriate segment are given
E,j
if I + J is
if I + J is even; rules for the choice of in Sect. 6.2.1.2. The values of C are listed in Tables 6.2 and 6.3 for several bifurcation orbits. These bifurcations belong to the set M (Table 6.4). 9.2.3.3 Bifurcation with Second
total bifurcations of type 2
pendicular crossing
Species, Type
(Sect. 6.2.1.3).
2. We consider next
The bifurcation orbit has
a
per-
happen only if a ! 1/2, i.e. IIJ > 2 -3/2 For given values of I and J satisfying that condition, there are two bifurcation orbits of type 2, one direct and one retrograde in fixed axes. When I + J is odd, these orbits belong to the family segment E,'j if I > J, and to E'j if I < J. When I + J is even, they belong to the family at xO
+1. This
=
can
.
Fj+j (Sect. 6.2.1-3).
segment
From the relation xo we
=
a(l
e)
obtain the
e
=
1
and from
1
a
I
(3.11)
C
+ a
(9.2)
eccentricity of the bifurcation orbit:
I -
1
--
(9-3) we
2c' 2
find the value of the Jacobi constant:
(9.4)
-
These bifurcations
a
belong
to the set 2TI
9.2.3.4 Bifurcation with Second
bifurcation orbit of type 3. It
(Sect. 6.2.1.4).
Species, Type 3. There is to the family segments E1+1 belongs to the set 3T2 (Table 6.7)_
belongs
This bifurcation
(Table 6.5). a
single ET,
and
9.2 First
Table 9.3.
Fragments
C
I
J
of
-
5
1-000000
1
3.000000
-,
L
E ,+,
bif
3T2
1
Hill-f
f
-
3
1.000000
1
3.000000
max
1
3T2
b
E2, I E2,
min
1.711013
2
1
8
2.970934
2
1
+
6
3.149803
2
1
Ist
-
8
-
2T1
-
2T1
h
species
E2ei 2
-
6
min
-2.233276
5
1
8
2.917266
5
1
+
3
3.761947
max
I-)
2T1 2T1
3
-3.077957
7
-0.513880
3
3.761947
min 0
1
5
M
max
2
-3.331388
min
8
-2.302638
6
1
8
2.908345
6
1
2
3.937095
max
E6'1 2T1
-
2T1
+
E6,
E6, -
7
-3.077957
8
E6i 1 1.889882
2
3
E51
ET, 9
177
family segments E (second kind).
E+ 9
Species Family Segments
E21
1.889882
min
0.406767
2
1
0
3.149803
2
1
Ist
1T2
species
E+ 31
2
-3.331388
mi n
E,:',
7
-0.534364
6
0
2
3.937095
1
,
M
max
E6'
E32
1
E'32
-
3
-2.403749
min
8
-1.984407
3
1
8
2.945907
3
1
3
3.365249
max
2T1
+
2T1
f
E3 1 -
2
-1.526286
min
8
-1.461139
3
2
7
-0.403687
3
2
0
M
8
2.987424
3
2
+
2T1
6
3.052571
3
2
Ist species
.
3
-2.403749
7
-0-450861
3
3.365249
E3e2
min 3
2T1
-
1
0
IT2
max
E'41
-1.526286
min
7
-1.422126
3
2
1
7
2.102200
3
2
-1
6
3.052571
3
2
1st
2
0.000000
min
6
3.174802
1
2
0.000000
min
8
0.302724
1
2
8
2.872078
1
2
6
3.174802
1
2
E4,
2
-2.777952
min
8
-2.135461
4
1
8
2.929161
4
1
2
3.571652
max
-
+
2T1
2T1
E4,
E4'1
E31 2
2
IT2 M
species
E1'2 h
-
E,'2 Ist species
2
Ej'2
2
-2.777952
min
7
-0-487019
4
2
3.571652
E4i, 1
max
0
E4,
1T2
-,
E12 -
+
2T1
2T1
Ist species
f
178
9.
Table 9.3.
Fragments
(continuation)
E+
E4:5
13
3
0.693361
min
2
-0.696238
min
3
3.466806
max
7
-0.668904
4
5
1-
7
-0.589224
4
5
3-
M
7
1.229779
4
5
1T2
7
2.744731
4
5
3+ 1+
6
3.017033
4
5
Ist
--+
E4 5 ,
E13 3
0.693361
3
3.466806
min max
-*--)
2
1.259921
min
2
3.779763
max
E14 El,
Ef14 2
1.259921
min
2
3.779763
max
Ejf
0.436790
-
0.296034
-
min
E23
2
3
1-M
2.087673
2
3
1+
6
3.057532
2
3
Ist
M
-
-
8
6
0.436790
min
0.350508
2
3
2.971249
2
3
3.057532
2
4
5
-- -
E45 2T1
-
7
-0.654354
4
5
2-
7
2.201452
4
5
2+
M
8
2.992994
4
5
+
2T1
6
3.017033
4
5
Ist
z
M
species
+
min
-0.735653
5
6
1-
M
7
-0.714983
5
6
3-
M
7
1.907027
5
6
7
2.831130
5
6
3+ 1+
M
6
3.011315
5
6
1st
0.752829
min
0.737044
5
6
0.730337
5
6
2-
7
-0.650269
5
6
4-
M
7
0 979150
5
6
M
2T1
2
2T1
8
species
Ist
-0.752829
7
-
7
-
2.481672
5
6
8
2.995530
5
6
+
4
1+M
6
3.011315
5
6
Ist
4
lst
4
7
2.567360
3
6
3.028534
3
E34 1-
species
E34
E3i4
2
-0.605707
min
8
-0.564634
3
4
7
-0.489624
3
4
2-M
7
1.574496
3
4
2+
1T2
8
2.987461
3
4
+
2T1
6
3.028534
3
4
Ist
-
-
species
E '6 -
7
3
M
,
-
M
min
0.554474
-
-
-
E56
2
4+ 2+
0.605707
7
min
-0.672200
2T1
E3'4 2
-0.696238
8
E5-6
E23 -
3
2
speciesz
E2-3
8
%
E,,
7
2
2
E14
E2'3
7
M
species
E45
EI'4
2
M
2T1
species
.
2T1 M
M
species
z
9.3 Second
9.3 Second
Species Family Segments
179
Species Family Segments
family segment of generating orbits of the second species is defined by a list arc family segments (Sect. 4.8). Conversely, any such list defines a second species family segment, provided that (i) the intervals of variation of C of the individual arc family segments have a non-empty intersection, and (ii) there A
of
two identical T-arcs in succession
are no
(Proposition 4.3.2).
9.3.1 Detection of Bifurcation Orbits a given second species family segspecies bifurcation orbits (Sect. 6-5), either total or partial bifurcations, of type
We need to find all the bifurcation orbits in ment.
They belong
to the set of second
described in Sect 6.2. -
1, 2,
or
9.3.1.1 occur
They
be
can
3.
Type
only
2. We search first for bifurcations of
at the ends of
families, and
arc
therefore
type 2. Arcs of type 2 at the ends of
only
arc
family segments. Any such end is also one of the two ends of the second species family segment. Therefore we need only look at these two end orbits. A necessary condition for an orbit to be a partial or total bifurcation orbit of type 2 is obviously that it contains at least one arc of type 2. Conversely, the presence of one arc of type 2 is sufficient, since total and partial bifurcation orbits of type 2 exist for all values of the order in the simplest case of a single basic arc of type
n
(Tables
6.5 and
6.9):
even
2, bordered by two non-zero deflection angles, we have a partial bifurcation of order 1, since the set 2P1 contains 4 branches (Table 6.9). Thus, the presence of an arc of type 2 is a
necessary and sufficient condition for
9.3.1.2
Type
3. The
condition for
an
bifurcation of type 2.
of type 3 is similar.
case
at the two end orbits of the second
orbit to be
that it contains at least
a
partial
a
or
of type 3.
one arc
Again
we
need
only look
A necessary total bifurcation orbit of type 3 is
species family segment.
Here, however, this condition is
not
no partial bifurcation of type 3 and order n = I sufficient, (Sect 6.2.2.3). Thus, if there is a single basic arc of type 3 (length M -- 1), bordered by two non-zero deflection angles, the orbit is not a bifurcation
because there is
orbit. We note from Table 6.6 that
m
=
1
family segments 5000 and S 1_1. Thus, the simply continued into another, with S 1_1 This is also
a
or
more
than
more
arcs
Restriction 4.9.1, such 9.3.1.3
substituted for
So"O
or
vice
versa.
minimum for C.
If there is where two
corresponds to one end of the arc second species family segment is
Type
one
isolated basic
reach
a case
an
more
arc
principle
of type
be left out of consideration.
Vanishing Angles. The detection of bicomplex. We note first that bifurcations of family segment (see Conjecture 6.2.1).
I- Detection of
furcation orbits of type I is type I occur only inside an
extremum
will in
3, we have a case simultaneously; according to
arc
180
Fragments
9.
We consider first the
where
case
the bifurcation. This is the
deflection
one or more
angles
vanish at
for all total
bifurcations, and also for partial bifurcations in which the bifurcating arc corresponds to more than one arc in the second species family segment. We detect these cases by monitoring the deflection angles. We examine all pairs of successive arcs in the orbit in turn. A deflection angle vanishes between two arcs of type I if they have the same supporting ellipse, i.e. if the following conditions are satisfied: The two The two
elliptic (Sh
must be
arcs
case
is
excluded).
must be
represented by the same point in the domain D of the (A, Z) plane. (In particular, in the case of D2, they must be in the same sheet D+ or D_.) 2 2 arcs
If the first
ends
arc
P
in
(resp Q),
the second
begin
arc
must
for
an arc
in P
(resp
Q). We recall from Sect. 4.3 that there are
-
-
-
easily
identified from the
name
are
of the
4
cases
corresponding
arc
of type
1; they
family:
Ingoing S-arc: PQ. This case corresponds to an arc family segment S.*,, with # ! 0 (Sect. 4.3.3.3). Outgoing S-arc: QP. This case corresponds to an arc family segment S*, with P < 0. Ingoing T-arc: PP. This corresponds to an arc family segment Tj , (Sect.
4.3.4). -
Outgoing This
by
QQ. This corresponds
T-arc:
gives
16
cases
for
a
to
an arc
sequence of two arcs;
family segment Tj'j.
only
8 of them
are
allowed
the third condition above. In order to
implement the second condition, we notice that the characterfamily segments in the (A, Z) plane are on straight lines. Two these lines have a single point of intersection, which is easily computed in
istics of the of
the various
arc
(Incidentally,
cases.
this shows the
sentation introduced in Sect. 4.3.2
W, 7) -) (1) Two S *' The ,,,2p,.
consecutive
Z=PIA-al and the
A
(ii)
point 02
S-arcs, belonging
characteristics
-
to
arc
family segments S*
(9-5)
,
of intersection is C11
z
=
02-01
'
Pla2
-
#2a,
02-01
A
Z
--
OA
(9.6)
=
S-arc, belonging to an arc family segment belonging to an arc family segment Tl*j. The point A
and
,,
are
Z=fl2A-a2
,
superiority of the (A, Z) reprerepresentations ('r, 71) and
the earlier
over
-
S,*,,,O,
followed
by
a
T-arc
is then
(9.7)
Second
9.3
(iii)
A T-arc followed
by
S-arc. The
a
Species Family Segments
representative point
is
181
again given
by (9.7).
(iv)
Two consecutive T-arcs. Since the two T-arcs cannot be
the second condition cannot be satisfied. The deflection in that
In the first three cases, the intersection
vanishes
point always exists,
i.e. the char-
parallel: this is obvious in cases (ii) and (iii), and in case fl, :A 02, because one of the arcs must be ingoing and the other
are never
have
we
identical,
case.
acteristics
(i)
angle
never
outgoing. The intersection
point should
is excluded since it contains this the case,
we
no
lie inside the domain D,
bifurcation
orbits;
first compute I and J from A
=
see
IIJ
D2 (Domain D3
or
-
6.2.1.2.)
Sect.
To
see
if
and the fact that I and
mutually prime (in cases (ii) and (iii), I and J are already known); then we compute L ZJ; and finally we check whether the combination (1, J, L) exists in Table 6.2 or 6.3. For domain D2, if the combination exists, we have two cases to consider, L+ and L-, in domains D+ and D-, respectively. 2 2 The intersection point should also lie inside the characteristics of the two arc family segmenis. This is checked with the help of the values of A for the J
are
--
critical
arcs
When in
a
Table 6.2
can
vanishing deflection angle 6.3. The value of
or
m
found, the value of C is read S-arc, which will be needed later,
has been
for each
computed with the help of (6.6), (6.14), (6.20), (6.22):
be
m
listed in Table 4.4.
I + 2
=
[J i
(9-8)
.
2. T-arcs, there is m Conversely, the presence of a vanishing deflection angle is sufficient for the existence of a bifurcation orbit'. If all deflection angles vanish, the whole orbit For
-_
contains
an even
M, 1T4,
...,
number of basic
of the bifurcations IP2, 1P3,
one
arcs
and
we
have
listed in Table 6.4. If not all deflection ..
.,
one
of the bifurcations
angles vanish,
we
have
listed in Table 6.8.
Type 1: Detection of Non-Basic Arcs. The above method fails a partial bifurcation when the bifurcating arc corresponds to a single arc of the second species family segment. (This corresponds to the branches formed of a single symbol in Table 6.8: 3, 7, ...). Thus, it is 2, 5, also necessary to monitor the individual arcs of the family (excepting the hyperbolic arc Sh, which is never involved in a bifurcation). 9.3.1.4
to detect
by the following observation: an arc which produces partial bifurcaiion of type I musi be a non-basic arc. If it is a basic arc, there is no other possible decomposition of the bifurcating arc, and therefore no other family and no bifurcation. (This corresponds to the fact that there is no partial bifurcation of type I and order n I in Table 6.8). Conversely, any non-basic arc gives rise to at least one other decomposition of the bifurcating arc, and thus corresponds to a bifurcation. This will be facilitated
by iiself
a
--
182
Fragments
9.
The
of non-basic
may be understood
graphically. In the an arc family is followed, the (x, y) plane, an arc begins shape of the arc changes. It may happen that somewhere along the family, the arc crosses the point M2, which is thus encountered not only at the two ends but also inside the arc. This corresponds to a non-basic arc. The method for finding non-basic arcs is different for S- and T-arcs; we occurrence
arcs
and ends in M2. As
consider them in turn.
S-arcs. It will be convenient to regroup temporarily the arc family segments into families: we look for non-basic arcs inside a whole arc family S,,,p. A bifurcation of type 1 At first
corresponds to a rational value of A (Sect. 6.2.2. 1). view, it might seem that we have a problem because there is an infinite
number of rational values of A
Fortunately, as we
show
value A
all but
11J,
=
From
now.
along
the characteristic of
finite number of these values
a
an arc
(6.14), (6.17), (6-18), belonging
to the
arc
we
correspond
find that at
family S,,a
family Sp.
an arc
a
to basic arcs,
given
rational
is the first basic
arc
iff 0 and
:! #
<
i
the second basic
is
-i <
0
<
We obtain the
If
# 0
>
In
is
arc
is
arc
a
=,31
Therefore the
arc
family SOO
non-basic
arc
iff
non-basic
arc
iff
(9.12)
Z
-
arc.
.
addition, for given
(A
non-basic
(9.11)
0, the
=
a
arcs.
.
aJ
=
for domain
a,
LIJ)
0, and J,
I must be such that the
lies inside the domain D,
or
representative
D2. The value of L
(9-13)
.
This value must be
again,
never
arc is a
11J, given by (6.9): L
is
rules:
non-basic
J < -0
point
iff
(9.10)
following
no
0
arc
-
0, the
i < 3. If 3 <
0
0, the
--
contains
If
(9-9)
,
one
of the allowed values listed in Table 6.2
D2, solutions
come
in
pairs L+ and 3. The 4, fl
or
6.3. Here
L-.
arc family S43 belongs example, consider the case a 3, domain D, (Table 4.1). From (9.11) we find that only the values J 1, 2, 3 have to be considered. This arc family lies in domain D1, and the allowed values of L are given simply by ILI < J. We easily find then that the 0. only solutions are J 2, 1 3, L 1, and J 3, 1 4, L An order-of-magnitude estimate of the number of non-basic arcs can be computed. We consider first the cases 3 and 4 in Fig. 4.10 and Table 4.1.
As
to
an
=
=
case
=
=
--
=
=
=
=
Species Family Segments
9.3 Second
183
forget for a moment the fact that I and J must be mutually prime. a given J, the number of allowed values of I is then on average (2J 1)/0. Summing over all values of J from I to 0, we find that the number of solutions is about p2 /# 0. Finally, we reintroduce the condition that I and J should be mutually prime; using the Lejeune-Dirichlet theorem (Knuth We
For
-
=
1981, Sect. 4.5.2),
find that the number of non-basic
we
In the other cases, the interval of variation of A is
of solutions is
but
correspondingly smaller,
is about
arcs
6,3/lr 2.
smaller, and the number
remains
of the
same
order of
magnitude. The value of C is found in Table 6.2 from
6.3. The value of
or
m
computed
is
(9.8).
Finally
we go back to segments. We must determine in which segment lies each non-basic arc which has been found; this is done with the help of the
values of A for the critical
listed in Table 4.4.
arcs
T-arcs. Here the detection of non-basic
arcs
is much
simpler.
Bifurcations of
are met on a family segment for all values of L permitted for the given values of I and J. These values are listed in Tables 6.2 and 6.3. In
T-arc
type I
these
bifurcations, the T-arc is always
9.3.1.5
Type
I-
Recapitulation.
bifurcations where all formed
arcs are
only of symbols
the branches
11
in
angles
non-basic arc, with
(This corresponds 11,
6.4). Thus,
use
use
9.3.2 Data
the
of both methods
-
happen
on
Are
arc
(looking for vanishing
setting
up tables of
For each
they play
a
arc
arcs
arc
as
some
has data
family segment; arc family segment.
inside the
family segment,
we
therefore collect these remarkable arcs;
We make
now an
inventory
of the
species family segments possible kinds of remarkable
We consider first the two ends. From Sect. 4.9
we
find that each end
be
happens only once, at the end of the hyperbolic arc family segment Sh corresponding to C -oo. A parabolic arc. This corresponds to a continuation from the elliptic arc family segment S O toward the hyperbolic arc family segment Sh, or conversely. This happens only twice, in these two arc family segments. A natural termination. This
>
---
-
deflection
role similar to the accidents in second
(Sect. 9.1.1).
-
and also
fragments
species, it will be convenient to first collect family segments. Specifically, we need information on
The non-basic
can
...'
of either method alone is not
Family Segments
The two ends of the
arcs.
to the branches
1111,
111,
been done for the first
-
2.
arcs) guarantees that all bifurcations of type I are that a bifurcation orbit is found by both methods.
9.3.2.1 Remarkable Arcs. Before
the
--
and for non-basic
found. It may
on
m
The second method fails to detect the arcs.
1 in Table 6.8:
Table
sufficient. However, the
basic
a
184
-
-
-
Fragments
9.
An
arc
of type 2. of type 3.
An
arc
of type 4. This
An
arc
(see Chap. -
corresponds
to
continuation toward
a
a v-arc
farmly
6.3).
5 and Sect.
An extremurn in C. The
arc
family segment
continues into another
arc
family segment. Finally,
we have the non-basic arcs of type 1, inside an arc family segment. distinguish thus 7 classes of remarkable arcs, which are listed in Table 9.4. They are in close correspondence with some of the accidents listed in Table 9.1, and we use the same identifying numbers (in the first column) to point out this correspondence. The right part of the table shows how these remarkable arcs are represented in the data tables below.
We
Table 9.4. The Class
seven
Remarkable
classes of remarkable
arcs.
Description
arc
1
natural termination
natural end
2
extremum in C
minimax
4
elliptic-hyperbolic
5
type 4: Hill continuation type 1
7 8
continuation Hill
type 2 type 3
9
1
J
L
1
J
c'
1
1
type type type
m rn m
Properties of Remarkable Arcs. We derive now some interesting properties concerning the possibility of simultaneous occurrence of different classes and different remarkable arcs. Many of these properties were proved by Bruno (1973; 1994, Chap. IV). 1) Is it possible for an arc to belong to more than one class? First, class 7 (non-basic arc of type 1) occurs only inside an arc family segment, while the other classes occur only at the end; thus class 7 is isolated. Class 1 corresponds to hyperbolic arcs; class 4 corresponds to a parabolic arc; all other classes correspond to elliptic arcs. Therefore classes I and 4 are 9.3.2.2
isolated from each other and from the other classes. Classes 5, 8, 9 (types 2, 3, 4) are mutually exclusive (Sect. 4.2). Finally, the extremums in C which concern us are those which divide
family
into segments
(Sect. 4.6).
an
Therefore
they lie inside one of the arc families found in Chap. 4, and they cannot correspond to arcs of type 2, 3, or 4, since these happen only at the end of an arc family. Thus: arc
Proposition 2)
9.3.1.
We consider
Each remarkable
now
arc
two remarkable
posite orbit, and therefore have the different classes?
belongs
arcs
same
io
which
one
and
belong
only
to the
value of C. Can
one
same
class, com-
they belong
to
9.3 Second
Class I
elliptic If
or
one
corresponds to C parabolic arcs, with C
-oo, while all other classes
-+
is of class 5
arc
Species Family Segments
-2v12 (4.8). Therefore 3 and all (type 4), then C >
--
185
correspond
to
class I is excluded. arcs
are
of type 4
(4.8). Therefore class 5 is also excluded. Bruno (ibid., Sect. 3.2, Theorem 3.1) showed that classes 7 and 8, 7 and 9, 8 and 9 cannot have the same C. He also showed (ibid., Sect. 3.4, Problem 4 and Theorem 3.3) that, provided that Schanuel's hypothesis is true, classes according
2 and
to
7, 2 and 8,
2 and 9 cannot have the
Classes 4 and 9
same
C.
-0.720283 and C -1, respectively. correspond to C The only possible remaining cases are classes 4 and 2, 4 and 7, 4 and 8. No proof of impossibility has been given in these cases. However, empirical evidence, provided by the computation of many remarkable arcs, suggests =
-_
that the value of C cannot be the
same
in these
cases
either. So
we
make the
following Conjecture 9.3.1. of the same class. Even if the
class
ing
conjecture
4, this would
because the is
Two remarkable
were
false in
which
one
belong
to the
same
orbit
are
of the last three cases, involv-
not have serious consequences for the
elliptic-hyperbolic
smooth continuation from
a
arcs
present study, there accident; truly hyperbolic arcs, accompanied by a
continuation is not
elliptic
to
an
smooth monotonic variation of C.
3) Finally
we
consider two remarkable
arcs
which
belong
to the
same
composite orbit, and therefore have the same value of C and are of the same class if Conjecture 9.3.1 is true. What can be said about them? Can they be different? Can they have different supporting ellipses? Class I is simple: for C -oo, the only existing arc family is Sh, and therefore the only second species family is fShl, which is made of only one -
arc
2, Bruno (ibid., Sect. 3.4, Problem 3 and Theorem 3.3) showed that, provided that Schanuel's hypothesis is true, different extremums in C correspond to different values of C. Thus we have For class
Conjecture
9.3.2.
belong
same
to the
Two remarkable orbit
are
arcs
of class
2
(exiremums of C)
which
identical.
Class 4 is also
simple: there exists only one remarkable arc, namely the parabolic joins the arc families S O and Sh. The orbit can include number of copies of this parabolic arc, together with other arcs. any If one arc is of class 5, then C 3, and all arcs of the orbit are remarkable arc
which
--
arcs
of class 5.
7, Bruno (ibid., Sect. 3.2, Problem 1; Sect. 3.4, Theorem 3.3) that, provided that Schanuel's hypothesis is true, different nonoriented supporting ellipses (i.e. supporting ellipses which differ in their (a, e, c') values; see Sect. 4.2) correspond to different values of C. Thus we For class
showed
have
186
Fragments
9.
Conjecture 9.3.3. Two remarkable arcs of class 7 (bifurcation of type 1) which belong to the same orbit have the same non-orzented supporting ellipse. For class
8, similarly, Bruno (ibid., Sect. 3.2, Theorem 3.1) proved that
different non-oriented supporting Thus we have
ellipses correspond
to different values of C.
Proposition 9.3.2. Two remarkable arcs of class 8 (bifurcation of type 2) which belong to the same orbit have the same non-oriented supporting ellipse. Finally, ellipse.
for class 9
Explanation
9.3.2.3
remarkable
(bifurcation
of type
3),
there is
a
single supporting
of the Data Tables. Tables 9.5 and 9.6 list the
for S- and T-arc
family segments, respectively. The format fragments (Sect. 9.1.2). Each subtable (between two thick lines) corresponds to one arc family segment, whose name appears in the heading. The first and last lines in the subtable represent the ends of the arc family segment. Intermediate lines correspond to non-basic arcs
is similar to that of the tables of family
arcs
of type 1. The first column indicates the class of the remarkable
arc.
The second column indicates the value of the Jacobi constant C. This value
always increases towards the bottom. The the arc; the nature of the
2, 'min'
or
given according
'max' is selected
In classes 2 and
4, the
family segment.
In classes 7 to
J. In class 7
column 5 to
(type 1),
gives
arrow
7
specified
give
information
to the nature of the extremum
is followed
by the
name
C.
in
continuing
of the
arc
9, columns 3 and 4 give the values of I and gives the value of L. In class 8 (type 2), In classes 7 to 9, column 6 gives the type 1
gives
c.
length
the
9.3.2.4 S-arcs. The data for all
other families which will be
rn
arc
needed,
of the
arc
families
are
(the :!
lal
number of basic
arcs).
4, Ifll
a
:!
4,
and
few
collected in Table 9.5.
The ends of the S-arc fan-Lilies have been studied in Sect. 4.5
Fig. 4.10).
on
in Table 9.4. In class
column 5
the value of
3, and column
next columns
information is
The nature of the two ends is
given by
(see
also
Table 4.1.
For type 2,
we compute c', 1* and J* from Table 4.2, (4.60) and (4.64), 1, J, rn from (4.57), where I and J are mutually prime. The value of C is given by (9.4). For type 3, there is I -1. The value of rn is given by 1, J 1, C
and then
=
(4.66), (4.68),
or
--
(4.70), depending
=
on
Values of C for the critical arcs, segments, are listed in Table 4.4.
Finally, the non-basic
arcs
of type I
9.3.2.5 T-ares. The results for all
the
case.
corresponding are
found
explained
family segments
collected in Table 9.6. The information for the
identical,
as
to ends of
T,j
Tlij arc
in
with
arc
family
Sect. 9.3.1.4.
1, J :!
4
are
family segments
is
and therefore not shown.
For given values of I and
T'j (Sect. 4.6).
There
are no
J, there
are
critical arcs;
two T-arc an
end of
family segments Tl'j and a segment coincides with
9.3 Second
Table 9.5. Data
C
S-arc
on
I
family segments.
L
J
t
rn
Sh -oc
S -2 2
8
-1.461139
3
2
-0.975496
max
2
1
2
1
2
1
2
2
2
2
-
S202
--+
S2+3
natural end
SO. 9
-1.000000
1
2
-0.399131
max
1
3 --+
1
S O
S O Sh
4
-0.720283
-
2
-0.399131
max
-*
9
-1 000000
2
-0 936944
.
.
1
1
max
8
-1.711013
2
2
-0.936944
max
3
3
--).S , ,
1
2
-
1
-SIO1
S+ 12 8
2.872078
1
2
2.874117
max
2
2
+ --*
1
S,02
so12 -0.870656
2
2,874117
-
8
-
S ,+2
max
0.870656
min
0,302724
1
S,02 2
2
-
1
0
S2 1 8 7
-
-
5
1.984407
3
0.406767
2
3.000000
-
1
-
0
1
2
1
1
3
-
max
3
+
S20`3
--+
S203 2
-0.963281
2
2.971299
min max
--+S2 , -
S2-i+3
--*
2
-0.963281
min
8
-0.350508
2
3
2
0
S23 -
S2+4
1.000000
1
0.975496
max
1
3 --
8
2.872078
1
2
2.872598
max
2
0.074233
min
2
2.872598
max
2
0 0742 3 3
min
8
0.302724
1
2
8
-2.135461
4
1
2
1
7
-0.450861
3
1
0
1
3
8
2.970934
2
1
+
2
1
2
2
1
3
3
7
+
S204
S204 S2 , -
S+ ,
S 4 .
()
S2 4 -
-
S302 'j
8
-1.711013
2
1
7
-0.4 036 8 7
3
2
5
3.000000
--+
-
0
Hill
S103
Hill
0 S2 2
-
2
2.971299
S 01
min --+Si.2
S12 2
2.971249
2
-
S11
2
8
S2-3
So00
so11
2
type.
=
-0.720283
4
9
t
Species Family Segments
S
2
5
9
-1.000000
1
2
-0.986987
max
1 ---->
S;3
187
9.
188
Table 9.5.
Fragments
(continuation) 0
S 3
sll
8
-1 342022
4
2
-0.986987
max
.
3
2
-
--+
1
So.
S34
2
-0 989951
2
2.992996
.
S4+
max
S45
8
2.987461
3
2
2.987469
max
4
2
+
1
S304
2
-0.989951
min
8
-0.672200
4
5
1
S+1
so34
-
2
-0.982706
2
2.987469
S34
min
S1+1
max
-
S405 -
2
1
2
1
1
8
2.970934
2
2
2.970940
max
+ -
S(-) 1-1
-
S+ 1-1
so-1-1
S3-4 2
-0.982706
min
8
-0 564634
3
.
S04 ,3
4
2
-
1
so41
9
-1.000000
2
2 970940 .
1
1
max
3
1
S- 1-1
8
-2.233276
5
1
2
1
9
-1.000000
7
-0.487019
4
1
0
1
3
5
3.000000
2.945907
3
1
+
2
1
8
SZ 45 ,
min
-
so
1 -*
1
3
2
2
1
2
1
2
1
Hill
1-2
0
S4 2 8
-1.871338
5
2
7
-0.406767
2
1
8
2.987424
3
2
2
1
0
1
5
+
2
1
-
0
S43 8
-1.558971
5
3
7
-1.422126
3
2
7
-0.423816
4
3
5
3.000000
--).
2
1
1
1
3
0
1
3
-
Hill
8
0.302724
1
2
8
2.872078
1
2
+
1
+
-
+
S-2-1 8
2.945907
3
2
2.945910
max
S
C) 2-1
S 02-1 2
-1.439479
2
2 945910 .
min max
S-2-1
S+ 2-1
0
S4 4
S-2-1
9
-1.000000
1
2
-0.991935
max
3
1
9
SZ -
S4-4
2
-1.439479
Min
9
-1.000000
1
1
2
SO-2-1 3
2
2
1
S+
-2-2
8
-1.271980
5
2
-0 991935
max
.
4
2
1
0
S4 4 -
S4+5
8
2.987424
3
2
2 987426
m ax
.
+
SO-2-2
so 2-2
8
2.992994
4
2
2 992996
max
.
5
2
+
1
0
-
S4 5 -
9
-1.000000
2
2 987426 .
1
1
max
3 + S -2-2
3
9.3 Second
Table 9.5.
S
(continuation) so-3-3
2-2
9
-1.000000
5
3.000000
so
2
-
1
1
8
2.971249
2
3
-
+
1
9
-1.000000
2
1
5
3.000000
so
0.302724
1
2
8
2.872078
1
2
-
+
2.929161
4
2
2 929162
max
.
1
So
1.785103
min
2 929162
max
.
S-
2
-
-
S+ -3
1.785103
min
1.711013
2
3
4
-0.296034
2
3
1-
1
2.970934
2
2.970936
max
SO-3-1 2
1
2
+
2
SO-3-2
2
3-2
-
2
1.283380 2 970936 .
Min
S
3
-
2
9
2
3
1+
1
3
3
4
+
2
1
1
+
2
1
S+ -4 -1 8
2.917266
5
2
2.917267
max
-
1 283380
mi n
1.000000
1
.
-
o
2
-2.002284 2 917267 .
3
4
4
2
2.992986
max
3
+ ---+
S
4-1
S+ 4-1
4-1
-
-
2.002284
min
1.984407
3
1
2
So -
4 -1
2
1
2
1
S+ -4-2 8
2.957105
5
2
2.957106
max
+
SO-4-2
2
SO-3-3
o 4-2
2- 1.544550
min
S
7- 0-406767
2
0
S2
-
2.992986
min max
S-
2.957106
2
S+ 3-3 8
4 -1
2
s
S) 3-2
1
SO-4-1
-
S-3- 2 2
2.087673
-
S+ 3-2
max
3
-
_
so
1
1
2.987461
s
3-1
-
1
2
7
8
2
-
8
2
8
6
3-1
2
-
5
3 -4
-0.564634
S+ 3-1
1
3
7
3-1
8
Hill
8
2
+
1
1 --+
2
3 -1
S-
3
S+ 3-3
2
o
-
1
2
_
2
1
inax
2
1
8
2
2.992986
2
4
8
s
-1.000000
-3-3
3
-
9 2
S2
S+ -3
4
3
-0.350508
-
3
Hill
--+
8
so 2
Species Family Segments
1
8
1
max
4-2
1
3
S+ 4-2
4-2
-
-
1.544550
min
1.461139
3
2
3
So-4-2 -
2
1
2
1
S+ 4-3 8
2.981728
5
2
2.981729
max
+
SO-4-3
189
190
9.
Fragments
(continuation)
Table 9.5.
so
S_ -4 -4
4-3
min
S- 4-3
2
-1.212991
7
2.102200
3
2
-1
2
2.981729
max
S_
1
3
4-3
4-3
9
-1.000000
1
2
-1.212991
Min
1
3
-1 000000
5
3.000000
so
-
S-
9
6
So 4 -3
S+ 4-4
1
.
--+
1
8
3
Hill
4-5
8
-0.672200
4
5
7
-0.554474
3
4
1-
7
2.567360
3
4
8
2.992994
4
5
2
1
1
3
1+
1
3
+
2
1
-
-
so
8
2 995527
5
2
2.995528
max
.
so
-
2
+
-1.000000
2
2 995528
S04-4
.
max
end of the T-arc
with
cos
7
=
1,
3
1
1
or
S_
5 -6
1
4-4
9
an
4
7
4-4
8
-0.737044
5
6
2
1
7
-0.668904
4
5
1-
1
3
7
-0.489624
3
4
2-
1
3
7
1.574496
3
4
1
3
7
2.744731
4
5
2+ 1+
1
3
8
2 995530
5
6
+
2
1
family. Therefore the
.
end values of C
-
given by (4.6)
are
(9.4).
The ends of the T-arc families have been studied in Sect. 4.5
Fig.
4.10 and Table
4-1).
For
10 J,
each end is
J 1, and c' is given by Table 4.2. For I 4, respectively. Type 3 corresponds to C 3. corresponds to C
rn
=
=
3 and
an
(see
also
of type 2. There is the ends are of types
arc
1,
-1 and
m
=
2.
Type
4
=
9.3.3
Fragments
We embark
now upon the decomposition of second species family segments fragments. Here we meet a practical problem: the number of these segments is huge, since an orbit may be an almost arbitrary combination of Sand T-arcs, in any number. Even if we limit the list to small values of the indices a, 0, 1, J and to a few arcs, it soon becomes unmanageable. Therefore we will not attempt a systematic study of the family segments. Instead, we include only some of the fragments (and the corresponding segments) which will be used in the next chapter to build generating families. If necessary, the decomposition of a segment not present in the tables will have to be done by following the algorithms described below.
into
9.3.3.1
Simple Family Segments. We begin
with simple family segments, family segment (Sect. 4.7): each orbit is made of a single arc. This arc cannot be a T-arc, because we would have identical T-arcs in succession; it must therefore be a S-arc, belonging to where the list contains
only
one
arc
9.3
Table 9.6. Data
C
T-arc
on
I
J
L
-1.000000
5
3.000000
1
t
--.).
3
1
m
2
Hill
.
T2,2 1 8
-1.711013
2
1
7
-0.406767
2
1
8
2.970934
2
1
2
1
0
1
2
+
2
1
-
191
family segments.
T1,1 9
Species Family Segments
Second
T3, 1
Tli: 3 8
-1.342022
4
3
7
-1.333769
4
3
2
1
2
1
2 2
-
7
-1.282752
4
3
1
1
7
-0.423816
4
3
0
1
2
7
1.596095
4
3
-1
1
2
7
2.568912
4
3
-2
1
2
8
2.992986
4
3
+
2
1
8
0.302724
1
2
8
2.872078
1
2
1.
T1, 2 -
+
2
1
2
1
T2'3
8
-1.984407
3
1
2
1
7
-0.450861
3
1
0
1
2
8
-0.350508
2
3
-
2
1
8
2.945907
3
1
+
2
1
7
-0.296034
2
3
1-
1
2
7
2.087673
2
3
1+
1
2
8
2.971249
2
3
+
2
1
8
-0.564634
3
4
2
1
7
-0.554474
3
4
1-
1
2
-
2.
T4, 1 8
-2.135461
4
7
-0.487019
4
1
8
2.929161
4
1
1
2
1
0
1
2
+
2
1
-
T3i2
T3'4 -
7
-0.489624
3
4
2-
1
2
2
1
7
1.574496
3
4
1
2
1
1
2
7
2.567360
3
4
2+ 1+
1
2
2
0
1
2
8
2.987461
3
4
+
2
1
3
2
-1
1
2
3
2
+
2
1
8
-1.461139
3
2
7
-1.422126
3
2
7
-0.403687
3
7
2.102200
8
2.987424
-
192
Fragments
9.
an arc
family segment
S,*,,p.
The second
species family segment
will be called
I S,-;'3 1. (It should be carefully distinguished from the corresponding arc family segment.) The interval of variation of C for the second species family segment family segment, and therefore non-empty. orbits, following the prescriptions of Sect. 9.3-1. The order n of a bifurcation is equal to the length m of the arc. If one end of the arc family segment is an arc of type 2, all bifurcation angles vanish and we have a total bifurcation of type 2. Similarly, if one end of the arc family segment is an arc of type 3, all bifurcation angles vanish, and we have a total is
obviously
the
same as
for the
arc
We look for bifurcation
bifurcation of type 3. Inside the second
species family segment, the deflection angle between vanish, since a S-arc starting with a negative radial ends with a velocity positive radial velocity, and conversely. So only the nonbasic arcs of type 1, listed in Table 9.5, give rise to bifurcations. The results for some simple family segments are listed in Table 9.7. These tables are similar to the Tables 9.5, showing data for the S-arc family segments; the lines are in one-to-one correspondence. The main difference is that the nature of the bifurcations can now be indicated: it is always a total bifurcation for the end lines of a fragments, and a partial bifurcation for the does not
successive arcs
intermediate lines.
Composite Family Segments. We consider now the case of composiie family segments, where each orbit is made of more than one arc. In 9.3.3.2
that case, the list may contain T-arcs. Tables 9.8 to 9.10 contain in an additional column
(second
from
left)
a
dash-dot sequence, indicating which of the arcs participate in an accident. Each symbol in the sequence corresponds to one arc; a dash stands for a
participating are, while a dot stands for a non-participating arc. The algorithm for processing a family segment and building its table of fragments is as follows. 1) We obtain the minimal value of C as the largest of the minimal values for the constitutive arc family segments, which can be read from Tables 9.5 and 9.6. We note the value of C and the
responsible
for the
ending (indicated
as
position of the
which
arc or arcs
dashes in the leftmost
are
sequence),
in
the first line of the table.
2)
Next
For this
we
determine the nature and the
we
look at the properties of the
Tables 9.5 and 9.6.
They
are
properties responsible arc
remarkable arcs,
to
excluded because it
classes 1, 2, 4, -
already
only happen
inside
a
than
to
class
one
bifurcation of
segment;
so we
given by
one
of the
according type 1, is
have to consider
5, 8, and 9.
Class I is not termination
can
arcs,
or
corresponding
(They cannot belong to more Conjecture 9.3.1.) Class 7, corresponding to a
classes in Table 9.4.
of this accident.
possible here: the only arc family segment which has a natural Sh, corresponding to the second species family segment I Sh
IS
shown
in
Table 9.7.
9.3 Second
Table 9.7.
Fragments
of
I
L
C
J
some
Species Family Segments
193
simple family segments ISI.
ISO'-,}
bif
jShj
2
-1 439479
2
2.945910
.
IS-2-1 1 IS + 2-1 1
i
m n
max
f
-
1
-oo
4
-0.720283
natural end M
S O1
o
So , 9
-1.000000
1
2
-0.399131
max
1
M
JS O 1
--+
M
ISO-01 jShI
4
-0.720283
-
2
-0.399131
max
S000 I
M
0 IS211 8
-1.984407
3
1
7
-0.406767
2
1
5
3.000000
--)-
-
0
IS-2 11 -
2
-1.439479
min
9
-1.000000
1
ISO 2-1}
1
3T2
1
3T4
f
IS-2-2} 9
-1.000000
5
3.000000
1 --+
9
Hill- 9+
ISO 2-31 8
-0.350508
2
3
8
2.971249
2
3
+
2T1
1
+
2T1
-
2TI
fS-+3-11
2T1 1P3 a
Hill-a
8
2.929161
4
2
2.929162
max
IS 3-1}
h
0
1 S321
SO-3 -1 1
8
-1.711013
7
-0.403687
5
3.000000
2
3 ---+
1 2
-
M
0
1P3 9
Hill-g'-
-
8
2.970934
2
2
2.970940
max
1
+ --+
2T1
f so-1-11
h
2.929162
IS-3-11 fS+3-11
Min max
h
2
-1.785103
Mill
8
-1.711013
2
1
fS-03-11 -
2T1
h
ISO 3-41
ISO -1 000000 .
2
-1.785103
2
IS-3-11
f s+, _ I
9
2
2 970940 .
9
-1.000000
5
3.000000
1
1
max
1 -+
M
--+IS+ 1-1 1
1
h
3T2 C
Hill-c
O
IS 1-21 8
0.302724
1
2
8
2.872078
1
2
-
+
2.945907
3
2
2.945910
max
1
+
-0.564634
3
4
7
-0.296034
2
3
1-
7
2.087673
2
3
1+
1P3
8
2.987461
3
4
+
2T1
1
+
2TI
-
2T1
1P3
I St, _1 8
2 917266
5
2
2.917267
max
.
--+
ISO 4-11
f
ISO 4-11
2T1 2T1
IS + 2-11 8
8
2T1
1 so-2 -1 1
f
2
-2.002284
2
2.917267
min max
-IS-,-,I JS+ 4_1 1 --*
f
194
Fragments
9.
(continuation)
Table 9.7.
fs-4-1 1
IS1
2
-2.002284
Min
8
-1.984407
3
P04-11
1
8
f
2T1
-
Is__ 31 -4
2
-1.212991
7
2.102200
3
2
2.981729
max
Is_ -4 9 2
-
min 2
-1
4
1-
1P3
7
2.567360
3
4
1+
1P3
8
2.992994
4
5
+
2T1
-5 -6
8
-
7
1
3
5
3
-
7
-1.000000
1
1
Min
-1.212991
3T6 g
IS04-31
-*
2T1
-
1
-
IS-+4-31
--+
4
0.554474
-
fSO
9
1P3
1
0.672200
7
ISO-4-3 1
4 -5
-
-
0.737044
5
6
0.668904
4
5
1-
1P3
0.489624
3
4
2
1P3
2T1
-
-
7
1.574496
3
4
7
2.744731
4
5
2+ 1+
1P3
8
2.995530
5
6
+
2T1
1P3
-Class 2: If
one and only one arc reaches an extremum in C, the simply replaced by its continuation (see Fig. 4.19).
If
than
more
reaches
one arc
an
arc
is
extremum, in principle the continuation
(Sect. 4.9) This is the case, for instance, in the 0 0 where four arcs reach an extremum segment ISGOO) 2-15 4JO; SO"01, simultaneously (Table 9.10). (These four arcs are identical, in conformity with Conjecture 9.3.2.) cannot be determined
SOtJOY So 0
In
some
cases,
however,
the situation
be saved
can
by considerations
of
symmetry. This happens, for instance, in the segment
fS 07 So 2-DS 01) at C -0.399131 (Table 9.9). Among the four present'b ranches, only two consist of symmetric orbits: the branch IS O) So on which we have 2-1) S 01 arrived at the extremum, and the branch fSO00i So in which both 2-D S0001, =
arcs
S O
have moved
branches must be Another of two in
case
arcs
we
come
have
a
same arc
back
reaches the value
so
=
family,
along
If the orbit includes
there,
segment
arc
an
extremum. This
(Table 9.8).
-1.439479 and
the
-
Therefore these two
they
no
same
--
family
-0.720283,
(see
ambiguity
arc
with
S O,
we
Sect.
happens
The two
is
made
for instance
arcs
belong
in
become identical at the minimum. So
one or more arcs
Cp
accident at all
there is
SOOO
be saved is when the orbit
can
same
standard reflection situation: each
we come
this is not
to the
for C
3
and -
where the situation
which
f S :2-1 So 2-11,
fact to the
to the next
over
joined.
is continued into the a
shift of
or one or more
have
9.1.1).
a
other, half-period. arcs Sh, and C
one
class 4 accident. In
C does not have
an
a sense
extremum
in the continuation. We obtain the
name
of
the next -
If C
=
a sense
segment simply by replacing all S O by Sh or vice versa. 3, all arcs are of type 4 and we have a class 5 accident. Again in this is not
smoothly
an
accident
continues into
a
(see
third
Sect.
9.1.1).
species family.
The second species
family
-
If
contiguous
one or more
(Sect. (class 8). total), depending on 9.3. 1. 1)
arcs are
Species Family Segments
Second
9.3
of type
2,
have
we
a
bifurcation of type 2
We determine the nature of the bifurcation whether all
or
arcs or
only
195
some
of them
(partial
participate
n of the bifurcation is obtained by adding the participating arcs. If there are non-contiguous arcs of type 2, we have a partial bifurcation with more than one bifurcating arc, and in principle the continuation cannot be determined (see Restriction 6.2.1). But this never happens is the examples in
the bifurcation. The order
lengths
of the
m
tabulated below. -
If
of type 3,
one or more arcs are
If there is this
not
is
single
a
arc
to
3,
I SOO) So 2
If there is
non-basic
-
1
1
have
a
there is
SOO
arc
minimum C
a
the segment a
we
of type
true accident. The
a
corresponds
This
basic
--
class 9 accident. no
(Sect. 9.3.1.2);
bifurcation
replaced by So 1 -1. This happens,
is
-
1,
conversely. instance, in
or
for
-
arc
of type
3,
or more
than
one
arc
of type
3,
we
bifurcation of type 3. We determine the nature and the order of the bifurcation as for type 2 above.
have
a
3)
The above steps 1 and 2 are the other end of the segment. The
repeated for the maximal value of C, i.e. corresponding properties are inscribed in
the last line.
4)
We look
Sect. 9.3.1.
The non-basic
for bifurcations inside the segment, as explained in we look for non-basic arcs.
now
They
are
bifurcations of type 1. First
in
arcs
given
a
arc
family segment
are
enumerated in Tables
9.5 and 9.6. Those which fall inside the interval of variation of C
bifurcation, segment IS', -1,S44'11, to
incorporated arc family segment S4, provides into the table. As
which is
a
the
-
for C
--
a
non-basic
rise the arc
Sect. 9.3-1.3. A bit of type an
first two
arcs
angles. We consider each apply the procedure described in angle corresponds to a bifurcation or-
vanishing
search for
we
of successive
As
give example, in
-0.487019.
5) Finally pair
an
in turn and
vanishing deflection incorporated into
which is
1, example,
arcs are
deflection
we
the table.
consider the segment I SOO) So 2 -13 S O I (Table 9. 9). The of the form PQ and QP, respectively, so that the deflection
between them may vanish. We have a, -_ 0, #1 -2,,32 = -1. 0) a2 From (9.6) we obtain A -- 2, Z = 0, from which we deduce I = 2, J 1,
angle
-
-
-
L
=
(1, J, L)
0. This combination
find that the value A
we
and
SO-2-1-
C
-0.406767.
=
The next
=
So the deflection
pair
of
arcs
So
exists in Table 6.2.
2 does lie inside the
angle
2-1)
S O
Finally, from Table 4.4 family segments S O
arc
indeed vanishes inside the segment, for also has
a
vanishing
a
In the last
and
PQ,
pair
of
arcs
S 0, S 0, however,
and the deflection
angle
the two
cannot vanish.
angle, for (This is simply
deflection
same values of the parameters and the same value of C. consequence of the symmetry of the orbit.)
the
arcs are
of the form
PQ
196
Fragments
9.
We have thus found
vanish, three
and three
arcs
bifurcation orbit in which two deflection
a
arcs are
implied;
basic arcs, with
are
m
this is therefore
=
a
partial
angles
bifurcation. All
1; therefore the order of the bifurcation
is3.
Tables
9.8, 9.9 and 9.10 give the decomposition into fragments for some family segments with two arcs, three arcs, and more than three arcs, respectively.
9.4 Third
Finally
we
Species Family Segments
decompose into fragments the family segments of v-generating in Chap. 5. It is sufficient to consider the segments found in
orbits described
Sect. 5.4 for
v
=
1/3,
be reduced to
can
The results "3rd
species".
they
are
i.e. for Hill's case; the cases of v = 0
asymptotic
are in
Table 9.11. There
values of
F,
not C.
only
is
Values in the first column
cases or v
are
in
v < 1/3 and V > 1/3 1/3 (see Sect. 5.8).
0 <
-_
bifurcation, indicated by parentheses to indicate that
one
9.4 Third
Table 9.8.
Fragments
C
of
I
some
J
family segments.
two-arc
bif
L
0 SO-2-11 1 SOO, 2
-
2
-
-1-000000
Min
-0.399131
max
ISO-1-1) S 2-11 S O) SO-2-1
S 0, S-2 -1 4
-
2
Sh, SO-2 -1
-0.720283
-
7
-
-0.406767
2
-0.399131
max
0
1
1T2
S1010, SO-2 -1
P121 S-1-1 1 8
-
2
-
0
fS 12, 2
S
1 -1
-
2
1
2.874117
max
-
2P1
+
- -1 0 S-1 S12,
1
-0.870656
-
7
2
2.872078
f S,-2, S-,-, 1
min
2.087673
2
2.874117
max
1+
3
1T2
IS121 S-1-1 I
Is-'s-1-11 12 -
2
-
7
-
8
I S102, S -1
-0.870656
min
-0.296034
2
3
0.302724
1
2
2.971249
2
3
2.971299
max
-
1-
1T2 2P1
-
P131 S-1 -1 1 8
-
2
-
2P1
+ 0
--*
fS23, S-1-1 I -
0
P2 3, S -1 -1 1 2 7
2
-
-0.963281
-
-
f S 3, S-1-1 1
min
2.567360
3
2.971299
max
1+
4
M +
I SLI S-_
IS2_31 S-1 -1 I 2 7
8
-
-
-
f S203, S-1 -1
-0.963281
min
-0.554474
3
4
-0.350508
2
3
2.987461
3
4
2.987469
max
1-
IT2 2P1
f S341 S-1 -1 8 2
-
-
+
2P1
JS304)S-1-11 -
-
Species Family Segments
197
198
Fragments
9.
(continuation)
Table 9.8.
0
f S14,S 1-11 2 7 2
-
-
-0.982706
-
-
IS,-4, S-,-, 1
min
2.744731
4
2.987469
max
5
1+ -
1T2
f S+ L)S-1-11
z
S341 S-1 -1 2 7 8
-
-
-
-0.982706
M'n
-0.668904
4
5
-0.564634
3
4
S304) S1-
IT2 2P1
-
f so 1-17S,011 0
0
f SOO, S41 1
2
-1.000000
min
7
-0.487019
4
1
0
1P3
8
2.945907
3
1
+
2P1
1
+
2P1
ISO-1-1,S+2-11 8
2.945907
3
2
2.945910
max
fSO-1-1) SO-2-1 I
fSO 1-1)SO-2-11 2
-1.000000
2
2.945910
min max
IS000, so '_1 I ISO 1-1 S+2-11
ISO 1-2,S-1-1} 8
8
-
-
0.302724
1
2
2.872078
1
2
-
+
2P1
2P1
fs-2-J) so 2-11 3
9
-
-
IS -2 2
-
9
-1.439479
min
-1.000000
1
1
3P2
a
so-3-21 -1.283380
min
-1.000000
1
Is__ I)S -3-21 -2 -
1
3P2
9
fS-2-1,S-3-21 2
-1.283380
min
9
-1-000000
1
fS-2-11 SO-3-21
1
3T6
fSO-2-3,S-1-11 8 8
-
-
-0.350508
2
3
2.971249
2
3
2P1 +
2P1
9
9.4 Third
Table 9.8.
Species Family Segments
(continuation)
ISO-1-1, S__1_11 8 7
7
8
-
-
9 5
3
4
-0.296034
2
3
1-
2.087673
2
3
1+
1P3
2.987461
3
4
+
2P1
-
-
Mil
I
2P1
-0.564634
-
1P3
Tie, 1
-
-
-
-
-1.000000
3.000000
3T4
1
1
9
Hill-g
-+
M21) SO-1-1 I ,.
f T2'1
L-0 I 1100 f
2
-1.000000
Min
7
-0.406767
2
1
0
1P2
8
2.970934
2
1
+
2PI
Table 9.9.
Fragments
of
some
I
J
-1.000000
1
1
-0.399131
max
C
1
three-arc
family segments.
bif
L
fSO00) SO-2-1) SOOO 9 2
-
-
3P2
f SO-0, so 2-I)SOO
a
ISO-01 so-2-1,S O -
4 7
2
-
-
-
jSh, So 2-1, Sh}
-0.720283
----1
-0.406767
2
-0.399131
max
1
1P3
0
0 SOOO) SO-2 -1 SO'O I
fSO00, SO-2-1, SO-1 -1 9 2
-
-
1
-1.000000
1
-0.399131
max
3P2
ISO-0, So-2-1,S01-11
fSOO) SO-2-1) SO-1-1 I 4 7
2
-
-
-
fSh) SO-2-1 So
-0.720283
-+
-0.406767
2
-0.399131
max
0
1
1-1
1P2
0 0 fSO0,S-2-1,S-1-11 0
-+
a
199
200
Fragments
9.
Table 9.9.
(continuation)
0 1 so00 so-3-2,S.01 '
9
-
7
-
2
-
ISO
-1.000000
1
1
-0.403687
3
2
-0.399131
max
3P2
0
1P3
0
-1-1,
S21, SO-1-1
2
-1.000000
min
IS000) S21) SOO 1
7
-0-406767
2
1
0
7
2.102200
3
2
-1
2.970940
2
max
0
-
0
1P3 1P3
0 Stj -1 1 IS-+1 -1, S21, U
I S--1-1 S12) S-1-1 8
2.872078
1
2
2.874117
max
2
2P1
+
0 I fS 1- ,S12,S_,-, -
0
fS_1 _1IS121 _
Min
2
-0.870656
7
2.087673
2
2
2.874117
max
3
fS-1- 1 S127 S-1-1 7
1+
fS
IP3 1 -1
+ S-1-1 S12,
I S-_I -I S12, S-1-1 I I
2
9
MI, so-3-2, S o -
-0.870656
min
7
-0.296034
2
3
8
0.302724
1
2
fS 1-
0 S12, S
1P3
-
2P1
-
2P1
fS-1-1 SO-1-2 S-1-1 1 i
8
0.302724
1
2
8
2.872078
1
2
+
2P 1
9
9.4 Third
Table 9.10.
of
Fragments
family segments with
some
I
J
1.000000
1
1
0.399131
max
C
Species Family Segments
more
than three
arcs.
bif
L
0 '0 0 15-1 -1, so00,S-4-1,SOOI
9
-
2
-
3P3 __+
ISO 1-1i S 0) S04-1i S 01
0 1 S-1 -1, Soo, so-4- 1, S O-1
4
--+
0.487019
4
0.399131
max
-
2
ISO-1-1, Sh, SO-4-1, Sh
0.720283
-
7
-
1
1P3
0
ISO 1-1) S0"O) So
SO0.1
4-1)
0-1-11S12)S-1- 17S 1-1} 8
2.872078
1
2
2.874117
max
IS
0
1
-1,S12,S
2
2P1
+
IS
0
-J,Sl2,S-,-J,S 1-11
liS 1-11
1-
IS _1-1)S12
S_1 -1
0.870656
min
7
2.087673
2
2
2.874117
max
0.870656
min
IS-, _"S"S 12
0.296034
2
3
1-
0.302724
1
2
2
-
2
-
7
-
8
I S-1 -1, So
1-2)
S -1 -1
3
1+
1
S-1-1
IP3
S 1-11 IS-,-,) S+) 12 S-1 -,I
t
-
-
1
1P3 2P1
S-1 -1 1
1
8
0.302724
1
2
8
2.872078
1
2
1.000000
1
1
0.399131
max
-
+
2P1 z
2P1
0 0 so00, so00, so-2-1, SOO, SOO
9 2
-
-
-
-
-
-
-
-
-
-
3P4
9
?
ISO -1jSOO,S0-2 -1iSOO) so-1-11 0
0
-1
9 2
-
-
1.000000
1
0.399131
max
3P4
1
ISO-1-1, S O, SO-2-1, Soo, so
9
-1-1
I so-1 -1i SO-0, SO-2 -1, Soo, SO-1-11 -
4
-
7
-
2
-
ISO-1-1, Sh, SO-2-1, Sh) So
0.720283
-
0.406767
2
0.399131
max
1
0
1-1
1P3
fSO-1-1,SOOO)SO-2-1) soOO)SO-1-11
9
201
202
9.
Fragments
Table 9.11.
Fragments
of Hill
r
Hill-g
Hill-a 5
3
family segments.
5
(-()0) (4.326749)
0 21
P I
10 a
1
00) (4.499986) (+00)
Tli I IT,,-, 3rd
species
natural end
9
g
Hill-c 5 3
C
(4.326749)
5
(-00)
10
(4.499986) 1
5
1
(-oo) (+oo)
9
Hill-g+
Hill-f 5
S302 3rd species
E,+, natural end
f
10
(-00) (4.499986)
-
fS-2-21
3rd species
9
10.
Generating
We have
now
Families
all the tools needed for
our
ultimate
goal: tracing
the
path
of generating families. We will reassemble in a different way the fragments found in Chap. 9, using the study of the branch junctions made in Chaps. 7 and 8.
Every fragment belongs
10.1
exactly one generating family. generating families with numerical computations
to
We will also compare the for a small non-zero value y
=
0.012155
(the
Earth-Moon
case).
Algorithm
We
explain here
We
assume
algorithm used to follow a generating family. initially we are on a given segment, at a given value of C, and moving in a given direction (C either increasing or decreasing). 1) We locate the fragment table giving the decomposition of the segment into fragments. This is Table 9.2 or 9.3 for a first species segment, Table 9.7 to 9.10 for a second species segment, Table 9.11 for a third species segment. If the segment does not appear in these tables, its decomposition must be computed, using the methods described in Chap. 9. 2) We read from the table the value of C corresponding to the next accident, and the properties of that accident. (Remember that C always increases in detail the
that
downwards in these
3) -
tables.)
We process the
accident, according
Class 1: natural termination. The end of the and the
-
-
to its class.
generating family
is
reached,
algorithm stops. Classes 2 to 5: simple continuations. The name of the next family segment is given by the fragment table. In the particular case of a reflection (class 3), the generating family comes back over itself, and the algorithm stops. For a second species family, if more than one arc reaches an extremum, in principle the continuation cannot be determined and the algorithm stops. In some cases, however, the situation can be saved by considerations of symmetry (see Sect. 9.3.3.2). Class 6: first species bifurcation. We use Proposition 7.1.1 or 7.1.2 to determine the next segment.
M. Hénon: LNPm 52, pp. 203 - 233, 1997 © Springer-Verlag Berlin Heidelberg 1997
204
-
Generating Families
10.
species bifurcation of type L fragment table the properties
Class 7: second
of the bifurcation: 1, J,
We read from the T
or
(total
P
partial),
or
partial,
If the bifurcation is
L,
n.
we
check whether Restriction 7.3.1 is
algorithm stops. identify the bifurcation ellipse
satisfied;
if not, the We
J, L (the value of C 0,0, P0
can
be used
in
as a
Table 6.2
check);
or
we
6.3 from the values of
read the values Of
1, A2, K,
-
We find the
A2 sign(AC). We symbolic sign of the incoming branch: s the branch. first of For a name species segment, symbolic E. For a second species segment, we use (6.21) and (6.22) to ---:
determine the this
name
is
determine the values of k and
m
for the S-arcs.
partial bifurcation, we note whether the starting point is P (the first arc is a S-arc with 13 f - 0, or a T' arc) or Q (the first arc is a S-arc with 0 < 0, or a T' arc). We find the incoming branch in the appropriate Table 8.4 to 8.11 or 8.14 For
a
(or
to 8.17
one
of the twin tables with all signs reversed). incoming branch belongs. If this subset
We consider the subset to which the contains
only
branches,
two
If the subset contains continued and the
We determine the the
new
more
outgoing branch. exploration cannot be
then the other branch is the than two
algorithm stops. sign of the new
branches, the
AC from A2 and the
symbolic sign
of
branch.
species family (symbol E), we find paragraph First species using If it each S-arc we compute second for to a families. belongs species family, of the and for the k, using(6.22), position partial bifurcations, the fact arc, that the starting point is P or Q; we obtain then the name of the arc family from (6.21). For each T-arc, we determine whether it belongs to Tj'j or T,j from the position of the arc, and for partial bifurcations, the fact that the starting point is P or Q. Finally, from the value A IIJ for the bifurcation orbit, and the values of If the its
new
branch
belongs
to
a
first
the rules laid down in Sect. 6.2.1.2,
name
=
A listed in Table 4.4 for critical arcs,
we
find the
names
of the
new
S-arc
family segments. -
Class 8: second species bifurcation of type 2. We read from the fragment table the properties of the bifurcation: I, J, T
or
P,
c',
n.
If the bifurcation
if not, the
is
partial,
we
check whether Restriction 7.3.2 is
satisfied;
algorithm stops.
symbolic sign of the incoming branch: s c'sign(AC). We the of branch. first For name a symbolic species segment, this name is E. For a second species segment, we use (6.23) and (6.24), which give: m 1011J, to determine the symbols of the S-arcs. We -find the
=
determine the
=
10.1
If I > J 8.12
or
(A
>
1),
we
8.18. If I <
find the
J,
we
incoming
Algorithm
205
appropriate Table appropriate table by
branch in the
must first construct the
exchanging i and e (Sect. 8.4.2). branches, then the other branch is the more than two branches, the exploration
changing all sides of passage and If the subset contains only two outgoing branch. If it contains stops. We determine the
sign
of the
new
AC fromE' and the symbolic
sign
of the
branch.
new
branch belongs to a first species family (symbol E), we find using the rules laid down in Sect, 6.2.1.3, paragraph Fivsi Species families. If it belongs to a second species family, for each S-arc we compute sign(#) from (6.26), and then we compute a and P from the appropriate equations (6.23) or (6.24). For T-arcs, the symbol i or e indicates directly whether it belongs to Tj'j or Tj'j. Finally, from the value A IIJ for the bifurcation orbit, and the values of If the its
new
name
=
A listed in Table 4.4 for critical arcs,
-
family segments. Class 9: second species We read from the
we
find the
names
of the
new
S-arc
bifurcation of type 3.
fragment
table the
properties
of the bifurcation: T
or
P,
n
partial, we check whether Restriction 7.3.3 is satisfied-, if not, the algorithm stops. We determine the symbolic representation of the incoming branch. For a first species segment, se use the rules laid down in Sect. 6.2.1.4. For a second species segment, we use Table 6.6. As a check, we verify that the sign of the branch is the sign of AC. We find the incoming branch in the appropriate Table 8.13 or 8.19. If the subset contains only two branches, then the other branch is the outgoing branch. If it contains more than two branches, the exploration If the bifurcation
is
stops. We read the
sign of the new AC, which is the sign of the new branch. species case, we read from Table 6.6 the names of the new
In the second arc -
family segments. species bifurcation. The
Class 10: third
4)
We note the
variation of
10.1.1
C, and
name we
Explanation
of the
go back to
new
step
new
segment is found from Sect. 7.2.
family segment
and the direction of
I)-
of the Tables
following tables describes one generating family. Successive lines correspond alternatively to a segment (cols. I and 3) and to an accident (cols, Each of the 2 to
8).
The order of the lines is not defined any more by the condition that C should increase, as was the case in the fragment tables. Instead, it corresponds
206
Generating
10.
to the
Families
progression along
the
generating family. Thus, C by ,- or
sometimes
and sometimes decreases. This is indicated The
name
of
a
increases
in column 3.
segment is indicated in the first column. In the
case
of
a
composite second species segment, the arcs which take part in the previous accident are overlined, and the arcs which take part in the next accident are underlined. The properties of an accident are in columns 2 to 8. Cols. 3 to 6 essentially reproduce the information on the accident from the fragment tables of
Chap.
9. The value of C is in column 3. A value in
parentheses represents
the value of F instead of C. In
gives
addition, the
name
for second
species bifurcations (classes 7, 8, 9), column 7 subset, as described in the tables of Chap. 8; of the incoming branch, and column 8 gives the
of the branch
column 2 gives the name of the outgoing branch.
name
10.2 Natural Families We will
determine, as far as possible, the generating families corresponding to natural families, which begin either as circular orbits of very small or very large radius, or in one of the Lagrange equilibrium points. Str6mgren (1935) studied these families in the particular case ft 0.5; however, they exist for all values of ft, and it seems natural and convenient to keep Str6mgren's for all cases. So we will use these names also for the generating names a, b, the i.e. 0. Broucke (1968) computed the natural families, limiting case p the
nine
=
.
.
.
,
-+
families in the Earth-Moon
computed
case
(it
-
PEM
=
0.012155).
natural families in the
Bruno
(1993a, 1996)
Sun-Jupiter case (p 0.00095388). Bruno (ibid.) also followed the corresponding generating families, using an approach on which no details are given but which is similar to the one developed here (personal communication, October 1996). The results presented =
below agree with those of Bruno, apart from a few small discrepancies. Families g, i, I are followed here for a longer time. Table 10.1 shows the correspondence between the names used by Str6mgren,
Broucke,
and Bruno.
Table 10.1. Names of the nine natural families.
Str6mgren
a
b
C
f
9
h
Broucke
I
J,
G
C
H1, H2
L2
D
Ll
A, IR+
Bruno
E +
1/1
2T,
z
I
BD
El
ID1
ID
-
M
F 1
IR-
10.2 Natural Families
FamUy
10.2.1
207
a
operations in detail for family a, which nicely illustrates algorithm. This family begins with orbits of vanishing dimension around the Lagrange point L2, which for /-t > 0 is situated on the x axis at the right of M2. For 1,L 0, this point tends toward M2; therefore family a begins with third species orbits. In Hill's coordinates (Sect. 5.4), the 3- 1/3 0.693361, with point L2 indeed has a definite non-zero abscissa, 4/3 r 4.326749. A family segment of generating orbits, which we have 3 called Hill-a, emanates from it in the direction of decreasing I' (Fig. 5.1). The family a has a reflection in L2 (see Sect. 2.5). The first fragment of family a is therefore taken from Table 9.11, and inserted in first position in Table 10.2, which describes the generating family We describe the
several steps of the
--+
=
=
--
=
a
In the limit r
family IS2'11,
-oo, this
-+
decreasing
with C
segment continues into the second from 3
(Table 5.2).
So the next
species
fragment
is
taken from Table 9.7. The next accident
at C
occurs
=
-0.406767, where the
arc
S201
becomes
non-basic arc, and we have a partial bifurcation of type I and order 3, with starting point P. The bifurcating arc coincides with one whole period of a
the
orbit; the complement is empty. Therefore Restriction 7.3.1 is satisfied. -1 for that bifurcation. We 0.988607, - 2 the bifurcation with AC > 0, therefore the symbolic representation
From Table 6.2
arrive at of the
we
branch is
incoming
Table 10.2. Now all branch
read K
signs
reversed).
lP3--S. This subset contains
the outgoing branch
Since A2 direction of
-1,
=
3
-
(Sect. 6.2.1.2); this is written (or rather the twin
consult Table 8.7
we are
-
--
is
we
increasing
in column 2 of
table in which
belongs to the subset the junction is established;
branch
incoming only two branches, The
so
I il.
-
have AC > 0 for the next
C.
(The sign
segment: it is followed in the
Of A2 is in fact irrelevant: the
same
result
+1, with both incoming and outgoing branches having then a + sign.) The branch starts in P, so it consists or 3 1. From (6.22) we find that k arcs PQ, QP, PQ, all of length rn +1, From Table read 0. From 6.2 we 0, 00 -1, +1, respectively. (6.21) we ao 2 for obtain then the names of the three arcs: SOO, S-2-1, Soo. There is A would have been obtained for
A2
=
--
_-
--
-_
=
the bifurcation
orbit;
from the values of A listed in Table 4.4 for the critical
names of the three arc segments, and finally the name of family segment: f S& so 2-13 SOWe turn to the description of that segment in Table 9.9. The next accident -0-399131. It is an end of the segment, provoked by the fact occurs at C
arcs,
we
obtain the
the next
=
that the the two
arcs
pursuit arcs
S O
of the
reach
the situation
reach
a
maximum of C. In
principle
we
should abandon
since Restriction 4.9.1 is violated:
generating family here, extremum simultaneously. However, in the present case still be saved by recourse to the symmetry, as explained in
an
can
Sect. 9.3.3.2, and the next segment is found to be
f soOO)SO2-1) soOO}U
U
208
Generating
10.
Families
We consult the
description of this new segment in Table 9.9. AC is now have passed a maximum. The next accident is at C -1. decreasing This is the end of the segment; the two arcs SO'O become of type 3, with a length m 1. Since these two arcs are neighbours, we have a partial bifurcation of type 3, of order n 2. The symbolic representation of the since
we
=
--
=
incoming branch is + 11 (Table 6-6). We consult the Table 8.13: this branch belongs to the subset 3P2+S, which contains only two branches, so that the
junction is established. The outgoing branch is family segment S-2-1 and C decreasing (Table is fS-2-1) So 2-11-
-
2, corresponding
6.6).
to the
arc
Thus the next segment
We find the is at C
description of this segment in Table 9.8. The next accident -1.439479, where both arcs reach a minimum for C. Here again
=
might seem that we are stuck because two arcs reach an extremum simultaneously and Restriction 4.9.1 is violated. Here again the situation can be saved, but for a different reason (see Sect. 9.3.3.2): the two arcs belong in fact to the same arc family, and they become identical at the minimum. So it
we
have
standard reflection situation.
a
This it
completes the tracing of family closed family with two reflections.
is a
Table 10.2.
C
(or F).
Generating family
a.
In the classification of Sect.
a.
2.5,
Col. 1: segment. Col. 2: incoming branch. Col. 3: or E. Col. 7: branch subset. Col. 8: outgoing
Col. 4: 1. Col. 5: J. Col. 6: L
branch.
in
segment
C
I
J
(11)
L
subset
out
6
(4.326749)
Hill-a
(-Oo) 3.000000
IS2011
3
I SOO SO 2 I
-
-
1
1
-0.406767
2
-0.399131
max
-1.000000
1
1
0
lP3--S
111
SOO I
0 P0001 so-2-1,so)J
+
11
Is_ -2-1,S(-)2-11
1
3P2+S
-
2
1--, -1.439479
10.2.2
This
Family
family
b
is described in Table 10.3. It
dimension around the orbits in
L3.
belong
to the
begins
with orbits of
Lagrange point L3, situated in x -1, y family ET, (see Sect. 3.3.1.2). The family has =
vanishing
=
a
0. These reflection
10-2 Natural Families
The
family segment E j
209
is described in Table 9.3. The next accident is
the other end of the segment, for C
--
-1; it is
a
total bifurcation of type 3
definiteness, we must decide where we take the origin; we point with a positive radial velocity . The incoming branch is then + E-+ (Sect. 6.2.1.4). It belongs to the subset 3T2 1, and the with identical branch is This + is E-branch outgoing (Table 8.19). outgoing Thus reflection. have a of shift we for a half-period; the ingoing branch exept a the family b has a simple structure. It is a closed family with two reflections. It coincides in fact with the family ET, and order 2. For
choose the intersection
---
*
Table 10.3.
segment
Generating family
C
in
b. See Table 10.2 for
I
J
1
1
L
explanation
subset
of columns.
out
3.000000
E+
10.2.3
This
E-+
Family
-1.000000
3T2
---
1
+
E--
c
family is described in Table 10.4.
It
begins
in
a
way similar to
family
which a, with orbits of vanishing dimension around the Lagrange point Li, coordinates Hill's In of left axis the at the situated for ju > 0 is x M2. on
-3-1/3 -0.693361, point Li indeed has an abscissa 34/3 with IF 4.326749. A family segment Hill-c emanates from it in the direction of decreasing IF (Fig. 5.1). (The family c has a reflection in Li.) In the limit F -oo, this segment continues into the second species with C decreasing. At C S1, this segment reaches a tot al segment I 1 1 1, bifurcation of type 3 and order 2. The outgoing branch + 11- is identical with the ingoing branch + 2 exept for a shift of a half-period; this is a reflection. Thus the family c has again a simple structure. It is a closed family with two
(Sect. 5.4),
the
=
=
=
=
-*
-
-
-
-
reflections.
10.2.4
Family f
family is described in Table 10.5. It begins with retrograde circular vanishingly small radius around M2. These are third species orbits, described in Sect. 5.6. In Hill's coordinates, they correspond to an asymptotic branch for IF +oo (see Fig. 5.1). The first fragment is thus read from Table 9.11, segment Hill-f. In the limit F -oo, this segment continues into the first species family
This
orbits of
--*
--+
segment
E1+1 (Table 5.2).
We turn to Table 9.3.
210
Generating
10.
Table 10.4.
Generating family
c.
C
J
in
segment
Families
I
L
(I')
subset
out
C
(4.326749)
Hill-c
3.000000 +
2
-1.000000
1
1
3T2
---
0
+-11-
11--)
The next accident is the other end of the segment, for C = -1; it is a total bifurcation of type 3 and order 2. We must decide where we take the origin; we choose the intersection point with the x axis at the left of M2. The incoming branch is then + E+- (Sect. 6.2.1.4). It belongs to the subset
3T2-+-O, and the outgoing branch is
-
2
(Table 8.19).
From then on, the family falls into a regular pattern: it follows successively the three segments V-2-11, IS02-11) then a part of the first then and species family E31 It be shown from the rules so on. can f S-4- 1, 317
fS+2-11,
1
Chap. 9 for the construction of fragment tables that this pattern continues indefinitely. This is a case of naiural terminaiion (Sect. 2.5): both the dimensions and the period of the orbit grow without limit. There is an infinite number of accidents. They all use the same entry in Table 8.18. Family f is thus an open family with no reflections. given in
Family
10.2.5
This
family
g
is described in Table 10.6. It
with direct circular orbits of In Hill's
vanishingly
coordinates, they correspond
begins
in a
way similar to
small radius around M2
family
(Sect. 5.6).
an asymptotic branch for I' +oo (see Fig. 5.1). The first fragment is thus read from Table 9.11, segment Hill-g. At r 4.499986, a third species bifurcation is encountered (orbit gl). From Proposition 7.2.1 we find that the generating family bifurcates toward
to
---*
--
the segment Hill-g+, In the limit r --
-oo, this
segment
continues into the second species
family segment IS--2-21 (Table 5.2). A bifurcation of type 3 leads the family segment fTl',,Tl'l I. This segment leads back into third orbits and into the segment Hill-9, with I' increasing from -oo. We time
come
we
next to
species
back to the bifurcation in gl from the opposite direction. This junction and we bifurcate toward Hill-g'
follow the other branch
In the limit I'
-*
-oo,
this segment continues into the second species
family segment f S3021 (Table 5.2).
10.2 Natural Families
Table 10.5.
Generating family
segment
in
C
I
J
L C
(+00)
natural end
3.000000
E+
'
+
IS-2
-
-1.1
E+-
-1.000000
1
2.945910
2
max
+
1
2.945907
3
1
-
E
-1.984407
3
1
+ -
2T1--0
-
E
2T1--0
+
1
2T1--0
-
E
2T1--0
+
1
1--,
-
min
-2.002284
ISO-, I
2.917267
max
1
2.917266
5
1
E
-2.233276
5
1
I +
+ j
The
-
,
Is -, ,I
-,
3T2-+-O
-A
E+ 31
I sf
1
min
-1.439479
IS+ 2-1 1
-
1
1-111
ISO-2-11
E
out
I
(F) Hill-f
subset
211
family
follows then
family segments comprising
a
+ -
complex
sequence of
bifurcations,
and exhibits
up to five S-arcs.
The subset 3T6-+-+-+-l does not appear in Table 8.19; it can be shown, it contains only the two branches -42-. First, Ta24 and
however, that
-
ble 8.3 shows that
only
arcs
with
rn even can
-
be present. From
that AC must be negative. Finally, using the positions of the crossings, we are left with the above two branches.
(8.82),
we
find
perpendicular
The pursuit of the family is stopped at the end of the fragment ISO, SOO, 0 so-2-1) so00) S0001, when four arcs SOO reach simultaneously a maximum in C
-0-399131. 16 branches arrive at the into
account,
we
are
still left with 4
maximum; even if we take the symmetry branches, and 3 possible continuations:
0 0 ISOO, SO-01 S-2-1) SOO, SOO1, IS O,SOOiSO2-1)SO07S 01; MO)S O)SO2-1)S 01 S 01. Restriction 4.9.1 applies: with our present tools, it is not possible to 0
-
0 U
0 'j
U
-
continue the
10.2.6
family unambiguously.
Family
h
family is described in Table 10.7. It begins with retrograde circular vanishingly small radius around M1, with large positive values of C (see Sect. 3.2.2). Thus the first fragment is taken from the segment 1, At C -1, there is a total bifurcation of type 3 and family h bifurcates into the simple family I S- 1 1 1.
This
orbits of
=
-
212
10.
Families
Generating
Table 10.6.
Generating family
g.
in
segment
C
I
J
(11)
(+C*)
Hill- g
(4.499986)
Hill-g +
L f
subset
out
3T4--+--O
+4
I
natural end
third
species
3 000000 .
fS
1
-
-
2-2
1-al +
ei
TTI T" 1 1
-1.000000
1
1
I
3.000000
Hill- g
(4 499986) .
Hill-g'
-
third
species
(-00) 3.000000
0
IS32 1 I S00013
1
3
-0-403687
3
2
11
-1.000000
1
1
-1.283380
min
-1 000000
1
-1.212991
min
0
lP3--S
+
ill
0
So
3
-
-
21
SOO +
C'O
IS_
-2
3P2+S
I
IS-2-1)S-3-21 -24
fS-4-3 1 O
fS -4-31
+
ISO 1-1 S20 1, So 1
-
ISO
-1
1
S 01 So
0 so I so-1-1 SOO 1
1
-
1-1
.
1
-
-42-
3
2.102200
3
2
-1
lP3++S
+
ill
3
-0.406767
2
1
0
lP3--S
-
ill
-0.399131
max
-1.000000
1
-0.399131
max
1
2 -1
SQ o So
2-1;
0 SOO so-1-1
1
,
.
+
-
11111,
1
I S1111O S(1))O ,so 2-1 Sooo, So()O I I
3T6-+-+-+-l
,
3P4 ... S
+
lill
Natural Families
10.2
213
From then on, family h falls into a regular pattern, similar to that of Here again it can be shown that this pattern continues indefinitely.
family f. Family h
is thus
Table 10.7.
segment
an
open
Generating family
h.
C
J
in
+00 +
so 1-11 S
+'
family with
1
-1.000000
I
L
no
reflections.
subset
out
natural end 1
1
2.970940
max
1
2.970934
2
1
E
-1.711013
2
1
-1.785103
min
M-0
1
+
_
+
E2' 1
fs-3 11 -
+ -
-
2T1--0
-
2T1--0
+
E 1
-
fSO-3-1 I fS
+
I-'
1
3 -1
E4',
fs-5-1 1
10.2.7
..
2.929162
max
1
2.929161
4
1
E
-2.135461
4
1
+ -
2T1--0
-
E
2T1--0
+
1
.
Family
i
family is described in Table 10.8. It begins with direct circular orbits vanishingly small radius around MI, with large positive values of C. Thus the first fragment is taken from the segment Idi. The first 76 accidents are listed in the table. The family appears to have a regular structure, although it is more complex than families f and h. The successive fragments of Idi are encountered along the family, in order of decreasing C. Each of these fragments is separated from the next by an excursion through fragments of families E and second species families; the length of this excursion and the number of segments visited grow regularly. We number 1, the successive fragments of Idi; fragment number i thus begins in the 2, first species bifurcation orbit I i i (except the first fragment which 1, J and at C ends in the orbit I i + I (see Table 9.2). begins i, J +cx)) Then the Table suggests the following properties: This of
...'
=
-
--
-
-
-
-
The
fragments
--
=
i and i + I
are
separated by
--
5i accidents.
These accidents consist of i extremums in C and 4i bifurcations. The
fragments belong either to family Ej,j+j fragments imvolve up to
The second species
Other
regularities
can
be noticed.
or
to
a
second
i S-arcs.
species family.
214
Generating
10.
Only four M
---
different branch subsets
are
used
again
and
again: 2Tl--O, 2P1S,
1P3--S.
likely that this structure continues indefinitely; true, family i is an open family with no reflections. It
is
1,
Families
seems
1-0.2.8
Family
if that
conjecture
I
retrograde circular M2, large large positive values of C. Thus the first fragment is taken from the segment Id, (Table 9.2). At C 3.149803, there is a first species bifurcation. From Proposition 7.1.2, we find that the next segment is E2'1 The family follows then a complex sequence of second species bifurcations. We reach a partial bifurcation of type 3 with the incoming branch + 11 1. The corresponding subset is 3P3++S, which contains 4 branches (Table 8.13), here we are stuck. The continuation of the family cannot be found with the This
family
described in Table 10.9. It
is
orbits of
begins
radius around both M, and
with
with
=
*
1
present tools. 10.2.9
Family
m
begins with retrograde circular M2, with large negative values of C. Thus the first fragment is taken from the segment -[,. At C -1, the the bifurcates into After a simple family fSool. generating family passage through a maximum, this family continues into the family Sh of hyperbolic -oo (see Sect. 4.1). arcs, which tends to its natural termination for C is thus with an open family no reflections. Family rn
This
family
orbits of
is described in Table 10.10. It
large
radius around both M, and
=
-4
10.2.10
Summary
Out of the 9 natural
generating family was completely exploration was stopped because the (a, b, c, m); continue to generating family appears indefinitely (f, h, i); a nd in 2 cases a point was reached where the continuation of the generating family could not families,
determined
in
in 3
be determined
4
cases
cases
the
the
(g, 1).
210.3 Other Families An endless number of other
generating
families could be
computed. We give family differs from the previous ones in that it is made of asymmetric orbits. The family as a whole, however, is symmetric. It closes over itself: the last segment of the table is identical with only
the
one
example
first, with We list
a
a
in Table 10.11. This
different choice of
few directions
ating families.
in
origin. might
which it
be of interest to
explore
gener-
10.3
Table 10.8.
Generating family
i.
in
segment
C
I
3.174802
1
0.000000
min
El,'2 E'12
ISO-1-21
+
E
-
1
Ef12 Idi
E2Z 3
1-11 0.302724 I-A, 2.872078
fS102, S-1
fSI+21 S-1
-
1
E
-1 I
S
1
1
2
Ist
-
species
2T1--0
-
2T1--0
+
1
1
2
+
3.174802
1
2
1st
species
3.057532
2
3
1st
species
2.087673
2
3
1+
1T2---1
11
2.874117
max
+
2P1S
1
E
2.872078
1
2
0.302724
1
2
-0.296034
2
3
-
2P1S
+
1
s_
-1-1 +
11
E23
1-
1T2---1
E
I--,
E23 +
fSo 2-31
E
-0.436790
min
-0.35 0508
2
3
2.971249
2
3
+
3.057532
2
3
1st
3.028534
3
4
Ist species
'
E2 3
1
Idi
2.567360
3
4
1+
1T2---1
11
2.971299
max
2.971249
2
3
+
2P1S
1
-0.350508
2
3
-0.554474
3
4
-0.605707 I--'I
min
-0.56 4634
3
4
E3'4 -
S -
-
2T1--0
1
-
-
L
out
1--, 1
,
subset
I--,
1 -1
-
L
I +
f S-O1 -21
P0231
2
1--, -
Is_ _12
J
natural end
+00 I--,
Idi
Other Families
1 -1
E
2T1--0
+
E
species
I
fS2+31 S-1-1 I +
1
fSO-2-31 S-1-1 I
-
2P1S
+
1
-
E
fs -3) S -1-1 I +
E3i 4
11
E3e4 +
f So 3-41 IsSlT21 -1-1 1
E
1-
-
1T2---1
2T1--0
-
1
-
S -
1
-
-3
-0.296034
2
3
1
0.302724
1
2
1
2.872078
1
2
2.874117
max
1-
1P3--S
111
+
1 +
-
2P1S
-
1
2P1S
+
1
fs- 1-1) so 1-2 )S- 1-1 1 S-1 -11 SI+21 S-1 -1 1
+
215
216
Generating
10.
Table 10.8.
Families
(continuation) in
segment
C
I
j
L
subset
ill
2.087673
2
3
1+
IP3--S
-
3
1
2.987461
3
4
+
2Tl--O
+
E
3.028534
3
4
Ist
species
3.017033
4
5
1st
species
2.T44731
4
5
1+
M
2.987469
m ax
2.987461
3
4
+
2P 1S
2.087673
2
3
1+
lP3--S
2.874117
m ax
+
2PlS
-
1
2PlS
+
1
out
IS- 1-11 S021S-1 -11 12
ISO 3-41
-
E'34
Idi E'45 -
E
1
---
11
I S3()41 S-1-1 I U
S3+t4l S
-1-1
1 +
fSO 3
4:
-
_
,
1
S-1 -1 1
1
1-11,
_
3
111
-
f S-1 -1, Sj2, S-1 -1, S-1 -1 1 fS
1
-
,IS,2IS
f S-1 -1 SO-1 -21 7
1
S
-
11S
-1 -1;
fS-I-,IS121S-1 -1IS
1-1 1
+
1
2.872078
1
2
-
1
0.302724
1
2
-0.296034
2
3
S-1 -1 1 -
1 -1 1
111
+
fSO-3-4,S-1-11
1
-
lP3--S
-3
1-.. -
1
-0.564634
3
4
11
-0.668904
4
5
-0.696238
min
E
1-11 -0.672200
4
5
3
-0.554474
3
4
1
-0.350508
2
3
1
2.971249
2
3
2.971299
max
2.567360
3
-
2PlS
+
1
f S:T41 S-1 -1 +
E4',5 E4', +
I S( 4-51 )
1-
-
M
1
---
E
2Tl--O
1
-
1-
lP3--S
111
+
S--1-1 S 31 S-1- 1 I
+
2PlS
-
1
+
2PlS
+
1
4
1+
IP3--S 2Tl--O
-
IS-1-11SO2-3,S -1-11 fs-
+
SL0 fs23)S-1- 11 -1-11 ill
ISC-)4-51
1
E4e5 Idi
2.992994
4
5
+
3.017033
4
5
1st
species
3.011315
5
6
Ist
species
2.831130
5
6
1+
M
2.992996
max
2.992994
4
E,z6 E
---
1
-
3
+
E
11
IS,04, S-I -I I SS+, -1-1} 45 +
1
5
+
2PlS
1
217
10.3 Other Families
Table 10.8.
(continuation) in
C
I
j
L
subset
-3
2.567360
3
4
1+
lP3--S
2.971299
m ax
+
2PlS
-
2PlS
+1
lP3--S
-3
segment
IS04-5''5-1-11 IS
I-1,S2031S-1-1,S-1-11
out
111
-
U
S -1-11 Is_ 23 S-1 -1) -1 -1 *s+, -
so
Is_
+
1
2.971249
2
3
-
1
-0.350508
2
3
111
-0.554474
3
4
1
1-.. -0.672200
4
5
11
-0.735653
5
6
-0.752829
min
1
2-3,S-1-1,S-
IS-1-1, S 3, S-1-1 S-1-1 I +
fSO-4-5,S-1-11 -
1-
-
2PlS
+
1
-
E
IS +
Ez 6 ,
1-
M
---
I
--V'
E' 56
E
-0.737044
5
6
3
-0.668904
4
5
1
-0.564634
3
4
3
-0.296034
2
3
0.302724
1
2
2.872078
1
2
2.874117
max
ill
2.087673
2
1
2.987461
3
1-41 2.987469
max
2.744731
4
2.995530
+
fSo 5-61 is -1 -1 S;4) S-1 -1
-
1-
2Tl--O IP3--S
-
1
Ill
+
1
+
IS-1-1)
so
IS_ -1-1,
S -1-11
-
2PIS
-
1
3-4)S-1-11
-
1
+
fs-1-11 S-1-1
I
1-
IP3--S
+
111
S-,S-1-1,S-1-11 12 2PlS
-
1
+
2PIS
+
1
3
1+
IP3--S
-
3
4
+
2PlS
+
1
5
1+
lP3--S
-
3
5
6
+
2Tl--O
+
E
3.011315
5
6
1st
species
3.008062
6
7
1st
species
-
1-11
so-1-21S-1-1 1S -
1
fsS-1-1 S-1-1 -1-11S- 1-1 S+, 12 ,
Is_ -1-11S
-
1-11
-
so121S
-
I
-
1-1)S 1-11 -
-
so-3-4) so Is_ -1-1)
fs-1-
I
+ S31, So
-
0 fs-1-11S3U4, So
ill
IS'-)1-6}
1-11 1
Ee .56
Idi
4 E67
...
218
Generating
10.
Families
Generating family
Table 10.9.
1.
C
in
segment
Ide
1
E, -
E
S0 2-1 1 I S-0) 0 -
+00
I
J
L
subset
out
natural end
-..
species
3.149803
2
1
1st
-0.406767
2
1
0
1T2---1
11
-- r -0.399131
max
-1.000000
min
IS.0 0, so 2-1 1 ISO-1-I) SO-2-1 1
ISO
S
-2-1
+
f so-1
-
1
7
ISO
0 S41
S O SO 4 I
-
ISO-1-1, SO O,
-
1
1
3
1
+
2P1S
1
3
-0.487019
4
1
0
1P3--S
111
-0.399131
max
-1.000000
1
3P3++S
?
in
1
1111
Generating family
C
I
-00
fShl
2.945907
so-4-1, SO0U O I
Table 10.10.
IS.-J
1
S OI +
IS0001
max
1
0
segment
2.945910
I
-1.000000
J
m.
L
subset
1
1
3T1++0
max
-0.720283
1--, 00
out
natural end
--11 -0.399131
1
natural end
+
1
10.3 Other Families
Table 10.11. A
new
generating family,
in
segment
+
C
ill
115,0 0, SO-2-1 SO-1-1 } )
I
J
-1.000000
1
1
1-111, -0.399131
max
S _o so 2-1 So 1 -1 1
asymmetric
L
orbits.
subset
3P2+A
out
111
+
1--,
1
I
made of
219
11
-0.406767
2
1
0
lP2-A
2.970934
2
1
+
2P 1A
-0.406767
2
1
0
lP2-A
-0.399131
max
-1.000000
1
-
2
Mill so-1-11 ITT, S-1 -1 1
-
e
I
-2
ISO 2-11 S I so-2-1,
So
1 -1
1
SO0U 0, S -1-1
1
O
I
0
, -
I I
1
1
3P2+A
-
+
11
111
fSC) 2-1i S?-J -J, S0o O J tJ
-
-The
study
asymmetric periodic orbits has been comparatively neglected part because they are more difficult to determine numerically. However, with the algorithm of Sect. 10.1, generating families of asymmetric orbits are determined as easily as families of symmetric orbits, as shown by the example of Table 10.11. -Each branch subset of a total bifurcation of type 3 which contains only two branches in Table 8.19 and which has not yet appeared in one of the above families provides a starting point for a new family. -Similarly, for type 2, we can start from any one of the many branch subsets which contain only two branches in Table 8.18. Only one of these subsets has been encountered so far: 2T1--0- Here we have additional degrees of freedom, because a bifurcation of type 2 is characterized not only by its order n, but also by the values of I and J (and c), which can be arbitrarily chosen. Thus, the number of potential families quickly becomes enormous as the numerical values of n, 1, J increase, and in practice it will probably only be feasible to explore the simplest of these families. -For type 1, we can start from any branch subset which contains only two branches in the Tables 8.14 to 8.17. Only one of these subsets has been encountered so far: M 1. Here, for any given bifurcation the values of 1, J, L can be arbitrarily chosen. -We can also start from a partial bifurcation of type 1, 2, or 3. Here the freedom of choice is again tremendously expanded, because any sequence of
up to now,
in
---
of
arcs can
-Instead of
be added to the
bifurcating
arc
to make up the total orbit.
junction as the starting point of a generating family, we can use any fragment which has not yet appeared. In the tables of Chap. 9, these fragments are distinguished by the absence of a generating family name in the last column. In particular, one might explore
using
a
220
Generating
10.
Families
The other
-
Other
-
fragments of family Id,: only one has been used (Table 9.2). fragments of the families Ejj (Table 9.3): most have not been
used. The unused
-
The
-
fragments of the simple families ISI in Table 9.7. fragments of simple families which do not appear in (Table 9.7),
-
The
'
However, nothing prevents this segment of hyperbolic -
for
ISI*11, IS2*21i IS30111 IS+2-2}) ISO 2-2b IS* 3-21) IS1*21) segment Sh appears only once in family m, as a simple family ISh}.
instance:
arcs
from
being
*
*
as-
sociated with segments of elliptic arcs S,*,,, in composite families. Other families of periodic orbits in Hill's problem could be computed and used
the
as
starting point
of
generating families (see Sect. 5-4).
Comparison with Computed Families
120.4
We compare now the generating families found in Sect. 10.2 with numerically computed families of periodic orbits for y > 0. A good choice is the value ft
0.012155, corresponding to the Earth-Moon case. This value is 0 meaningful; yet enough to make a comparison with the limit y
fLEM
:--
small it
is
--
---
not
so
small that differences cannot be
of obvious
practical interest. A large
computed
for the Earth-Moon
This value of P is also collection of periodic orbits has been seen.
case by Broucke (1968). Here these orbits recomputed, and the families have been extended in a number of cases (families a, f, g, h, i, 1). Regularization was used, so that there was no difficulty in continuing the families through collisions with either of the
have been
primaries.
We will compare the characteristics in the (C,x) plane (or the plane in the case of third species segments). Each orbit is represented two
points, having
as
abscissa the Jacobi constant C and
as
by
ordinates the
positions of the
two perpendicular crossings of the x axis. This is essentially representation as the one used by Broucke (1968): Broucke's energy is equal to -C/2; thus, one of Broucke's energy diagrams is transformed into the corresponding figure in the present section by a counterclockwise 90'
the
same
rotation.
The characteristics of the families of
generating
orbits
are
computed
as
follows: First
species, first kind:
x=a
C
=
we
obtain the characteristics from
2c'Va- +
I
First
form,
and
(3.2): (10.1)
.
a -
(3.8)
species, second kind: the characteristics are obtained in parametric with 0 as parameter, from (3.16) and either (3.22) or (3.23). Extremal
values of 0
are 0 or 7r for a first species bifurcation; for a type 2 bifurcation, they are computed from I sin 0 1 11 I/a 1; for a type I bifurcation, they are computed from C with the help of (3.16). =
-
Comparison
10.4
-
Second
species: there
-In the limit y
are
possible
two
with
cases
for
Computed
a
Families
221
perpendicular crossing:
0, the perpendicular crossing becomes a junction between two arcs, i.e. a collision. In that case we have simply x = 1. Incidentally, the crossing is not perpendicular any more in the limit; there
is
--)-
angular point and the slope is undefined. perpendicular crossing lies in the midpoint of an arc. In that case the arc family is computed numerically, as in H6non (1968), by solving the implicit equation (A.17) for the old variables r and q. an
-The
-
Third species: the families are computed by integration of Hill's equations (5.28).
10.4.1
Fig.
Family
as
(1969),
in 116non
a
10.1a shows the characteristic of extension of
is an
numerically,
Fig.
family
a
for y
--
0.012155;
this
12 in Broucke 1968. The closed nature of the
figure family
is evident.
Fig.
10.1b shows the characteristic of the
in Sect. 10.2.1.
Accidents, corresponding
generating family
to the lines of Table
a,
10.2,
dots. Each part of the curve between two dots corresponds to one the name of the corresponding segment is indicated on the figure. as
as
derived
are
shown
fragment;
The segment Hill-a consists of orbits of the third
species and reduces 1 in Fig. 10.1b. In the next segment JS011, one point C 3, x 2 perpendicular crossing takes place at the midpoint of the S201 arc; this corresponds to the upper right part of the characteristic in Fig. 10.1b, extending from the point (3, 1) to the point (-0.406767,3.034227). The other crossing takes place at the collision with M2. It corresponds to the horizontal line 3 C 2! -0.406767, x I in Fig. 10.1b. to the
--
_-
=
At C ues on
-_
-0.406767
the segment
at C
=
ment
is
-0.399131,
we
reach
I S 0) So 2 on
N30,
bifurcation. The
a
angle, sponds now corresponds
which
At C
is
to the
to
a
plainly midpoint
collision
contin-
-
very short and not visible
an
generating family
and after passage through an extremum 1) S o 1, S 2-11 S0'01, down to C = -1. (The first frag-
visible
on
on
of the
(between
the
Fig.
S02-1
figure.)
The characteristic makes
10.1b. The upper crossing correarc, while the lower crossing still
the two
S000 arcs).
-1 there is another bifurcation and the
generating family conperpendicular crossing points, corresponding to the midpoints of the S02-1 and S-2-1 arcs. The corresponding parts of the characteristic are at the left of Fig. 10.1b. They finally meet at C -1.439479, where the family has a reflection. The two characteristics in Figs. 10.1a and 10.1b are very similar (note that the (C, x) frames are identical). One difference is that the angular points in Fig. 10.1b are replaced by rounded corners in Fig. 10.1a. The most conspicuous difference is the bulge at the lower right of Fig. 10.1a. This part of the characteristic corresponds to third species orbits (in a loosely extended sense; see Sect. 2.10), i.e. orbits which stay close tinues
on
=
fS-2- 1) So 2-1 1.
=
There
are now
two
222
10.
Generating
Families
3
b
3
iso2 11 Is00 O'S0-2-1' S0001
x
x
2
2
Is- 2- 1'S02-11 0
0
0
1 s00's -2-11SO01
z -2
0
2
x
-2
C
0
Is 211
2
0
C
d
C
ffI 1111 a Hill-a -
1.2
Hill-a
1
0
2.8
3
C
3.2
-2
0
4
2
r
Fig. 10.1. Comparison of numerically computed families and generating families. (a) characteristic of family a of periodic orbits in the Earth-Moon case (g gEM 0.012155); (b) characteristic of the generating family a; (c) enlargement of the dotted rectangle of Fig. a; (d) characteristic of the v-generating family Hill-a. -,::::
P
--:::
Comparison
10.4
to
(see
M2
the
beginning
of
Fig.
11 in Broucke
with
Computed Families
(1968)). Eqs. (5.26)
show that the dimensions of this bulge should be of order and of order p 1/3 0.23 in x. In the generating family of
corresponds
to the limit 1i
j2/3
and
223
(5.27)
0.05 in C
,-
Fig. 10.1b,
which
0, this bulge is reduced to a point. A better representation of that part of the family is obtained by using the v-generating orbits introduced in Chap. 5. Fig. 10.1c is an enlargement --+
of the dotted
rectangle of Fig. 10.1a. Fig. 10.1d shows the characteristic of family segment Hill-a, in the (IF, ) coordinates; this is similar to part of Fig. 5.1, with the difference that here both crossings are represented. The (r, ) frame has been adjusted so as to correspond to the (C, x) frame of Fig. 10.1c through the change of coordinates (5.26) and (5.27). The shape of the bulge is now satisfactorily reproduced. the
10.4.2
Fig.
Family
b
10.2a shows the characteristic of
in Broucke
(1968)).
This is
a
very
0.012155 (see Fig. family b for ft closed simple family.
0
=
15
0
x
x
-2
-2 -1
0
1
2
3
-1
0
2
1
C
Fig. 10.2. (a) family (b) generating family
Fig.
periodic
orbits in the Earth-Moon
case
(JU
=
0.012155);
b.
10.2b shows the characteristic of the
in Sect. 10.2.2
and
b of
3 C
(3.23)
with
(see
Table
1,
10.3).
so
--
-1, it is
1)2 + (X + 1)2
=
I
a
--
generating family b, as derived family E 1. From (3-16) deduced that the equation of the easily
It coincides with
characteristic is 1 4
(C
_
.
(10.2)
224
In
of
10.
Fig. Fig.
Generating
Families
10.2 the scale ratio has been taken
10.2b is
perfect
a
equal to 2,
so
circle. In contrast, the closed
that the characteristic curve
of
Fig.
10.2a
is
somewhat distorted circle.
a
10.4.3
Flamily
c
Family c is another simple closed family. It includes Hill-type orbits. Therefore comparison is done, as in the case of family a, both in (C, x) coordinates
the
(Figs. Fig.
10.3a and
10.3a is
10.3b)
and
in
Hill's coordinates
essentially identical with Fig. 9
(Figs.
10.3c and
10.3d).
in Broucke 1968.
x
0
n
0
0
0
2
2
C
C
Hill-c
x
0.8 Hill-c
d
C
2.8
3
Fig. 10.3. (a) family (b) generating family generating family
C
c
c;
Hill-c.
of
3.2
periodic orbits
(c) enlargement
-2
0
in the Earth-Moon of the dotted
2
case
rectangle
(P of
4
=
r,
0.012155); a; (d) V-
Fig.
Comparison
10.4
10.4.4
with
Computed
Families
225
Family f
is more complex (Fig. 10.4). C oscillates back and forth while x indefinitely. After an initial fragment of Hill-type orbits, the generating family consists of an alternating sequence of E and f S1 fragments (Fig. 10.4b). Fig. 10.4a is an extension of Fig. 31 in Broucke 1968.
Family f
increases
10.4.5
Family
Family species
orbits around C
g
complex (Fig. 10.5),
g is also
The Earth-Moon shown in
Fig.
--
family
3,
is in
in
in the region of third
particular
1, which is shown enlarged in Fig. 10.5c. good agreement with the v-generating families x
10.5d. There is
=
notable difference: in the latter
one
figure,
in
Hill-g' intersect in the critical orbit gl, while bifurcation. The Earth-Moon family passes twice
put forward by Broucke
(1968,
Hill-g
the two families
and
Fig. 10.5c, there is a through this bifurcation (see Table 10.6), first coming through the right Hillg branch and emerging through Hill-g+, and later coming back through the left Hill-g branch and emerging through Hill-g' Incidentally, the conjecture .
p.
71)
is verified: the two families H, and
family g presented Fig. 10.5c is identical with Fig. 35 in Broucke 1968. Only a few segment names could be written on Fig. 10.5b. The other segments can be identified with the help of Table 10.6. H2 which he computed
are
parts of the
same
family, namely
the
here.
10.4.6
Family
h
The evolution of family h
Fig.
10.6a is
10.4.7
an
Family
(Fig. 10.6)
extension of
Fig.
is similar to that of family
f (Fig. 10.4).
18 in Broucke 1968.
i
family i of periodic orbits for the Earth-Moon case (Fig. 10.7a, which nearly identical with Fig. 21 in Broucke 1968) initially follows closely the generating family (Fig. 10.7b), as in the case of the previous families. At the fifth accident, however, i.e. the bifurcation from f S-2 1 21 to E1e2 (Table 10. 8), the characteristics diverge. This is clearly seen at the top of the figures: the characteristic of the Earth-Moon family goes back towards the left, while the characteristic of the generating family moves towards the right. A more extended computation of the families (not shown here) confirms that the two families follow completely different paths from that point on. 0.012155 of the Earth-Moon case is sufficiently Thus, the value I-LEM different from 0 to produce large-scale changes in the family of periodic orbits. For smaller values of y, the agreement with the generating family should be better. Fig. 10.8 compares the family of periodic orbits for the Sun-Jupiter The is
-
=
226
10.
Generating
Families
6
6 b
E5+1
x
5
5
4
4
,3
3
2
2
Is 04-11
-
E+1
-
0
Is 2-11
-
E+
is--2-1 //T
n
n
-2
0
2
4
-2
0
2
4
C
C
1.2 x
Hii Hill-f I-f
1
0
Hill-f 0.8
3
C
3.2
Fig. 10.4. (a) family f of periodic orbits (b) generating family f; (c) enlargement generating family Hill-f.
0
in the Earth-Moon of the dotted
2
case
rectangle
4
(A of
=
r
0.012155); a; (d) v-
Fig.
10.4
Comparison with Computed Families
227
3
x
2
0
0
0
1.2
2
2
0
C
C
1
1
Hill-9' 0.5
Hill-g -
gI
Hill-g
0
gl -0.5
-
HiI11-g'I
C 1
0.8 2.8
3
3.2
C
Fig. 10.5. (a) family g of periodic orbits (b) generating family g; (c) enlargement generating family Hill-g.
Hill-g
-2
0
in the Earth-Moon
of the dotted
Hill14 4
2
case
rectangle
(IL of
1d 6
r,
8
0.012155); Fig. a; (d) V=
228
10.
Generating
Families
C1 x
x
0
0
-2
-2
-3
-3
-4
-4
-5
-5
-2
0
2
is
0 1-1
1
E%21 is 03 -1 1
E'41
4
-2
0
2
4
C
Fig. 10.6. (a) family (b) generating family
C
periodic orbits in the Earth-Moon
h of
case
(p
0.012155);
h.
C
E
0
is -1-21
12
b
el2
x
Ell 2 E
12
Idi
T-JS01-21 -AIE12 2
1
0
1
2
3
4
5
1
0
e
1
2
3
C
Fig. 10.7. (a) family i (b) generating family %'.
of
periodic
orbits in the Earth-Moon
5
4
C
case
(IL
=
0.012155);
10.4
Comparison
with
Computed
Families
229
with the generating family. Both families have been longer interval, and for better clarity each family is represented in two parts. Thus, the Sun-Jupiter family begins in Fig. 10.8al and continues in Fig. 10.8a2 (the arrows indicate the passage from the first to the second part). Similarly, the first part of the generating family, including the first four fragments and a part of the fifth fragment (which belongs to the segment E12), is represented in Fig. 10.8bl (which is identical with Fig. 10.7b); the second part of the generating family, including the remainder of the fifth fragment and the following fragments up to a part of the fifteenth fragment (which belongs to E23), is represented in Fig. 10.8b2. (The segments whose names are not indicated on Fig. 10.8b2 can be identified with the help of case
0.00095388)
computed
over a
e
10.8.)
Table
before
Sun-Jupiter family i was recomputed here. This family is of interest study of the asteroids, and parts of it were computed and published by other authors (Message 1966, Colombo et al. 1968, Deprit 1968,
Bruno
1993a).
The in
the
expected, the family of periodic orbits follows the generating family longer time in the Sun-Jupiter case. Also the agreement between the
As for
a
characteristics
is
closer than in the Earth-Moon case,
smaller value of y. However, the two families
accident,
company, this time at the fifteenth
again part
i.e. the bifurcation from
consequence of the
as a
ISO 2-31
to
E23 (Table 10.8). e
Fig. 10.9 (the scale generating family (Fig. 10.9b)
for better
visibility).
The
This
is seen
magnified
has been
in the continuation of the families in
describes
a
part of
joins the segment Idi of circular orbits for the third time. The of family periodic orbits (Fig. 10.9a), on the contrary, goes back toward the left. The shape of the orbits (not shown here) never comes close to circular. e and then E23
One may try to compute periodic orbits for y = 0.00095388 in the vicinity of the generating orbits Idi. Such nearly circular orbits are indeed found. However, when their family is followed, it is found that these orbits do not
belong
to
family i,
but to
a
(lower curve). (This family
small closed
family, which is shown
Colombo et al. 1968, Fig. 2, and family ID3J in Bruno It seems likely that, if y is further decreased, the orbits will follow the generating before
diverging
on a
on
has been found before: it is the Thule
different
family
course.
for
an ever
larger
There should exist
Fig. 10.9a Group in
1993a.) family
i of
a
sequence of critical
values of /-t, possibly infinite, at each of which the behaviour of the periodic orbits becomes qualitatively different.
10.4.8 Families I and
periodic
number of fragments,
family
of
m
Figs. 10.10 and 10.11 show the comparison for these families, which begin as large circular orbits (segments Id, and 1,). Fig. 10.10a is an extension of Fig. 25 in Broucke 1968. Fig. 10-11a is identical with Fig. 28 in Broucke 1968.
10.
230
Generating Families
a
1 12
b1
Ee
0
12
iS-1-21
'di E"
12
E'&
12
'di
1
1
0
1
2
3
4
5
1
0
1
-
2
21
_.)IEl?
2
3
4
5
C
C
x
0
-1
0
1
2
3
5
4
-1
0
1
2
3
10.8. (al, a2) family i of periodic 0.00095388); (bl, b2) generating family i.
Fig.
5
4
C
C
orbits in the
Sun-JuPiter
case
(p
10.4
Comparison with Computed Families
ISO-2-3)
x
x
0.5
0.5
E'23 Ee E' E
di
e Ez3 4 34
b
a
3
2.8
231
3
2.8
C
C
10.9. (a) upper curve: family i of periodic orbits in the Sun-Jupiter case (continuation); lower curve: a closed family (Thule Group); (b) generating family i (continuation).
Fig.
5
5
-
-
G
-
x
x
4
4
,3
3
2
2
Is.-1-1'S 0 -1"34IJ
W,_40
21-1 ,
b
is00 1-11 S0 2-11
E Ee2l e
Ide Is 0O'S -2-11 0 0
0
0
EEe::
-
E
-3 -1
0
1
2
4
3
21
0 so0 2 1 1 s -1-11S-2-11 JS 1 1,
-2
-2
e
0 SO0 -11 S 1 is so 41 1J'S41 -11
-1
0
1
2
of
periodic
4
3
C
C
Fig. 10.10. (a) family I (b) generating family 1.
'[de [de
-
orbits in the Earth-Moon
case
(IA
0.012155);
232
Generating
10.
Families
b
a
2
2 X
X
0
jS001
IS001
ishi 0
(SO01
0
ishl
is001 0
Ir
-2
-2
-3
-2
-3
0
-1
-1
-2
0 C
C
Fig. 10.11. (a) familyrn (b) generating family Tn.
periodic
of
orbits in the Earth-Moon
0.012155);
case
110.5 Final Comments
Figs. is
periodic orbits for p PEM generating families in most cases. A divergence family i, and then only after a common initial part
10.1 to 10.11 show that the families of
0.012155
are
observed
_-
close to the
only
for
(Fig. 10.7). Instead of
comparing only
the characteristics of the
families,
one
could
also compare the shape of individual orbits in the (x, y) plane. This is not done here because it would take too much space. But the results show again that the
The
periodic orbits for small y are very similar to study of generating families thus fulfills some
the Introduction. It accounts well for the existence and
periodic
the
generating
orbits.
of the aims stated
properties
in
of observed
o-rbits for small y. It can also be used to discover and describe periodic orbits (one example has been given in Sect. 10.3).
new
families of
Generating orbits provide more than a simple qualitative explanation; they are a good first approximation of the periodic orbits for small p, and in many cases they might reduce or eliminate the need for numerical computations. (It may be observed in this respect that the restricted problem already involves a number of approximations: the eccentricity of the primaries is neglected; the motion of the third body is assumed to be in the orbital plane of the primaries; the gravitational pull of other bodies is ignored.) Clearly, the agreement observed in the above figures for y PEM should =
hold for all values
in
the interval 0
<
U
<
UEM
-
It
seems
likely
that the
agreement will also persist for a while for larger values of y. As an extreme example, consider the oscillating structure exhibited by the generating family
10.5 Final Comments
233
(Fig. 10.6b). This structure was found to persist for the small value P (Fig. 10.6a). More surprisingly, it is apparently still present at P 0.5, i.e. the case of two identical primaries (H6non 1965a, Fig. 3). What h
--
0.012155
--
is
more, if
we
corresponding
continue to increase p, we reach the value p = I to the Earth-Moon problem with the two primaries
-
0.012155,
exchanged. The sign of x is then changed, and family h becomes family f. Fig. 10.4a shows that the oscillating structure is still present! Finally, going to the limit 1, we obtain Fig. 10.4b. We have here a case where the analysis of the p generating families, in principle applicable only to small values of P, in fact -4
explains the structure of a family of periodic orbits for all values of the mass ratio, i.e. in the whole interval 0:! p < 1. One hope which has not been realized is finding, via the generating families, a systematic classification of periodic orbits and their families. No governing principle, no particular order can be discerned in the structure of the generating families described in Tables 10.2 to 10.10. From a modern point of view, however, this is not too surprising: it has now been realized that any non-integrable system has an inexhaustible complexity of detail, and that there can be no hope of a complete description of its solutions, even if the scope is limited to periodic orbits.
Between Old and New
Correspondence
A.
Notations
correspondence
We establish here the
between the notations used for
elliptic
S-arcs and S-arc families in previous papers (H6non 1968; Hitzl and 116non 1977a, 1977b) and the notations of the present work.
A.1 Arcs
by the two quantities -r and 'q, and c", which can take the values +1 or -1 (116non 1968). 27- and 2 are respectively the duration of the arc and the variation of the eccentric anomaly E; these two numbers are positive by definition. It will be convenient to introduce two algebraic numbers r* and q*: 2-r* is the time of passage at the Q end of the arc, minus the time of passage at the P end; similarly, 271* is the value of E at the Q end, minus the value of E at the P end. Thus, notations,
In the old
-r* with
S-arc is described
a
three numbers c, E,
by
also
T
--
71*
,
signs for
plus
=
an
77
(A. 1)
,
ingoing S-arc and minus
signs
for
an
outgoing S-arc;
and T
-_
IT* I
77
,
=
177* 1
signs hold
In order to find which
(1968) r
so
in
(A. 1),
we
note that from
(26)
in 116non
the distance MiM3 is
--
a(l
-
c"e
cos
(A -3)
E)
that dr =
dE In
(A.2)
.
aE//e
particular, dr
dE
dE/dt
is
sin
at the
ac// e
(A.4)
E
sin
beginning 71
of the
arc
there is E
and
(A-5)
.
always positive;
therefore the
sign
of the radial
velocity
at the be-
if ginning of the arc is the sign of -c" sinq. It follows that the arc is ingoing c// sinq > 0, outgoing if 0 sin 77 < 0, and (A. 1) can be more precisely written
M. Hénon: LNPm 52, pp. 235 - 242, 1997 © Springer-Verlag Berlin Heidelberg 1997
T* 2T*
TE
=
//
sign(sin 77)
T*
7ra
=
At time t
77*=
,
//
qE
I
sign(cos 7) arccos
-
(A. 6)
.
-
V2 COS2
we
have
'Y
(A-7)
given by (4.15), with -7r < Eo < 0-1 0 to the next passage at pericenter
0, the value of E is E0
=
therefore the variation of E from t
=
E from t = 0 to the next passage in Q is into account the definition Of t4o and its interpretation
-Eo; and the variation of
is
-2EO. Taking
now
(see (4.23ff)),
we
-2Eo arc,
t
sign(sinq)
given by (4.20); using (4.22),
identical with t2
is
Between Old and New Notations
Correspondence
A.
236
27rH(I
-
-
find that the variation of E from t
a). Finally, (4.23)
M3 describes 0 full revolutions 40 to t 14 is 27ro. Therefore
0 to t
=
shows that from t40 to the the
on
ellipse,
Q
=
t40 is:
end of the
i.e. the variation of E from
=
-
2q*
-2EO
--
-
27rH(I
-
a)
+
27ro
V,
i
(A.8)
,
or
7ro
+
V I
With these formulas
(T-,,q,
the set and q
are
C
("
c', c").
can
--
V2(2
-
one can
-r* and
obtained from
sign(cos -y)
=
(A.6b)
c,
q*
(A.2).
V2) COS2 7
go from the set of
--
(A.6a) f
V)
c' is
(A.9)
parameters (V,
computed (A.7) given by (4.13b):
and
-y, c,,
(A.9);
0)
then
to 7
(A. 10)
.
be written
sign(SIn'q*)
(A. 11)
,
(-1)13sign(V
-
combined with
E/Slgn(Sin 7*)
-_
-
from
are
which becomes upon consideration of term lies between 0 and 7r,
f"
7rH(I
arccos
1)
(A.9),
and
using
the fact that the
arccos
(A. 12)
.
(27b)
in H6non
(1968) gives (A. 13)
,
which becomes upon consideration of
(A.7)
and
(A.10) (A. 14)
Conversely,
(29) can
in
(7,
if the set
H6non
(1968);
c,C") given, a co-eccentricity e' is
YI) c,
the
then be obtained from
(4.28);
-r* and
are
I IT
,F*
a
(4.19);
A and Z
obtained from
if 'E'
-
+1
and
e
can
be
computed
from
(3.9); V and -Y from (4.12b) and
obtained from
are
obtained
(A.6); (A.7) gives
,
(A. 15)
=
if 7r
is
A. 2 Arc Families
and
(A.9) gives,
11
+
q*/7r
i.e.
tions
is
as
easily verified,
(A. 16)
,
2
237
rounded to the nearest
Thus the old and
integer.
sets of nota-
new
entirely equivalent.
are
A.2 Arc Families The characteristics of the old shown in H6non
Fig.
and
are
in the
reproduced
-
[7](1
77
0
-
CEII
COS 7 COS
71)
-
sin
77(cos 77
IECII
-
(A, Z) plane.
COS T
The corresponhelp of
established with the
easily previous Section. Consider family Ao (H6non 1968, Table 2), for
first the
the relations derived in the arc
are
(A. 17)
4.12 shows the characteristics in the
elliptic
were
They
A.I.
.
dence between these characteristics is
of the
('r, 77) plane
Fig.
in
implicit equation
COS 7 COS
-TI sin q 13
Aj, Bj, Cij
families
(1968, Figure 4),
the solutions of the Cc"
arc
0 <
<
r
beginning
7r/2.
Then
1; and (A.6), (A.15), (A.16) give r], 7-, q* 0. On the other hand, A decreases from +oo to 1. Therefore this a 0, 0 part of the old arc family Ao corresponds to the entire new arc family Soo. This is represented in the (A, Z) plane in the lower Ao frame of Fig. A.2. The end point of Soo for A oo corresponds to a parabolic arc; there Soo joins with of hyperbolic arcs (H6non 1968). arc Sh smoothly family Consider next the part of Ao which corresponds to ir/2 < 7 < 27r. Then -1. -1; (A.6) gives r* r/2 < 71 < 7r, e" -r, y* -77; (A.16) gives # 0 < 77 <
7r/2,
c'
-1, 0
=
r*
=
=
=
=
=
---
=
=
=
=
-1 in both -r > 7r, so that (A.15) gives a hand, A increases from 1 to 2. Therefore this part of the old family Ao corresponds to the entire new arc family S-1-1. Proceeding in the same way, we find that the continuation of the Ao family corresponds to the following infinite sequence of new arc families: S-1-1, S31, i.e. an infinite sequence of arc families Sp defined by S-5-1, S71,
c' is -1 for cases.
and +1 for
7r
On the other
-
a
<
-r
=
--
-
-,
(71)'(2i
-
(i
1)
This sequence is shown in the upper
S-arc families
are
also indicated
which separate these families The other
arc
families
are
on
1,2
)
....
.
.
Ao frame of Fig. A.2. The
Fig. A.1,
marked
Aj, Bj, Cij
--
are
as
also
and the
arcs
of
names
(A. 18) of the
types 2, 3, 4,
dots.
represented
in the
(A, Z) plane,
D2, respectively. Each arc family Aj, lies of it consists two in branches, which are represented entirely Dj; j.> 0, branch for The first separately (lower frame in Fig. A.2; lower greater clarity. in
Figs.
A.2 and A.3 for domains D, and
branch in
Fig. A.1)
consists of
an
infinite sequence of
arc
families
S,,3,
with
A.
238
Correspondence Between Old
and New Notations
S
S-2-5
C25
5
S
5
-3-5
C,35
S25
S35
S-2-4
S-3-4
/7T
SS5 5-5 -5 5
-4-5
-
3 E5 5
C45 S
A4
45
77
4
C24
-
S
,3
C
-
S
S-
B2 A
12
S-
I
21
Sil
S
-2-1
S-1-1
Soo
Fig.
S-4-4
S-5-4
43
S53
S33
S
32
S-5-3
S-4-3
S22
S52
42
S-5-2
S-4-
S-2-2 S
1-1
A0
0
S 2-2
S44
12
Bi
1
3-3
A2 S-3-3
23
S-1-2 2
S-
B3
C23 S
A,3
34
S-2-3
S54
4-4
B4
C34
S24
-
S-
S-5-5
S31
S
S-3-1
S-4-1
S51
41
S-5-1
1
I
I
I
I
I
0
1
2
3
4
5
A.I. Characteristics of
1968).
The
names
arc
A., B, Ci,
families in the
-r/7.r
(-r, 77) plane (adapted
of the old families
are
in bold. Dots
from H6non, correspond to
arcs of types 2, 3, 4, p,and mark the ends of the new S-arc families. (Note: families C'12 and A, are separated by a gap. A similar gap exists generally between C,,,+,
and A, although increases.)
it
becomes smaller and
ceases
to be visible
on
the
figure
when 1
A. 2 Arc Families
239
0
1
2
3
4
1
5
0
2
3
4
5
2
3
4
5
1
2
0
3
4
5
3
4
5
1
2
3
4
5
4
5
A2
0
0
2
3
A,
A0
1
2
0
0
1
1
4
5
1
2
3
4
5
1
2
3
B, Fig.
A.2. Old
arc
4
5
1
2
3
D2 families Ai and Bi in the
4
5
3
-V"vV-
0
1
2
1
2
3
B3 (A, Z) plane (domain Di).
4
5
A.
240
Correspondence Between
-(-I)'(j
+ 2i
2)
-
and the second branch
Old and New Notations
(i
(upper
frame in
Fig. A.2;
=
1,
(A. 19)
2 ....
upper branch in
Fig. A.1)
consists of another infinite sequence with a
(-I)'(j + 2i
=
1)
-
3
,
=
(-I)i(j + 1)
(i
(A. 18) is a particular case of (A. 20) with j 0. The Aj is symmetrical of the first branch of family Aj+j =
1,
(A.20)
2 ....
second branch of
--
Z
--
family
with respect to the axis
0.
Each
family Bj consists of one finite central piece lying in D2, made arc family segment S:j-j; and two infinite branches lying in Di. The piece in D2 (Fig. A.3) belongs to case 9 (Fig. 4.10): it begins at 1 point (1, 0) in D2+, rises (upper frame in Fig. A.3), crosses 17 and enters D 2 then goes down and ends at (1, 0) in D2 (lower frame in Fig. A.3). The two infinite branches in D, correspond respectively (lower and upper frames in Fig. A.2; lower and upper branches in Fig. A.1) to sequences of arc families of the
S,p
arc
single
with
(-1)(j
+ 2i
(-I)ij
1)
-
(i
--
1,2
....
(A.21)
1,
(A.22)
and
-(-I)'(j
+ 2i
These two branches z
--
-
are
1)
'3
--
-(-I)ij
(i
--
2 ....
symmetrical of each other with respect
to the axis
0.
Finally,
each
family Cij
lies
entirely
in D2 and
corresponds simply to correspond respectively to cases 8 and 10 of Fig. 4.10. Each of them starts at the point (IIJ,O) in D+, rises then down crosses and ends at point (upper frame), F, 2 goes (J J' 0) in D2 (lower frame). For arc families C12 and C23, one of the points of intersection with IF seems to be at (1, 1) in Fig. A.3; in reality it is close to but distinct from that point, because the slope of IF tends to infinity when the (1, 1) point is approached (see Appendix B). The true situation is clearly the two
seen
arc
arc
families
Sij
and
S-i-j.
These
arc
families
for C24-
Thus,
the
tion of the
arc
representation in the (A, Z) plane provides a natural explanafamilies found earlier in the (7-, 77) plane (Fig. A. 1) and of their
properties. It can be verified that all arc families S,,p and A.3; therefore the new representation serves also to no
other
arc
appear in
Figs.
prove that there
A.2 are
families than those described in the 1968 paper.
A.3 Critical Arcs Values of -r, 77, c, H6non
c',
(1977a) (c, c',
c" for critical c"
are
arcs
have been tabulated
called o-o, 0-1,
0-2
in that
by Hitzl
paper).
and
From these
A.3 Critical Arcs
D
+
2
0
0 0
0 0
0
D2
0
0 0
0 0
B,
D
B2
B3
+
2
0
0
0
D
0
0
0
0
2
0
0 0
1
C
Fig.
A-3. Old
1
0
C23
12
arc
0
0
families Bi and
Cij
in the
(A, Z) plane (domain D2).
C24
241
A.
242
Correspondence Between
Old and New Notations
values, as explained in Sect. A.1, one can easily derive the other quantities interest, and in particular ce, 0, A, Z, C shown in Table 4.4.
of
The
arcs
Aj(0) (i
f -:
0),
in the notation of HitzI and H6non
(1977a), belong
type 3. Similarly, the arcs Bj(0) (i ! 1) belong to type 4. These arcs are not critical arcs according to our present definition (Sect. 4.6), which requires a to
critical our
arc
present
to be of arc
namely Ai (1) type I and
with
are
type 1. They correspond, in fact,
families. All other
10 0, Bi (1)
critical
arcs.
arcs
with
to end
points of some of
in Table I of HitzI and H6non
10 0,
and
Cij (1)
with I
--
(1977a),
1, 2, belong
to
The Domain D2
B.1 The Curved
study here
We
Boundary
the upper
to the two sheets
common
computing aZla7 equation results: V2 COS3 -y
+
from
V(2
_
boundary D+ and 2
(4.28),
F IF of the domain
(see
D2
and then
V2) COS2,y
_
COSy
D2; this boundary is It is obtained by
4.3.2). writing aZla-y Sect.
=
_
V
=
0. The
following
(B. 1)
0
was first published by Bruno (1973; 1994, Equ. (IV.16)+), with equivalent variables a, e, E'. Bruno also tabulated its solutions (ibid., Table II). If we introduce the quantity
This relation
the
q
(B.2)
Vcos7
--
(which, incidentally, be solved in
can
V2
q(q
=
2
is the
parametric
+
q2
2q
-
angular form
momentum of
(ibid., (2.32+)):
1)
q
(B -3)
V
+ I
(The quantity
Bruno is the opposite of
here
of type 1, V and 7 must
E used by supporting ellipses
V
2
:,
0
I
,
This is satisfied -1 < q < 0
Cos
M3 in fixed axes), (B. 1)
-YI
< I
q.)
Since
we are
considering
satisfy (Sect. 4.2):
(B.4)
.
only in the interval
(B.5)
-
It follows that c'
-1 on the F boundary. Table BA lists values of V, 7, a, along the IF boundary, computed from (B.3), (4.12), (4.13), (4.6), (4.10), (4.28). Limiting values for q -1 and q 0 are also given, although the corresponding points do not strictly belong to IF. The shape of r near its ends is of interest. Expanding the equations, one e,
C, A,
--
Z
--
finds for q A
-
-+
--
0:
2-3/2
3q 27/2
Z
and therefore
M. Hénon: LNPm 52, pp. 243 - 249, 1997 © Springer-Verlag Berlin Heidelberg 1997
-
7r-lv/-
q
,
(B.6)
244
B. The Domain D2
Table B.I. Parameters v
'y
-1.000
1.000000
-0.999
0.999999
-0.998
0.999998
-0.997
0.999995
-0.996
along
the 17
boundary.
a
e
c
3.141593
1.000000
0.000000
-1-000000
1.000000
1.000000
3.096879
0.999999
0.044699
-0.997999
0.999998
0.999980
3.078368
0.999996
0.063182
-0.995996
0.999994
0.999943
3.064172
0.999991
0.077344
-0.993991
0.999986
0.999894
0.999992
3.052209
0.999984
0.089264
-0.991984
0.999976
0.999835
-0.995
0.999987
3.041676
0.999975
0.099751
-0.989975
0.999962
0.999768
-0.994
0.999982
3.032157
0.999964
0.109217
-0.987964
0.999946
0.999692
-0.993
0.999975
3.023409
0.999951
0.117909
-0.985951
0.999926
0.999608
q
A
z
-0.992
0.999968
3.015270
0.999936
0.125987
-0.983936
0.999904
0.999518
-0.991
0.999959
3.007629
0.999919
0.133563
-0.981919
0.999878
0.999420
-0.990
0.999950
3.000406
0.999900
0.140718
-0.979900
0.999849
0.999316
-0.980
0.999798
2.942255
0.999596
0.198020
-0.959596
0.999394
0.997954
-0.970
0.999543
2.897858
0.999087
0.241331
-0.939087
0.998631
0.996072
-0.960
0.999184
2.860613
0.998371
0.277301
-0.918368
0.997557
0.993722
-0.950
0.998718
2.827961
0.997444
0.308525
-0.897438
0.996169
0.990931
-0.900
0.994738
2.701615
0.989612
0.426025
-0.789503
0.984458
0.970821
-0.850
0.987843
2.606969
0.976405
0.509942
-0.675835
0.964817
0.941123
-0.800
0.977802
2.529002
0.957944
0.576110
-0.556098
0.937584
0.902610
-0.750
0.964365
2.461817
0.934579
0.630971
-0.430000
0.903492
0.856239
-0.700
0.947267
2.402333
0.906878
0.678000
-0.297315
0.863621
0.803234
-0.650
0.926234
2.348671
0.875587
0.719351
-0.157909
0.819311
0.745046
-0.600
0.900980
2.299552
0.841584
0.756462
-0.011765
0.772052
0.683257
-0.550
0.871218
2.254025
0.805815
0.790319
0.140979
0.723358
0.619464
-0.500
0.836660
2.211319
0.769231
0.821584
0.300000
0.674660
0.555171 0.491695
-0.450
0.797019
2.170750
0.732729
0.850668
0.464761
0.627213
-0.400
0.752009
2.131649
0.697115
0.877771
0.634483
0.582046
0.430111
-0.350
0.701335
2.093296
0.663073
0.902914
0.808129
0.539936
0.371215
-0.300
0.644668
2.054833
0.631152
0.925961
0.984404
0,501420
0.315520
-0.250
0.581580
2.015138
0.601770
0.946647
1.161765
0.466816
0.263253
-0.200
0.511408
1.972598
0.575221
0.964604
1.338462
0.436267
0.214347
-0.150
0.432907
1.924628
0.551696
0.979396
1.512592
0.409779
0.168375
-0.100
0.343252
1.866415
0.531299
0.990544
1.682178
0.387266
0.124286
-0.050
0.233962
1.786168
0.514070
0.997565
1.845262
0.368581
0.079206
-0.040
0.207526
1.764757
0.511004
0.998433
1.876933
0.365289
0.069362 0.058805
-0.030
0.178170
1.739981
0.508064
0.999114
1.908256
0.362141
-0.020
0.144165
1.709975
0.505250
0.999604
1.939216
0.359137
0.046999
-0.010
0.100985
1.669983
0.502563
0.999901
1.969802
0.356275
0.032526
-0-009
0.095711
1.664969
0.502301
0.999919
1.972839
0.355996
0.030791
-0.008
0.090150
1.659655
0.502040
0.999936
1.975873
0.355719
0.028967
-0.007
0.084246
1.653983
0.501781
0.999951
1.978903
0.355444
0.027038
-0.006
0.077920
1.647874
0.501523
0.999964
1.981928
0.355169
0.024978
-0.005
0.071062
1.641216
0.501266
0.999975
1.984950
0.354897
0.022753
-0.004
0.063497
1.633833
0.501010
0.999984
1.987968
0.354625
0.020307
-0.003
0.054936
1.625433
0.500756
0.999991
1.990982
0.354355
0.017548
-0.002
0.044811
1.615444
0.500503
0.999996
1.993992
0.354087
0.014297
-0.001
0.031654
1.602393
0.500251
0.999999
1.996998
0.353819
0.010088
0.000
0.000000
1.570796
0.500000
1.000000
2.000000
0.353553
0.000000
B.2 S-arc Families in Domain D2
Z
--
2 7/4 3- 1/2 7r
Similarly, I
-
A
for q 3 -
2
-
1VA
245
2-3/2
-
(B -7)
-1,
-+
(1+ q)2
I
Z
-
25/2 3-17r-1 (I
--
+
q) 3/2
(B.8)
and therefore 1
-
Z
--
213/43 -7/47r- 1 (1
A)3/4
-
(B.9)
Thus the slope of r is infinite at both ends.
B.2 S-arc Families in Domain
D2
We prove here the assertions made in Sect. 4.3-3.2. (i) 0 = 0: the characteristic A is then a horizontal line in the
with
an
ordinate
this domain lies no
entirely
an
A has
< 0 a
(we
/0
>
a/0.
=
1, then for A
--
I
No such line =
0 and Z
before
case
case
(ii)
=
1. Therefore
because it is
and intersects the horizontal axis Z Therefore it does not intersect D2 if we
(A, Z) plane,
intersect D2 since
can
case.
consider this
negative slope,
with abscissa A
integer.
between the two lines Z
S-arc families exist in this
(iii)
a
which is
a
have Z
=
0 -a
>
0,
or, since
a
=
a/#
# 2-3/2
and
simpler):
0 at
a
point
< 2 -3/2.
are
If
integers:
< a / 0 :! I, 1, and again A does not intersect D2- Conversely, if then Fig. 4.9 shows that A intersects D2, and the intersection consists of a single segment of straight line. (ii) 0 > 0. A has now a positive slope. Figure 4.9 shows immediately
Z
that A does not intersect D2 if
a/0
f 1. We must show that there is
no
2-3/2 This requires a little work; essentially we have to show that a situation such as is represented by Fig. B.1 cannot arise, in spite of the fact that the curve r begins with an infinite slope at the point (12 -3/1,0). We consider the expansion of 2 -3/2 as a continued fraction:
a/)3
intersection either if
2-3/2
<
.
I
(B. 10)
-
2+
1
4+ 1 +
The convergents
fj with
Aj Bj
are
(
.
4+...
(Abramowitz
>
0)
and
Stegun 1965,
p.
19)
B. The Dornain D2
246
Ao
Al
0,
A2i
A2i-1
A2i+l
=
+
4A2i
=
BO
1
A2i-2
+
=
B2i
B,
I
B2i-1
=
B2i+l
A2i- I
+
4B2i
=
2,
B2i-2 +
(i
B2i-1
=
1,2....
(B. 12) There is
A f2i
and
f2
<
<
<
f2i
2-3/2, f2j+1
--+
<
...
<
2-3/2
--+
2 -3/2 for i
...
f2i+1
<
...
<
f.3
<
h
(B. 13)
oo.
z
r
A
Fig.
B.1. A forbidden situation.
We show
now
Proposition there Z'S a -
an
<
If 0 <
a
and
a
::
<
2-3/2,3, (B.13)
f2i-2
a
and
such that
> 0 and
<
0
(B. 14)
B2i
shows that there exists
an
<
132i-2a,
+ I :!!
i
=
0. If
i > 0 such that
OCI =< f2i
AN-O integers, A2i-20
-3/2 < 2
a1g
0, the proposition is immediately verified by taking
Then are
given
B.2.1. For
! 0 such that
i
f2i
13
that
(B. 15) or,
B2i-2a
.
using
the fact that all numbers in this
inequality
(B. 16)
We have also
13
<2 -3/2
<
A2i-1
B2i-1
(B. 17)
B.2 S-arc Families in Domain D2
from which
derive in the
we
B2i-10 :! A2i-10
Combining (B.16)
I
-
same
way
(B. 18)
-
(B.18),
and
we
have
B2i-1 + A2i-2B2i-O :!! A2i-iB2i-20
A2i-lB2i-2
B2i-2
+
B2i-1
(B. 19)
that
(B.19)
B2i-2
-
we
have
(B.20)
I
(B.12)
and from
so
A2i-2B2i-1
-
-
of continued fractions
general properties
But from the
0 2! BN
(B. 2 1)
BN
=
reduces to
(B.22)
Q.E.D.
i
OA A, of equation: Z part A+ of A which lies in the half-plane Z > 0. It corresponds A such that A > o/p. For points on A+, we have then We consider
the characteristic
now
Z=OA-a=O(A-O') Thus, A+
which
we
Aj, for i
B2i
A2i
-
-
A-
0, 1, 2,
=
2-
v
(B.23)
straight
line
(B.24) intersects D2
.,
..
5/2(q-1
alues of
C,
call Ai. Therefore it will be sufficient to prove that --
a, and the
to
A2i
the
or on
-
i
q)
-
B2i
,
none
of the lines
-
A2j and B2i
Direct formulas for
AN
above
=
!B2j
B2jA
-
always
is
B2jA
=
A2i
A
B2j
Z
247
are
--
easily deduced
2-1(q-1
+
q)
from
(B.12): (B.25)
,
with V"2-
q
(B.24) Z
I
-
(B.26)
)2i
becomes =
2-1(q-l
Consider the
+
q)A
family
of
-
2- 5/2
(q-1
-
q)
straight lines obtained by giving to q family includes all the Ai lines. by deriving (B.27) with respect to q:
the interval 0 < q :! 1. This
the
family 0
and
is
A
=
is obtained
2-1(1
-
given in
=
2 -3/2
q
-2
)A
+ 2-
parametric + q
q2
5/2(l + q-2)
form
by
Z
2- 1/2
=
all real values in The
env
elope of
(B.28)
q
1
(B.27)
.
q2
(B.29)
B. The Domain D2
248
By elimination of
Z2
A
=
q
one
obtains
1
2
(B-30)
_
8
*
The interval 0 < q :! - I corresponds to the branch A > 0, Z > 0 of the hyperbola (B.30). It is represented as curve A on Fig. B.2 (dotted line) along
region and the straight lines AO, A13 A21 A3 17A 6, 3A 1, Z A, Z respectively: Z (the therefore A has and all Z The curvature curve a negative Ai lines 99A-35). lie above it. A and IF intersect for A -- 0.8; below that value of A, however, A remains constantly above F This is shown by numerical computations, and also by the fact that for A 2- 3/2 A is approximated by boundary
with the
r' of the D2
equations of these lines
are
--
=
-
-
-
=
,
Z
--
while F Z
2- 1/4 is
--
VA
2-3/2
0.841v/A
-
2-3/2
(B .3 1)
,
approximated by (B.7), which gives
0.61WA
It follows that
-
none
2 3/2
(13 .3 2)
of the Ai intersects D2-
proved that there is no intersection for a/0 2: 1 or a/0 < 2-3/2. Conversely, if 2-3/2 < C, /0 < 1, Fig. 4.9 shows that A intersects D2 It remains to show that the intersection consists of a simple segment of straight line. This is not entirely obvious because the boundary IF of D2 is Concave between A -- 0.8 and A I (see Fig. 4.13); so conceivably we could have the situation sketched on Fig. B.3, where the intersection of A with D2 consists of two disconnected segments. In fact, however, this cannot happen because We have thus
-
the intersection of A with the vertical line A Z in the range 0 < Z < 1; and this is Z -- 3 a, an integer value. -
=
I would then have
impossible
since for A
an
=
I
ordinate we
have
B.2 S-arc Families in Domain D2
A3
A2
Al
A
249
0
0.5
0
-
0
Fig.
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0.5
B.2. Relation of the IF and A
curves.
z
0 0
A
Fig. B.3. actly.)
Another forbidden situation.
(The boundary
r is not
represented
ex-
C. Number of Branches
We derive here formulas for the number of branches in given
type and order. This will allow
forgotten
to
us
verify
that
no
a
bifurcation of
the lists established
in
the number of branches grows
a
branches have been
by hand in Sect. 6.2. It will also asymptotically for large n.
show how
CA Partial Bifurcation We begin with this simpler. Also, some case
case
which is
easier, because the formation rules
are
of the formulas established here will be needed for the
of total bifurcation.
C. 1. 1
Typ e
1
The formation rules
are given in Sect. 6.2.2.1. For a given type and for a order let BS, be the set of branches which end with a S arc (an odd given n, and BT the set of branches which end with a T arc (a symbol 2). symbol),
We define the sizes of these sets of branches is
b,
=
bs
-
n
JBS I, n
bT n
JBT 1.
=
n
The total number
bT n
+
n
bs
as
recursively. We consider first the set Bs. A branch 1, or with 3 or more. In the first case, if n 2! 3, the branch of BS-1 or BT the addition of an arc 1. I by
We build the branches of
B.S
ends either with
branch derives from
a
n
a
n
n-
T or Bn-1 can be extended in this Conversely, any branch of BS n-1 way. In the second case, if n 2! 4, the branch derives from a branch of B,,S-2 by adding 2 to the length of the last arc. We obtain
bs n
=
bs-1 n
+ b T_ 1+ n
bs n-2
n
)
=>=
4
(C. 1)
.
We consider next the set BT. A branch n
branch of BS and n-2,
bT- bs n-2 n
we
n
)
-> 4
a
2 derives from
a
(C.2)
Successive values of bs and b n n
starting from the values for bs 2, bT2, bs 6, bT 2 2 3 3 -
with
.
T
=
ending
have
n
-
-
-
-
=
can
2 and
n
be =
computed from (C.1) and (C.2),
3 which
2. Values up to
M. Hénon: LNPm 52, pp. 251 - 264, 1997 © Springer-Verlag Berlin Heidelberg 1997
n
=
can
20
be read from Table 6.8:
are
given
in Table C.1.
C. Number of Branches
252
partial
Table C.1. Number of branches for
bs
n
n
bn
T
b, 4
2
2
2
3
6
2
8
4
10
2
12
5
18
6
24
6
34
10
44
7
62
18
80
8
114
34
148
9
210
62
272
10
386
114
500
11
710
210
920
12
1306
386
1692
13
2402
710
3112
14
4418
1306
5724
15
8126
2402
10528
16
14946
4418
19364
17
27490
8126
35616
18
50562
14946
65508
19
92998
27490
120488
20
171050
50562
221612
The asymptotic behaviour for form
a
bs
set of linear =
n
with
_
bT
cSAn
cS, J,
A3
recurrence
A2
A
_
I
0
=
1.839286755.
=
A21 A3 The
=
general
n
-
and
..
(C.2) (C.3)
0.
Substituting
and
eliminating,
we
obtain
(CA)
equation
are
,
-0.419643378
0.606290729
...
...
(C.5)
i.
solution has the form 3
1: Cjq A '
bT n
j=1
where the
(C.1)
.
3
b'5
also be derived.
relations. We look for solutions of the form
A constants, and A _
n can
CTAn
-
n
The three roots of this
A,
large
bifurcations of type 1.
=
E CjTy
(C-6)
j=1
9 and J
are
constants. The total number of branches is
3
b,,
E cj Ajn
(C-7)
j=1
with cj
9
+
values of b,, for For
large
cT (j 3 n
=
--
1, 2, 3). The
constants cj
can
be obtained from the
2, 3, 4.
n, the last two terms of
(C.7)
tend to 0 and
we
have
CA Partial Bifurcation
b,,
cIA-1
=
Numerical
(C 8)
o(l)
+
-
computation gives:
Type
C.1.2
ci
1.128767221...
=
collisions in the
b+ n
2n-I
=
bifurcating
Next
sign.
+
a
There
adding
consider the branches with
a new arc
-T
S
b,-,
by extending by by adding a new arc
1
a
-
n
-
A branch of B
from
n
-T -
2b-S, n-
b,T
=
Bn-11,
is
sign.
We define
branch of
a
Bn-S,
B-51, n-
etc.,
as
either
by
of the last arc;
length
or
-T
n
2. Values up to
n
=
n
=
12
2 2
(C 9) -
a
arc
i
or e.
a new arc i or e-
Thus
(C-10) be can
given
computed from (C.9) and (C.10), S be read from Table 6.9: bi 1,
in Table C.2.
partial bifurcations
b+
b-s 1
2
3
4
2
2
4
4
8
10 28
of type 2.
bn
bn
3
4
12
12
24
4
8
36
36
72
80
5
16
108
108
216
232
6
32
324
324
648
680
7
64
972
972
1944
2008
8
128
2916
2916
5832
5960
9
256
8748
8748
17496
17752
10
512
26244
26244
52488
53000
11
1024
78732
78732
157464
158488
12
2048
236196
236196
472392
474440
The asymptotic behaviour is
by adding
.
n
n
n-
different
I which
Table C.2. Number of branches for
b-T
B-S1,
bnTcan
are
from
1. Thus
branch of
a
by adding
+ b n-
from the values for
n
corresponds
2.
n
Successive values of b,-, S and
starting
I intermediate
-
derives from
branch of
a
bn
-T
I the
or
-T 2bn--Sl + b1
=
n
or
-
(Table C.2).
we
branch of Bn-I ,
a
n
Therefore the number of branches
arcs.
derives from Sect. C.I.I. A branch of B-S n
in
are
Each subset of these collisions
arc.
possible decomposition into
one
.
2
We consider first the branches with
to
253
easily obtained. We look for solutions
of the
form
bn
S
=
We obtain
C-SAn
bn
T =
C-T An
(C.11)
C. Number of Branches
254
A
The
3
-
0
=
solution has the form
general
bnS
-
=
(C. 12)
-
c
SP
bn
C-T 3'
T -
From the numerical values for small
-
8
x
3
(Table 6.9)
n
number of branches with
Thus, the total
bn
(C-13) a
sign
-
C-T
find C S
we
4/9.
=
is
n-2
(C. 14)
and the total number of branches is
bn
8
--
large
For
3
x
+2
n-1
(C 15)
2! 2
n
-
n, the first term dominates.
Type
C.1.3
n-2
3
We consider first the branches with
a
-
consider the intermediate collisions of subset of these collisions
corresponds
Therefore the number of branches Next
+T B n-1
or
n-
branch of B+S n
b+s
=
n
a
an
B+S2
branch of
n-
or
2b+s n-1
from
b+T
=
n
+T
+ 2b n-1 +
2b+s n-2
a
+T branch of Bn-2
2b+s n-2
+T + b n-2
in that
I
B+S, n
1'. A branch of
or
by
We
case.
Each
arc.
into
arcs.
(Table C-3). We define
V derives from
or
+T B n-2
a
b+S 2
=
+T 5, b 2
=
n
)
a
a
as in
branch of
B s ending
the addition of from
etc.,
an arc
branch of
2! 4
2. A
B+S n-2
(C 16)
.
-
-
branch of
by adding
,
n
)
a
B+S n-2) by adding
different
arc
i
or e.
a new
n
2! 4
=
2, b+S 3
arc
i
Thus
(C 17) -
-
from the values for
ble 6.10:
sign.
+
arc
+T + b n-2
+T Successive values of b+S and b n n
starting
-
ending with 3, 3', 4, or more, derives length of the last arc. We obtain
n
e;
2n /2-1
a
even
bifurcating possible decomposition
with I
or
must be
2 to the
A branch of B +T derives from or
n
rank in the
one
bn
Bts ending
by the addition of
with 2 derives from
by adding
to
consider the branches with
we
Sect. C.1.1. A branch of
B+Sj
Is
sign.
even
can
2 and =
be n
computed from (C-16) =
+T 18, b 3
3 which -
can
and
(C.17),
be read from Ta-
4. Values up to
n
--
12
are
given in Table C.3.
The asymptotic behaviour for
solutions of
b+S
=
n
(C. 16)
c+s ,n
and
(C. 17) b +T n
-
large
n
can
also be derived. We look for
of the form
C+T M
n
(C 18) -
We obtain 3
-2_/,2
(C. 19)
-3/-t-2--O.
The three roots of this equation
are
C.2 Total Bifurcation
Table C.3. Number of branches for
b+5
bn
n
+T bn
n
partial bifurcations of type
b+
2
1
5
2
7
8
0
18
4
22
22
4
2
56
12
68
70
5
0
176
40
216
216
6
4
556
124
630
684
7
0
1752
392
2144
2144
8
8
5524
1236
6760
6768
9
0
17416
3896
21312
21312
10
16
54908
12284
67192
67208
11
0
173112
38728
211840
211840
12
32
545780
122100
667880
667912
--
3.152757602.
[12, P3
The
:--
..
,
-0.576378801
general solution
0.549684246
...
(C.20)
i.
3
E Ci+S YJ
b+T
n
--
n
...
has the form
3
b+S
3.
b,,
n
3
p,
255
I
n
j-1
E Cj+T Pj
n
(C.21)
j=1
The total number of branches with
a
+
sign
is
3
EC
b+ n
+
n
(C.22)
I-Ij
j=1
with
t
Ci
cts
=
+
3
tT
For
bn+
large =
n
--
=
1, 2,3). The
+
0
(1)
constants
Ci
can
be obtain ed from
2, 3, 4.
n, the last two terms of
c+pn 1 I
Numerical
(j
Ci
the values of bn for
(C.22)
tend to 0 and
we
have
(C.23)
-
computation gives:
c,+
=
0.692486803...
The total number of branches is
bn For
-::::
large
C+/,n 1 1
/2-1 + 2n
+0(1)
(C.24)
n, the first term dominates.
C.2 Total Bifurcation We will consider
only
second
species branches. We ignore the first species 2 (Table 6.4); particular cases: type 1, n I (Table 6.5); type 3, n I and n 2 (Table 6.7). type 2, n As specified by the formation rule No. 4 (Sects. 6.2.1.2 to 6.2.1.4), the branches which
occur
in
a
few
=
sequence of
--
--
arcs
must be minimal.
(Note:
--
a
minimal sequence is also called
a
C. Number of Branches
256
primdive
sequence
by
1958, Gilbert and Riordan 1961, in the
Fine
context of
problem periodic sequences of symbols.) However, it turns out to be easiest to ignore that rule at first. In other words, we count also non-minimal sequences. The number of rooted branches thus obtained another enumeration
for
will be called d,,.
branches, excluding now the appeared as a n. Conversely, any minimal sequence for n' dividing n can be repeated so as to give a nonminimal sequence for n. Thus, e,, can be computed by subtracting from dn the value of en, for every proper divisor n' of n (that is, every divisor smaller than n itself). (For type 1, only even divisors should be considered.) Finally, we compute the number f,, of free branches by Next
we
obtain the true number e,, of rooted
non-minimal sequences. A non-minimal sequence has already minimal sequence for a lower order n' < n, with n' a divisor of
A
do
for type 1, for types 2 and 3.
(C.25)
I
need to go back to the case of partial bifurcation (Sect. C.1.1) and detailed computations. We call the set of branches which
we
Bnss
some more
begin a
(
Type
C.2.1
First
2en/n e,, /n
-
-
T
and end with
arc
a
S
arc.
and end with
a
S
etc., the sizes of these
TS
We call
arc.
Bn Similarly
sets. Note that
the set of branches which
begin
with
define BST and BTT. We call n n
bss,
we
bss + bTS
bs and bST
-
-
n
n
n
+
TT
bn
where bs and bT have been defined in Sect. C. L L Also, for symmetry n
n
b TS n
bnss
n
T
bn,
reasons,
ST
bn Proceeding =
-
=
as in
bssl
STJ
+ b n-
n-
C.1.1,
Sect. +
obtain for
we
4
n
bss n-2
S b s=b TS 1+b TT 1+ bn_2 n
bST n
TT
bn
n-
=
=
n-
bss n-2
3
TS
bn-2
(C.26)
-
Successive values of
bss, bTS, bST, bTT n
n
n
tions, starting from the values for
n
n
can
2 and
n
be =
computed by
3 which
can
these rela-
be read from
Table 6.8:
bss 2
=
2
bss 3
=
4
TS
,
b2
,
JS 3
=
-
-
0
bST 2
0
bTT 2
=
2,
2,
bST 3
2
bTT 3
=
0
(C.27)
Values up to n -_ 20 are given in Table CA. The asymptotic behaviour be derived as in Sect. C.1.1; we find that bss, etc., are of the form n 3
bss
3
qs
cj
n
j=1
V'
-
j
TS
bn
E cjTsM 3
j=1
can
C.2 Total Bifurcation
3
3
ST
bn
cj
T
TT
Aj'3
cjTTA?3
bn
j=1
where
We consider
now
the total bifurcation. The
d2
=
n
2 will be treated
=
two sequences
11
the number
:
2.
n
origin of the
total bifurcation branch
the orbit at that
point,
obtain
at the
is
junction of two
arcs.
By cutting
linear sequence of arcs, which represents partial bifurcation branch. Conversely, out of a partial bifurcation branch
we
can
build
branches Next
is inside
belonging bss
TT to B n
consider the
T
arc.
.
Thus,
case
which
consider the
we
tively the
an
even
which
an
odd and
belongs
S
arc.
even
belongs TS is bn
to B
where the
case
It separates the S
number of basic
TS
to
Thus,
.
n
origin arc
arcs.
ends, except for
the number of total
of the total bifurcation
in two parts,
We subtract
to make it odd. We have then
part,
the two
where the origin of the total bifurcation branch at the origin of this T arc, we obtain a
bifurcation branches of this kind branch is inside
by joining
the number of total bifurcation branches
By cutting the orbit
partial bifurcation branch,
Finally,
a
TS ST + bn + bn
n
we a
we
total bifurcation branch
a
of this kind
of
are
case
! 4, we build the branches for the total bifurcation of order n (n out of the branches for the partial bifurcation. Suppose first that the
For
a
by (C-5).
given
are
Table 6.4 shows that there
of branches is
even)
(C. 2 8)
j=1
Al, A21 A3
separately.
257
a
one
having basic
respec-
arc
from
partial bifurcation branch,
BSsl. Conversely, we may add one basic arc to either end B.SS1 and join the two ends, thus obtaining
partial bifurcation branch of
a
two different total bifurcation branches
The
only
case
consists of
a
where this is not
single symbol
possible
cation branches of this kind is
-
(the origin
Ix In the
2(bssl
end,
=
bss n
From
TS
+ 2b n
(C.28)
we
ST
+ bn
+
find that
2bSS1
-
n-
4
the number of total bifur-
2).
-
n
Adding the three contributions, we obtain the branches, including non-minimal sequences: dn
differs
by one position). partial bifurcation branch
is when the
n
,
total number
d" of rooted
(C.29)
4
d. has the form
3
dn
--
E Cij' Aj'
-
4
n
4.
(C-30)
j=1
Numerical
large
computations show that, remarkably: c',
n, the second and third terms of
d,
=
2An1
(C.30)
== c2 = c3 tend to zero, and
-4+o(l)
(C .3 1)
with A, given by (C.5). Values of dn are listed in Table CA. The next column en of rooted branches, excluding the non-minimal
gives
sequences
The last column
gives
the number of free
2. For
=
branches,
obtained
the number
(see
C.2). 2Cn/n.
Sect.
by fn
=
C. Number of Branches
258
Table CA. Number of branches for total bifurcations of type 1.
bss
bTS
bST
bTT
d,
6n
fn
2
2
0
0
2
2
2
2
3
4
2
2
0 18
16
8
74
72
24
258
240
60
882
880
176
2994
2904
484
10138
10136
1448
34306
34048
4256
116066
115992
12888
392658
391760
39176
n
n
n
n
4
8
2
2
0
5
14
4
4
2
6
26
8
8
2
7
48
14
14
4
8
88
26
26
8
9
162
48
48
14
10
298
88
88
26
11
548
162
162
48
12
1008
298
298
88
13
1854
548
548
162
14
3410
1008
1008
298
15
6272
1854
1854
548
16
11536
3410
3410
1008
17
21218
6272
6272
1854
18
39026
11536
11536
3410
19
71780
21218
21218
6272
20
132024
39026
39026
11536
Type
C.2.2
2
We consider first the branches with
a
sign.
+
There
decomposition
of arcs, with the
number of rooted branches
(n
'
collisions
are n
corresponds
bifurcation orbit. Each subset of these collisions
to
exception of the empty subset.
(including
non-mmimal
sequences)
one
in
the
possible
Therefore the
is
d!
-
2'
-
1
1).
We consider next the branches with to the
case
of
-
sign.
(Sect. C.1.2)
bifurcation
partial
a
First
and do
need to go back some more detailed we
SS the set of branches which begin and end with a computations. We call Bn T=T ST We call Bn S arc (a symbol i or e). We define similarly B.-Ts and Bn .
begin and end with identical T arcs (two arcs T7 T the set of branches which begin and two arcs e). Finally, we call Bn with two different T arcs. We call b. SS, etc., the sizes of these sets. Proceeding as in Sect. C.1.2, we obtain for n ! 2 the set of branches which
2b; Sj + b,-, ST
b;ss bn b
2b-ss n-1
ST
bn-I
given
in
n
=
-TS
-T=T
2b n-1 + b n-1
-T?4-T
+ b n-1
ST
+
Successive values values for
-TS n
or
+ bn-i
-TS
-T=T n
b
n-1
n-1
n
i
end
T? T
bn-I can
I which
Table C.5.
be
can
-T91-T
bn
computed by
-TS
bn-1 these
+
-T=T
(C-32)
bn-I
relations, starting
be read from Table 6.9. Values up to
n
from the =
12
are
C.2 Total Bifurcation
we
The asymptotic behaviour of bn find that
bn-S,5
c-SS3'
-
ST
bn
SS
and
bn
ST
is obtained
259
in Sect.
as
C-ST 3n
=
C.1.2;
(C-33)
We define
bn
TT
(C.32)
Then from
and
2bn-1
=
n
T
T
(C.34)
have -TT
-TS
-TT
bn-1
+
2bn-I
bn
-TT
(C.35)
bn-1
+
obtain
we
bn
bn
+
we
-TS
-TS
b
T=T
bn
-
TS
C-TS3 n,
-
Finally, (C.32d)
can
-TS
-T=T
bn
bn-1
bn
TT
-
C-TT 3n.
(C.36)
be written
+
-T=T
-TT
bn-1
(C-
bn-I
-
TS
+ c-
TT
)3
n-1
-T=T
bn-I
(C-37) This is neous
linear
a
for which
-b
=
n
general
We consider first the
homoge-
T=T
-T=T
(C.38)
solution of
=
(C.40)
equation (C.37)
CIT=T 3n -TS
+ C2
=
has solutions of the form
T=T(_l)n
are
ST
_
n
(C.41)
found from the values for small
-TT
bn bn bn bn b-T=T 3' -2 (_,)n =
is of the form
T=T(-,)n
The numerical coefficients SS
equation
(C-39)
=C2
T=T
(C-38)
.
Therefore the full
bn
bn
n-1
obtain the characteristic
we
A+ 1 _-O
bn
relation for
equation
b-T=T
The
recurrence
T=T
2
3
x
b-T5,'T n
n.
We obtain
n-2
=
3'
-2
+
(_l)n
n
2.
(C.42) Now
the
we
partial
build the branches for the total bifurcation out of the branches for
bifurcation.
Suppose first that the origin of the total bifurcation junction of two arcs. By cutting the orbit at that point) we obtain a linear sequence of arcs, which represents a partial bifurcation branch. Conversely, out of a partial bifurcation branch we can build a total bifurcation branch by joining the two ends, except for branches belonging T=T to Bn Thus, the number of total bifurcation branches of this kind is TS SS ST + bn + bn T?4-T. + bnbn branch is at the
.
Next
we
is inside
an
consider the
S
arc.
case
where the
origin
By cutting the orbit
of the total bifurcation branch
at that
point,
we
obtain
a
partial
C. Number of Branches
260
partial belonging to B.-SS, we can build a by joining the two ends, except when the partial bifurcation branch consists
bifurcation
both ends of which
branch,
S
are
out of
Conversely,
arcs.
a
total bi'itircation branch
bifurcation branch
-
b;SS
of this kind is
Adding
both
branches with
dn or,
=
a
2bn-SS
-
1.
contributions, sign is
find that the number of total bifurcation
we
-
+
bn
TS
+
bn
ST
bn T?
+
T
(C.43)
using (C.42) dn
-
(-I)n
3" +
-
I
2: 2
n
In fact this relation is true also for
the branches with
(C.44)
.
n
=
1.
sign,
Adding branches, including non-minimal +
a
rooted
dn
=
d+ + d-
Values of d. of rooted
3n + 2
n
n
are
n
+
we
obtain the total number dn Of
sequences:
(_I)n -2,
n
(C.45)
2! 1
listed in Table C.5. The next column
branches, excluding
gives
the number
non-minimal sequences. The last column
en
gives
the number of free branches.
Table C.5. Number of branches for total bifurcations of type 2. 55
TS
ST
T=T
bn T: 6
T
n
fn
2
2
2
12
10
5
30
10
21
n
d+n
0
0
2
0
1
2
3
2
2
2
0
2
9
3
7
6
6
6
4
2
25
32
_bn
bn
bn
bn
d,,
dn
e
4
15
18
18
18
8
10
81
96
84
5
31
54
54
54
28
26
241
272
270
54
6
63
162
162
162
80
82
729
792
750
125 330
7
127
486
486
486
244
242
2185
2312
2310
8
255
1458
1458
1458
728
730
6561
6816
6720
840
9
511
4374
4374
4374
2188
2186
19681
20192
20160
2240
10
1023
13122
13122
13122
6560
6562
59049
60072
59790
5979
11
2047
39366
39366
39366
19684
19682
177145
179192
179190
16290
12
4095
118098
118098
118098
59048
59050
531441
535536
534660
44555
Type
C.2.3
3
We consider first the branches with must consist of
place
at
at odd one
even
an even
-
number of basic
positions, counting
positions.
a
sign.
must be even, and each
Thus, origin, or
arcs.
from the
n
into arcs, with the
place corresponds to
all collisions take
In each case, every subset of the positions
possible decomposition
arc
either all collisions take
exception of the empty subset.
C.2 Total Bifurcation
Therefore the number of rooted branches
(including
non-minimal
is
d,-,
2 ( 2 "/2
=
_1)
> 2
n
sequences) (C.46)
.
We consider next the branches with to the
261
a
sign.
+
First
we
need to go back
of
partial bifurcation (Sect. C.1.3) and do some more detailed computations. As in Sect. C.2.2, we call B+SS the set of branches which begin and end with a S arc (a symbol i or e). We define similarly B+TS and case
n
+ST Bn We call
T
(two
arcs
which
arcs i or
begin
n
Bn+T=T
.
the set of branches which
two
e). Finally,
arcs
and end with identical
T the set of branches call B+T7 n
we
and end with two different T
begin
We call
arcs.
b+ss,
etc., the sizes
of these sets.
Proceeding
as
in Sect.
> 4
2b +TS + 2b +T-T + 2b +T94-T + 2b +TS + n-1 n-I n-I n-2
b+ST
2b+SS n-2
+T=T
bn
+
+
+TS
bn-2
=
+
Successive values values for 12
n
2 and
--
given
are
2b+ST n-I ST n-2
b+
+
2b+SS n-2
bn+T;kT
bn-2 can
be
n
3 which
+TS
bn-2
+
b+T? T n-2
+
+T=T
bn-2
(C.47)
computed by these relations, starting from the can
be read from Table 6.10. Values up to
in Table C.6. n
(C.47c)
as
in Sect.
C-1.3;
we
n
3
1: ctssy '
--
b+ST
-
n
j=1
n
=
b+TS
-
n
b+TT
-
n
given by (C.20).
are
b+T=T +b +T? n
n
(C.48)
We define
T
(C.49)
n
(C.47b), (C.47d),(C.47e),
Then from
E Cj+ST I-Ij j=1
where 1-ij, P2) P3
b+TT
n
find that
3
b+SS
we
b+T=T n-2
I
+T;4T
--
+ST + b n-2
asymptotic behaviour of b+SS and b+ST is obtained from (C.47a)
The
and
n
b+TS n
and
obtain for
2b+SS n-I
n
=
we
b+SS n
n
C.1.3,
+TS
+TT
have
we
+TS
2b n-1 + 2b 2b n-2 +b +TT n-I + n-2
i
+TS +TT 2b n-2 + b n-2
(C-50)
obtain 3
b n+TS
3
CtTS117
+TT bn
j=1
Finally, (C.47e)
r Cj+TT lIj
n
--,
(C-51)
j=1 can
be written 3
b+T 4T n
=
+TT b+TS +b n-2 n-2
-
b+T?4-T n-2
E( Ci+TS + Ci+TT),,n-2 i
b+T36T n-2
j=1
(C-52)
C. Number of Branches
262
This is neous
linear
a
T
b+T
+T;4T
homoge-
for which
A2
obtain the characteristic
we
+ I
=
general b+T#T
(C .5 3)
-b n-2
-
n
The
We consider first the relation for b+'3". n
recurrence
equation
0
(C.54)
-
(C.53)
solution of
C+T#T(i)n
=
+
4
n
equation
Therefore the full
is of the form
C+T? T(_i)n
(C-55)
5
equation (C.52) has solutions of the form
3
b+T:?'-T
1: cj+T7'-T Yj
n
-
n
+
C+T7 T (i)n 4
+
C+T:p'-T (_i)n 5
(C .5 6)
j=I
The values of for small
+T;6T
c+T;z-T 5
and
4
can
be obtained from the numerical values
by noting that
n
b+T7 T
c
+T7 T
-
n
2b n-1
+TF T 3b n-2
-
-
+T? T 2b n-3
4C+T56T W 4
-
-
n
P
4C+T? T(_i)n 5
(C .5 7) We obtain
C+T? T 4
=
Thus, (C.56)
C
+T 6T 5
(C.58)
2
can
also be written
3
+Tt-T
b+T? T
Cj
n
7rn
n
Yj +
2
j=1
We consider For
2!
n
the total bifurcation. The
now
from Table 6.7
separately:
2,
we
(C-59)
Cos
find that
we
d+1
=
case
n
1 will be treated
2.
build the branches for the total bifurcation out of the
partial bifurcation. Suppose first that the origin of the total junction of two arcs. By cutting the orbit at that point, we obtain a linear sequence of arcs, which represents a partial bifurcation branch. Conversely, out of a partial bifurcation branch we can build a total bifurcation branch by joining the two ends, except for branches belonging to B +T=T Thus, the number of total bifurcation branches of this branches for the
bifurcation branch is at the
.
n
kind
is
b'+' SS
Next is
a
inside
partial
we a
+
bn+TS
+
bn+ST
consider the
T
arc.
+
case
bn+T?
T
where the
origin
the orbit at the
By cutting branch, which belongs
bifurcation
of the total bifurcation branch
origin of this T arc, we obtain B.+TS or Bn+T;4T Thus, the
to
.
+T9'--T number of total bifurcation branches of this kind is b TS + bn n
Finally we consider the case where the origin is inside an S arc. We call corresponding number of total bifurcation branches. Adding the three contributions, we obtain Xn the
C.2 Total Bifurcation
d+ n
b+SS
=
We
+TS +ST + 2b n + bn +
n
2b+TI-T n
+ Xn
n
,
263
2! 2
(C.60)
evaluate xn
recurrently, by reference to lower values of n. S arcs single kind, while S arcs of odd length come in two kinds (with and without prime). Therefore, to obtain the correct number of arcs, we should put into correspondence only S arcs with the same parity. Accordingly, we will make reference to values for n 2. The S arc is separated by the origin in two parts, of lengths ii and t2- We distinguish four subcases, depending on the values of i and 4. In each subcase we remove the two basic arcs on either side of the origin, thus obtaining a branch of order n 2. I (the original arc has a length 2), after removal and t2 (i) If tl have we a partial bifurcation branch of length n opening 2, which can be The number of b+SS branches for this sub is therefore b +TS arbitrary. case n-2 + n-2 + of
now
length
even
come
in
a
-
-
=
-
-
+ST +TT b n-2 + b n-2
(ii) If i > 1 and 12 1, then after removal and opening we obtain a partial bifurcation branch ending in a S arc which has the same parity as the b +TS original S arc. The number of branches is therefore b+SS n-2 + n-2 If and I find in the i same way that the number of 4 > 1, we (iii) =
=
branches is:
(iv)
i
If
bn-2 12S
b 52
+
n-2
and 4 >
> I
1,
and with the
origin
inside
a
S
two basic
we remove
not open the orbit. We obtain
a
total bifurcation
arcs
as
branch,
before but with
do
we
length
n
-
2
So the number of branches for this subcase
arc.
is Xn-2-
Adding Xn
:--
all four
Xn-2 +
3b+SS n-2
Successive values Of
values
X2
-
1)
contributions,
X3
Xn
-
We
+TS + 2b n-2 + can
be
6 which
then be obtained from
we
2b+ST n-2
computed
b+TT n-2
+
(C. 6 1)
*
from this
relation, starting
from the
be read from Table 6.7. Values of
can
(C.60).
obtain
These values
are
study the asymptotic behaviour Of
Xn.
d
can
listed in Table C.6.
homogeneous part
The
of
(C .6 1) Xn
has
:---
(C.62)
Xn-2
characteristic
a
A2
_
I
-
0
equation
(C.63)
.
Therefore the full equation
(C.61)
has
a
general
solution of the form
3
Xn
:::::
E Cj/-ijn + C6 + C7(- 1)n
(C.64)
j=1
The values Of C6 and C7
can
be obtained from the values Of Xn for small n,
by noting that Xn
-
2Xn-1
-
3x,,-2
-
2x,,-3
-
-6C6
+
2C7 (_l)n
.
(C.65)
C. Number of Branches
264
We find C6
-1,
=
Substituting
C7
in
-
-L
(C.60),
we
find that d+ is of the form n
3
cj+I-,,n 1
d, , n
+ 2
7rn cos
computations
Numerical
I
-
2
show
that, remarkably:
n, the first term dominates and
d+
+
-
n
=
c+2
=
c+3
2. For
a
sign,
-
are
we
obtain the total number dn of
listed in Table C.6. The next column
branches, excluding non-minimal
the number en of rooted
gives
large
(C.67)
rooted branches. Values of d, last column
c+1
have
we
0(l)
the branches with
Adding
(C.66)
(-I)n
-
gives
sequences. The
the number of free branches.
Table C.6. Number of branches for total bifurcations of type 3.
n
dn
b+S5
b+TS
n
n
b+ STb +T=T b+Ti6T n
n
n
Xn
d+
d,,
Cn
fn 2
n
1
0
2
2
2
2
2
5
0
0
2
0
1
6
8
6
3
3
0
14
4
4
0
0
6
32
32
30
10
4
6
46
10
10
0
2
18
98
104
96
24
5
0
144
32
32
4
4
64
312
312
310
62
6
14
454
102
102
12
10
198
978
992
954
159
7
0
1432
320
320
36
36
632
3096
3096
3094
442
8
30
4514
1010
1010
112
114
1990
9762
9792
9688
1211
14232
3184
3184
356
356
6280
30776
30776
30744
3416
44870 10038 10038
1124
1122
19798
97026
97088
96770
9677
0 141464 31648 31648
3540
3540
62424 305912 305912 305910 27810
12 126 446002 99778 99778
11160
9
0
10
62
11
11162 196806 964466 964592 963504 80292
C.3 Conclusions The number of branches
Several checks -
can
increases
be made
on
always exponentially. the computations:
The number of
computed from formulas in this Appendix agrees by hand in Chap. 6. branches (b,, for partial bifurcations, e,, for total bifurca-
tions)
even,
The number of branches
with the lists of branches derived -
-
is
by
n
always
as
it should be
(Proposition 6.0.1).
bifurcations, the number en of rooted branches for types 2 and 3, and by n/2 for type 1.
In total
is
always divisible
Index of Definitions
This index refers to the section where
a
word
or
expression
and defined. abnormal
8.2.1
arc
accident
9.1.1
antinode
8.1
arc
1, 2.10,4.1
v-arc
5.3
arc
arc
family family segment
basic
arc
basic
arc
4.1
4.6 4.1
half
bifurcating
7.3.2
6.2.2
arc
bifurcation bifurcation
1
ellipse
6.2.1, 6.2.2 1, 6 1,6 1, 8
bifurcation orbit
branch Broucke's
principle
ceiling
2.1
1,4.1
characteristic
class class
(accident) (remarkable arc)
closed
9.1.1 9.3.2.1
family
2.5
co-eccentricity
3.3
collision
2.10,4.1
complement composite family segment composite orbit
4.8
7.3.1
4.7
continuation
5.1
critical
4.6
arc
critical orbit
deflection
(first
and second
angle
direct exterior circular orbit
D1,
D+, D-, 2 2
2.8 4.1
direct interior circular orbit
domains
kind)
D3
3.2.2 3.2.2
4.3.2
is first encountered
Index of Definitions
266
duration
4.1
elliptic-hyperbolic family fragment family of arcs, see arc family family of generating orbits family of v-generating orbits family of periodic orbits family segment
continuation
first basic
9.1
1, 2.9 5.1
1, 2.3 2.6
6.2-1.2
are
first kind first
9.1.1
3.1
species
bifurcation
6.5
first species family first species orbit
2.10, 3.4 1, 2.10, 5.3
fixed
axes
2.2
floor
2.1
formation rules
6.2
fragment,
family fragment
see
free branch
6.2.1.1
generating family v-generating family generating orbit O-generating orbit
1, 2.9 5.1
1, 2.9 5.1
v-generating orbit
generating
5.1
solution
2.9
Hill continuation
9.1.1
Hill's coordinates
5.4
family Hill's problem
5.4
ingoing
4.3
Hill
2.10, 5.4
arc
Jacobi constant
2.2
junction keplerian keplerian
arc
2.10
orbit
2.9
I
kind
2.10, 3.1
maximal
6.2.2
arc
midpoint minimal period
4.3.1
minimal sequence
4.7
multiple-periodic orbit
5.4
natural families
10.2
2.3
(principle)
2.5
supporting ellipse
4.2
natural termination node
8.1
non-oriented normal
arc
8.2.1, 8.3.1
Index of Definitions
open
2.5
family
orbit space orbit with consecutive collisions order
(of
a
bifurcation) orbit
ordinary generating origin outgoing arc partial bifurcation period period-in-family periodic orbit periodic solution phase (in junction of branches) phase space principle of natural termination principle of positive definition
6.2.1
1,4.1 2.3 4.3 6.2
2.3 2.4
2.3 2.3 6.2.1.1 2.3
2.5 2.4
2.5
reflection remarkable
retrograde
I
2.10
9.3.2.1
arc
circular orbit
3.2.2
rooted branch
6.2.1.1
rotating
2.2
axes
4.3, 5.3.2.1
S-arc S-arc
4.3
family
second basic
arc
6.2.1.2
3.1
second kind
species species family second species orbit segment, see family segment
6."5
second
2.10,4.8 1, 2.10, 5.3
side of passage
8.1
simple family segment simple orbit simple-periodic orbit singular perturbation problem
4.8
second
bifurcation
4.7 5.4 2.9
solution
2.3
species stability stability index supporting ellipse supporting keplerian orbit
2.10
2.8 2.8 4.1 4.1
surface of section
2.8
symmetric symmetrical
2.7
symmetry
2.7
2.7
267
Index of Definitions
268
T-arc
4.3, 5.3.2.2
family
T-arc
4.3
third species bifurcation third species family third
species
orbit
6.5
2.10, 5 2.10, 5.3
total bifurcation
6.2
T-sequence
4.3.4
type 1
to 4
type I
to 3
type I
to 4
(arcs) (bifurcation orbits) (supporting ellipses)
1,4.2 1, 6.2, 6.2.1, 6.2.2 4.2
Index of Notations
This index refers to the section where
notation
a
is
first introduced and
defined. Notation
Section
Definition
a
3.2
seM17major
a
5.4
a, b
8.2
family subscripts for
a,
5.6
orbit radius in
A
4.3.1
a
Aj, Bj
B.2
numerator and denominator of
b,,
6.2.2.1
number of branches in
b+, b-, b5, bTbSS n
BS n,
n
n
Bj,
..
n
C
ST
S
,C,C3
T
,
Cj
,
Ci
,
( , 71)
coordinates
3/2
partial
convergent
bifurcation
number of branches Hill
C
(subsets)
subsets of branches
5.4
C
after and before collision
arcs
...
C.1.1' C. 1. 1,
.
axis
Hill
+
,
Ci
family
SS
,
C3
C.1.1'
coefficients
C
2.2
Jacobi constant
CP
4.1
value of C for
dn
C.2
number of rooted
dt, d;
C.2.2
number of rooted
D
2.5
dimension of orbit
parabolic arc branches, including
non-MInimal sequences n
n
D, D1, D2,
D+, D-, 2 2
branches,
for
a
particular
sign
D3
domains in (A, Z) plane eccentricity 4.3.4, 6.2.1.3 symbol for arc T' 3.3 co-eccentricity 4.4 eccentricity for type 2 number of rooted branches 2 C. 6.2.1.2, eccentric anomaly 3.3.1, 4.3.1
4. 3.2 3.1
e e
el CO en
E
Eo
4.3.1
value of E at t
3.3.1.1
family
=
in total bifurcation
0
Eii, Ej'j, Ejj e
of second kind
(symmetric orbits)
Index of Notations
270
Ej j E j
3.3.1.2
Ea
3.3.2
,
ll
f
5.4
fj
B.2
C.2 f" F, Fo, Fj, G Go 1,2.4,2.10
family of second family of second Hill family
kind kind
(symmetric orbits) (asymmetric orbits)
convergents number of free branches
,
characteristics of families
91 9
5.4
Hill families
gl
5.4
critical orbit
h
7.3.2
number of basic
H (X)
2.1
step function
6.2.1.1
rooted branches
Hi, H2
Hill-a,
etc.
i,
arc
halves
5.4
Hill family segment auxiliary integers (local use only) 4.3.4, 6.2.1.3 symbol for arc T' number of revolutions made by M2 3.3,4.3.4
1, 2.3, 2.4,
or
T
orbit
arc
4.4
number of revolutions made
Idi, Ide, 1,
3.2.2
circular families
1
3.3,4.3.4
number of revolutions made or
in second kind
by M2
in
by M3
in second kind
by M3
in type 2
type 2
arc
T-arc orbit
P
4.4
number of revolutions made
k
6.2.1.2
index of S-arc in type 1 bifurcation values of k before and after collision
ka, kb
8.2.1
K
8.2.1
quantity characterizing
Ki, -K4 el; 2
5.3.1
constants of
C.2.3
parts of
L
6.2.1.2
numerator of Z in bifurcation orbit
L, L, to L5
6.2.1.2
maximum of L
3.2.1
Lagrange points
M
4.4
number of basic
arcs
M
6.2.1.2
number of basic
arcs in an arc
Ml, M2, M3 M4
2.2
three bodies
4.1
fictitious
n
3.2
mean
a
arc
bifurcation of type I
integration
S-arc
an
body
on
in
(types 2 (type 1)
an arc
and
supporting ellipse
motion
n
4.1
number of
n
6.2.1
order of bifurcation
n'
C.2
proper divisor of
O(X) OW
2.1
vanishingly
vanishing
deflection
angles
n
2.1
small with respect to of the order of x
P
4.1,4.7
number of
P
4.3.1
P
6.2.1.2, 8.3.1
auxiliary quantity numbering of successive
PO
8.3.1
value of p for final collision of
x
arcs
collisions in P an
arc
or
Q
3)
Index of Notations
intersections with unit circle
P, Q
4.3
q
4.5,
r
2.2
distance from M, to M3
ri, r2
4.2
R
4.3.1
B. 1, B. 2
auxiliary quantity (local
use
only)
S
8.2.1
pericenter and apocenter midpoint of a S arc phase space vector sign of initial abscissa - 2 sign(AC) sign of branches
sign(x)
2.1
sign
S, Sh
4.3
arcs
R, R'
2.3
SO
3.3.1
T
S+
arcs
z*
'z-
a07
I
-ap
family segments
4.6
S-arc
2.2
time
4.1
initial and final time for
6.2.1.2
time of first collision in
5.6
scaled time for
to
2.3, 3.3,
particular
t2
4.3.1
120
4.3.1
time of passage of M2 at particular value Of 12
t1.1
t1.
v
>
an arc
Q
1/3
time
Q
t2P) I2Q
8.3.1
t4
4.3.1
time of passage of M2 in P and time of passage of M4 at Q
140
4.3.1
particular
t4Pi t4Q T, T'
8.3.1
time of passage of M4 in P and
2.3
period
TO
2.3
minimal
T*
2.4
Tri,
for type I
x
family of hyperbolic 4.1,4.5 S-arc family 5.3.2.1 4.3.3,
S'P so
=
of
Q
value Of t4
Q
period period-in-family
T', T' Tj'j, Tj'j 4.3.4, 5.3.2.2 T-arc family ,
6.2.1.1
T
origin shift unspecified arc velocity at collision, in rotating axes (relative velocity of M2 and M3)
Ui U111
4.1,4.7,
V
8.2
VY V
4.2
vertical component of v velocity at collision, in fixed
V
4.2
modulus of V
VY
8.2.2
vertical component of V
X,
2.2
rotating
Xo' X1
3.3.1
abscissa of intersection with
X,
C.2.3
auxiliary quantity
X, Y X0, X,
2.2
coordinates in fixed
3.3.1
abscissa of intersection with X axis
2.8
stability index
I
z
8.2.2
axes
axes x
axis
axes
271
Index of Notations
272
variable
z
4.3.1
new
zll
4.3.2
maximum of Z
0 0,0, go Oa; Ob
4.3.1
parameters of S-arc family
a,
and
# for first basic arc 0 before and after collision
6.2.1.2
a
8.2.2
values of
7
4.2
7
4.5
angle angle
'YM
4.3.2
F
4.3.2
value of -y for maximum of Z upper boundary of D2
11
5.1
Jacobi constant for
F
5.6
Jacobi constant redefined for
6X
8.2
variation of
of V with vertical axis for T-arc
a
enlarged Vicinity
quantity
x
v
>
Of
M2
1/3
from before to after
collision A
A, Ax
A0
Ai
4.3.3, B. 2 straight-line characteristic of S-arc family 6.2.1.2, 8.2.1 variation of a quantity x away from a bifurcation 8.3.1 angle 3 3 1 sign of pericenter abscissa .
.
3 2
direction of rotation
.
A. I
old
71
4.1, A.1
half-variation of E
77
5.1
coordinate for
A. 1
77
C,
EE
*
T1
signs on arc
enlarged vicinity
of M2
0, 0, Oa) Ob
8.2
angle
8.2
values of 0 before and after collision
E)(x)
2.1
strictly
A
2.3
parameter along family
Ao
2.4
particular value of
Aj
C. I. I
roots of characteristic
A2
8.2.1
sign[_(19Z1aC)A
A
B.2
auxiliary
P
I
mass
Yo
2.9
tiEM
10.2
maximum of IL 0.012155 (y for Earth-Moon
Yj
C.1.3
roots of characteristic
1/1 1/11 1/2
5.1
5.1
of
with vertical axis
v
of the order of
x
A
COS
equation
01
curve
of M2
case)
equation exponent for enlarged vicinity of M2 coordinate for enlarged vicinity of M2 3.14159...
zu
3.3 4.3.1
P
2.2
argument of pericenter distance from M2 to M3
2.10
minimum and
0,
8 1
side of passage
Y,
2.7, 7
symmetry
,
I//
P A ,P P
T-1
.
Tj
maximum
4.1 , 5.3.2 , A. 1 half-duration of
arc
of p
Index of Notations
A. I
00 0 Q, Q, 01) 22
8.3.1
lead of M3
3.2
angle (circular orbits) parameter for family of the second
3.3.1
1, 2.5, 2.7,
3.3.2,
over
M2
initial
...
orbit
particular asymmetric u 54 0
QA
2.9
orbit for
1XI 1XI
2.1
floor of
2.1
ceiling
x
of
x
orbits
kind
273
References T"Ib
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