Generating Families in the Restricted Three-Body Problem (Lecture Notes in Physics Monographs)

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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

Notes in

ISBN 3-540-63802-4 Edition

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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

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

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


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
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

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


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

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
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

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


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


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 =


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

-

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


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


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


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


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

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

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


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


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


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.

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
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
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

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

(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

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


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


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


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


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


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


+

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


+ +

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


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.


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.


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


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


+ +

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.


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


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

(8.20)

142


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

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
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.


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


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


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

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


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


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


8.

Table 8.16. Total

principle,

Principle

bifurcation, type


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.


Table 8.18. Total

principle,

2Tl++O

bifurcation, type


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


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


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

--

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


179


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


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


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

:! #

--

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


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


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


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


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


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


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
J 8.12

or

(A

>

1),

we

8.18. If I
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


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


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


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


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


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

-


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


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

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


n

=

can

20

be read from Table 6.8:

are

given

in Table C.1.


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


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)


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


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


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

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)


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


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


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


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


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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Bruno,

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Guillaume,

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(1975a):

Linear

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