Thes

is-1

972—

Gum

owsk

i‘

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

CERN/SI/Int . BR/72-l

CERN LIBRARIES, GENEVA

HIIIIIIIIIIIHIIIIIIIIIlllllllllllllllllfilIlllll?llllilllllllllllllCM-P00081051

SOME PROPERTIES OF LARGE AMPLITUDE SOLUTIONS

OF CONSERVATIVE DYNAMIC SYSTEM

Part 1 : Quadratic and cubic non—linearities

I . Gumowski

GENEVA

1972

ABSTRACT

Two large-amplitude effects are described which appear to constitute possible mechan—isms for slow and fast particle losses.

Note to readers with a moderate background in non-linear analysis

The present report tries to explain certain notions and to describe certain methods> which are useful in the interpretation of a particular class of non—linear phenomena: theoscillations of systems which conserve some quantity. In this paper the quantity conservedin the area in phase space.

The specialized terminology, such as, for instance, phase space in the preceding sen—tence, is introduced in two stages, first to get each particular notion on the "inventory"list, and then in the context where it is actually used. In the first stage no familiaritywith or understanding of this notion is required.

A similar attitude was adopted towards references. No familiarity with the contentsof the cited papers is assumed. Authors are cited primarily to show that a large body ofmethods and results exists already which can be drawn upon, and secondarily to permitinterested readers to go readily beyond the scope of the present paper without having tomake first a personal evaluation of the rather extensive existing literature.

I. INTRODUCTION. STATEMENT OF THE PROBLEM

In present accelerators and storage rings there are several phenomena, such as, forexample, slow particle losses just after injection into an AG proton synchrotron, whichare difficult to explain in terms of a linear or weakly non-linear theory of beam dynamics.It is therefore possible that these phenomena involve strongly non—linear interactions,i.e. that they are described by large-amplitude solutions of non-linear dynamic systems.In order to test this conjecture, a study of the nature of large-amplitude solutions wasundertaken, with particular emphasis on aspects which are absent in the theory of linear orweakly non—linear oscillations.

Since the time of Poincarél) it has been known that solutions of autonomous fourth-

order conservative or third—order non—conservative dynamic systems

do:if = XGC) , 1(tu) = 3C, x = vector (1)

can be described by an autonomous second-order point mapping (a surface transformation or asecond-order recurrence). Second-order non—autonomous dynamic systems

olx dg; =X(x,3,t) ) fi=Y(z.j.t) (2)

Xfloj = 1,, , 3(a) -_= 030 'X. "a= scalars

can also be reduced to such a form”.

It has also been known since the time of Poincaré and Birkhoffz) that the phase planeof a second-order autonomous recurrence may have features not found in the phase plane ofsecond-order autonomous differential equations [i.e. in formula (2) when BX/at = BY/Bt E 0],such as, for example, intersecting invariant curves (trajectories) in what are now lmovm as‘Birkhoff's instability rings. The pattern of the intersecting trajectories is in general socomplex that at first sight it appears to be stochastic. Of course there is no real stocha—sticity, the word stochastic, or one of its synonyms, being used merely as a matter ofconvenience .

As stochastic solutions, i.e. deterministic solutions of a very complex nature, aredescribed by correspondingly complex transcendental functions, it is understandable that inthe last 80 years analytical progress has been rather slow. Some reasons for this slowprogress will become clear in this paper, which uses a mixed 'analytical-nmnerical approach.Analytical expressions are used to define the solutions, but numerical methods are used to

determine the corresponding functions or constants. In contrast to the non—conservativecase, a conservative recurrence is usually badly conditioned numerically. Systematicaccuracy checks coupled with a certain amount of redundancy are essential. Detailed cal-culations are readily possible with present computers which have a high speed and a workingword of 15 or more digits.

Similarly to differential equations, one may try to divide the phase plane of a second-order recurrence .into cells,.,each cell filled with trajectories of the same type. ’But since

-2-

there is no counterpart for recurrences of the Andronov—Pontryagin theorem for differentialequations”, such a procedure is not likely to be completely successful. It is, however,reasonable to look first for singular solutions, as if a division into cells were possible.

The recurrence chosen for study, which possesses many of the properties of conserva-tive recurrences in general, is

xhn ‘ M“ + F(°‘n) = 39% “3“) ‘

”m1 = ~95“ + F(3¢nu\ -’- {(xn r3") , (3)

F(o)=0 n=0,:1,~:,....')

where F(x) is in principle an arbitrary real-valued function. In order to retain theapplicability of the existence theorems established by Poincaré and Birkhoff, it is neces-sary to assume that F(x) and a certain number of its derivatives possess at least an asymp-totically convergent McLaurin series. In this paper only the particular case

FM = Ikx+(1-,~)x°‘, -1

-3.—

of Eqs. (5) is said to be singular if it is somehow in conflict with a known existence oruniqueness theorem. The simplest singular solutions of Eqs. (5) are fixed points, defined

by

xnu 7- In = x , 30m” = 3n = 3 ) (9)

i.e. the coordinates (x,y) of the fixed points are given by the real roots of the (algebraic)equations

gum—x = 0 , Hun—g = 0 - (10)

Fixed points are classified according to the eigenvalues A of their characteristic equation

ark b

.c d—AII 0 . (11)

with the derivatives evaluated at (x,y).

For a conservative recurrence one has A1 - A2 = 1 because J E l. The fixed points arecalled saddles if the A are real (of type 1 if M > 0 and of type 3 if A1 < 0), and centresif

I ,_ 1' '1‘? a“- .12. {7 , = integers . (12)A1,: a ‘f g? 1’ ’ W)

The "critical" cases A1 = kg = i1 and the "exceptional" cases ¢ = 211(p/q) require a specialexamination, because the nature of the fixed point is then determined entirely by the non-linear parts of g,f in Eqs. (5).

If the fixed-point (x,y) is a saddle and b 75 0, its eigen directions p [slopes of theeigenvectors at (x,y)] are

P1 3 (Ana)? , p‘=()\,.-a).b" . (13)

If Eq. (7) is satisfied, the "direct" recurrence (5) admits a unique "inverse" recurrence

1,, = EH”, 33,...) ,3“ = {(xm, ,3“) , n = 0,-1, -a,.... (14)

The direct iterates of a point (xo,yo) are called its consequents and the inverse iteratesits antecedents. The kth consequents (k > 1) [or antecedents] are given explicitly by therecurrence

in”. = mama.) , 3m: = (human) , *h 2.3,...- (15)

obtained from Eqs. (5) [or from Eqs. (14)] by the elimination of the intermediate conse-quents (or antecedents). If k = l, by definition f1 = f, g1 = g.

-4-

A cycle (periodic point) of order k of (5) is defined by

imk= xn=°‘ , l3H*1

-5.—

For conciseness a cycle saddle (or centre) of order k and rotation number r is desig-

nated by saddle (or centre)'k/r or k r)‘ Consider a family of two cycles, consisting of a

cycle saddle k/r of type 1 and a cycle centre k/r. By tracing the two invariant curves

(four segments) passing through each point of the cycle saddle k/r, one finds that these

invariant curves either merge smoothly with the invariant curves passing through a neigh-

bouring point of the cycle saddle k/r or they intersect the latter curves. In the first

case the curve configuration so obtained is called an island structure, and in the second

an instability ring (in the sense of Birkhoff). In both cases the invariant curve—segments

emanating from the saddles k/r surround the points of the cycle centre k/r. The inter-

section points of invariant curves have been classified and named by Poincarél) . Some of

this nomencluture will be introduced in the following sections.

According to Birkhoffz), the phase plane picture of a conservative recurrence (5), pro-

vided x = y = 0 is a fixed point centre with (I) 7‘ 2n (p/q), is in general as shown in Fig. l:

the centre (0,0), is first surrounded by a set of regular trajectories (i.e. non-singular

trajectories), then by one or more island structures, and finally by one or more instability

rings, each separated by a set of closed regular trajectories.

The main objective of this paper consists in determining whether the solution structure

shown in Fig. l is sufficient to account for all features of the recurrences (3), (4). The

answer turns out to be negative“). At least two new features appear: a region of

"diffusion" and a region of stochaStic instability. In the next three sections it is shown

hdw these features arise.

DETERMINATION OF FIXED POINTS AND CYCLES

Consider the particular recurrence (3), (4)

t = 59V? FOL") , gun = 'xn * F(xn+1)

(22)

"T! A R v I!

d

PX*(I”L)1 r -1

For x = 0, y = 0 the eigenvalues are

A” = path-F. (26)

independently of on. The fixed point (0,0) is thus a centre, except possibly if

|,L = m 21r-% - (27)

For a = 3 the fixed points (1,0) and (-l,0) are saddles of type 1. There are no cycles fork = 2 if, as supposed, lu] < 1. For k between three and about six it is possible to findexplicit, although increasingly complex expressions for some of the real roots of Eqs. (17).Such expressions will be discussed in Section 5. For larger values of k the purely analyti-cal method becomes impossible in principle. The determination of the distribution of cyclesin the phase plane thus unavoidably involves the use of numerical computation. Unfortunately,

even a theoretically trivial problem such as finding the roots of Eqs. (17), when theirexistence is assured, constitutes for k large (in practice k z 10 for u z 0 turns out to bequite large) an unsolved problem of numerical analysis [see, for example, RabinowitzSU.By the combination of various methods and some trial and error, a sufficient number of cycleswere found to establish some patterns of their distribution.

Consider first the quadratic non-linearity (on = 2). A partial list of cycles, startingwith the lowest value of k, is given in Tables 1 to 4 for u = 0.8, 0.6, 0.5, and 0.125.Because of symmetry p1 = -p2 if y = 0. From Eq. (4) it is obvious that the parameter

-7-

some exploratory computations have been published only a few years ago”) . The particulartype of intersection points visible in Figs. 3 to 5 have been called by Poincaré "homoclinicpoints' ' .

From an inspection of Figs. 2 to 5 it is readily seen that there are no cycles outsideof the "separatrix" and that at least some of the cycles are aligned on certain curves(showed in the figures) which terminate at homoclinic points. It is well known thathomo-clinic points are enumerable and that any neighbourhood of a homoclinic point contains aninfinity of cyclesz).

Let the "first" homoclinic points be numbered as shown in Fig. 6a. This numberingpermits one to assign a measure Ai,i+1 to the "amplitude" of a "loop" formed by the invari-

i+1' The homoclinic points Bi’ 1 = l, 2, will be called ofrank 1, because other homoclinic points are possible when the Ai,i+1large and lead to additional intersections (see, for example, Fig. 5). The homoclinicpoints so createdvmay be called of rank 2, 3, .. . , and numbered by additional subscripts.

ant curves between Bi and B

become sufficiently

From a study of the data obtained, some of the properties of the distribution of cyclesof Eqs. (22) with a quadratic non-linearity can thus be suImnarized as follows“):

If r = l, for every value of u there exists a cycle of minimum order k. Let thisminimal value be kgg.

. The cycles (k + j)(1), k 2 kggg, j = 0, l, 2, form an infinite sequence. Thesequence of the coordinates of the cycles (k + j) (1) is bounded by the coordinates of theset of homoclinic points B1 of rank 1.

Let (£1571) be the coordinates of the homoclinic point Bi and (i’yki) the coordinates

of a point of the cycle k0), numbered similarly to the Bi; then

,{m xi; = 52; , [Wt '1“ = E; . (28)k—VU *9”

Consider now the cycles kCr)’ r > 1. An additional set of cycles for the rather strongnon-linearity 6 = 1 is shown in Table 5. This set, originally compiled by Laslett”), hasbeen completed by the saddle of type 3-saddle of type 1 families, resulting from thebifurcation of the centres. From the Tables 2 to 4, and especially from Table 5, it can beseen that there exists a subset of cycles k(r) verifying the inequality (symbolic)

12(1) < 7%” km < (1+k)(1) ‘ (29)

For this subset one has furthermore

(0) (a)k z 7 km . (30)'1'

The estimate .(30) will be made more precise in Section 5.

-8-

An inspection of Tables 2 to 4 together with Figs. 3 to 5 suggests that the cycles

k(r) also converge to the homoclinic points Bi of rank 1, but in a less ”direct" manner,

because the additional points arising for r > 1 form additional curve segments, as illus-

trated in Fig. 7.

The preceding conclusions remain valid for a cubic non—linearity [a = 3 in (22)]. The

first cycles k(r) for a weak (6 = 0.2), moderate (6 = 0.4 and 0.5), and strong (6 = 0.875)

non—linearity are given in Tables 6 to 9 J. The position of the cycles drawn together with

the "separatrices" passing through the saddles (x = l, y = 0) and.(x = -1, y = O) are shown

in Figs. 8 to 11. The intersections of invariant curves emanating from different fixed

points, or from points of different cycles, have been called by Poincaré "heteroclinic

points”. Similarly to the homoclinic points Bi in the quadratic case, the heteroclinic

points C1 in the cubic case can be numbered as shown in Fig. 6b. These heteroclinic points,

visible in Figs. 9 to 11, are called heteroclinic points of rank 1. The coordinates of the

cycles converge to the heteroclinic points of rank 1. With this slight change in vocabulary

Eq. (28) retains its validity.

The main difference between the quadratic and cubic non—linearity consists in the fact

that in the latter case the centre—saddle bifurcations are more complex. The centres may

turn into saddles of type 1 as well as into saddles of type 3. Comparing the data of the

quadratic and cubic non—linearity for the same value of 6, it is seen that the crowding of

cycles near the separatrices is more pronounced in the latter case, but the cycles of the

same k(r) are closer to the origin. This property is consistent with what one would expect

from.the relative shapes of the non-linearity.

Taking into account a group property of recurrencesz), Eq. (28) can be used as an

analytical characterization of homoclinic or heteroclinic points of rank 1 in terms of

continuous variables. In fact Eq. (28) holds for k continuous, at least if [kl is suffi-

ciently large, provided (i’yki) are suitable branches of the curves defined implicitly

by Eqs. (17) for continuous k.

DETERMINATION OF INVARIANT CURVES

In the study of the distribution of cycles it was found that centres turn into saddles

both with an increase of non-linearity 6, k being constant, and an increase of k, with 5

constant. Let us examine what happens to the invariant curves in such a case. Consider

for this purpose the quadratic non-linearity and the cycle 9(1) for 0.4 S 6 S 0.5

(0.6 2 u 2 0.5). The invariant curves emanating from two adjacent saddles are shown in

Figs. 12 to 16.

An inspection of these figures shows that an increase of the strength of the non-

linearity 6 leads to a simultaneous increase of the eigenvalues 11 of the saddles of type 1,

the angles between the eigenslopes p1 and p2, and the amplitudes Ai,i+1 of the loops between

two successive homoclinic points Bi’ B.1+1 of rank 1. In Fig. 16 the amplitudes Ai,i+1 are

already sufficiently large for the appearance on the first loop of homoclinic points of

rank 2. This appears to happen simultaneously with the centre-saddle of type 3 bifurcation

of the centre-saddle of type 1 family.

*) Because of symmetry one has p1 = -p2 for cycles at (x,0) and (0,y).

-9-

Figure 17 shows the invariant curves of the cycle 11(1) for u = 0.6 and Fig. 18 of the

qualitatively similar cycle 19(“3 for u = 0.125. A comparison of Figs. 12 and 17, both

having the same strength of the non-linearity, to Figs. 12 and 16, which have the same k,

shows that an increase of the distance 5 of a cycle from (0,0) plays the same role as an in-

crease of the strength of the non-linearity for a fixed distance. The qualitative equivae

lence between the strength of the non-linearity and the distance from (0,0) is even more

apparent from a comparison of Figs. 17 and 18, where roughly the same amplitude of the loops

A. . occurs at s z 0.9 and s w 0.5, respectively.i,1+1 .

Taking into account all results for a constant strength of the non-linearity, the

structure of solutions in the phase plane of (22) with a = 2 can be summarized as follows:

using the centre (0,0) as a reference, Birkhoff's picture (Fig. 1) applies to cycles up to

a critical order k(r) = kC1'

Since the amplitudes Ai,i+1 of the loops formed by invariant curves increase while the

distance between the cycles k(r) and (ki-m)(r) (m being an integer between 1 and r) dimin—

ishes indefinitely, a situation is reached when the invariant curves of two adjacent cycles

intersect. The appearance of heteroclinic points so created may be used as a definition of

kcl'

If k(r) > kcl’ a second critical value k(r) = kc2 is reached when the loops of the

invariant curves have a sufficiently large amplitude Ai,i+1 to fall into the influence

domain 0 of the invariant curves passing through the saddle at x = 1, y = 0 (see, for

example, Fig. 5). Since the latter invariant curves have two unbounded branches, the

invariant curves emanating from the cycles k(r) > kC2 become also unbounded.

A.sufficient condition for the unboundedness of an invariant curve passing through a

cycle k(r) appears to be the existence of saddles produced by a bifurcation of centres,

i.e. the existence of homoclinic points of rank 2 on the first loop (see Fig. 16).

In.the region defined by kCTJ < kC1 the distance sn between (0,0) and the antecedents

or consequents of an initial point (xo,yo) is bounded. One has therefore the analogue of

orbital stabilityz) of the theory of differential equations.

For kt1 < k(r) < kc the distance sn grows in the mean, but at a less than exponential2

rate. This region may be described qualitatively as a region of diffusion.

For k(r) > kc2 the mean of the distance sn grows exponentially. Since the antecedents

and consequents of (xo,yo) form geometrically a complicated sequence of points, this region

(Q) is well described by the name stochastic instability, already used elsewherelz).

Finally, the total region containing intersecting invariant curves, Birkhoff's

instability rings included, may be described as the region of stochasticity. As the

strength 6 of the non-linearity increases, the region of stochasticity may coincide with

the whole phase plane. For a = 2 this happens in the case of Eqs. (22) only for u = -%.

For a = 3 in Eqs. (22) the subdivision of the phase plane into the regions of (orbital)

stability, diffusion, and stochastic instability remains unaltered, except that the whole

phase plane is never fully stochastic if —1 < u < +1.

-10..

CRITICAL AND EXCEPTIONAL CASES

5.1 Statement of the problem

The preceding analysis of the properties of Eqs. (22) was carried out with the assump-tion that the fixed point (0,0) does not have an eigenvalue of the form A = 11 or A = e1¢with ¢ = 2n(p/q), p, q being integers. The first assumption is obviously verified in theinterval —1 < n < +1, but the second is not. In fact there exists an infinite set of p, q

for which it fails.

_ The distinction between critical cases (A = fl) and exceptional ones [¢ = 2n(p/q)] is

a matter of convenience and not one of mathematical necessity. It is quite easy to see thatif the recurrence (5) presents an exceptional case at a fixed point (x,y), then the iteratedrecurrence (15), which is not independent of (5), presents a critical case at (x,y) whenk = q or a multiple thereof. Hence, in principle the study of an exceptional case canalways be replaced by the study of an equivalent critical case. Suppose that Eqs. (5) or

(15) present a critical case at (0,0). It was already established by Liapunovla) that under

such circumstances the solution of (5) or (15) may differ qualitatively from the solution

of the linearized recurrence

xrm = ax“ "' by" , gnu = 59% + d'jn r (31)

a, b, c, d given by Eqs. (6), even in an infinitesimal neighbourhood of (0,0). In other

words, in a critical case the non-linear parts of f, g dominate the linear parts.

At present there are two known methods of studying critical and exceptional cases, a

local one based on a reduction to certain canonical forms, some of which were worked out

by Levi-Civital”), Cigalals), and Birkhoffz), and a global one, based on the bifurcationtheory of Poincarél). In practice the reduction of an exceptional case to a critical one

cannot always be used, because the iterated recurrence (15) is hard to establish in an

analytically explicit form, even with the help of the "algebraic" computer programming

systems now available. Such a mathematically trivial problem as the explicit formulationof fk, gk in terms of f, g is still a practically unsolved problem of algebraic computerprogramming, even if the f, g are low-degree polynomials. The study of critical andexceptional cases is therefore lengthy and laborious. In order to preserve some trans-

parency in the link—up of ideas, it was necessary to omit many theoretically straightforward,

but practically difficult intermediate steps. L

The essence of the method of reduction to a canonical form consists in identifying and

isolating "dominating terms" of the non-linear parts of f and g. One tries to attain this

objective by means of a sequence of non—linear transformations, which successively remove

non-dominating terms. If by means of the successive transformations one has managed to

remove completely the non-linear parts of f and g, i.e. no dominating terms have appeared,

the recurrences (5) and (31) become qualitatively equivalent. Since the recurrence (31) is

readily solved explicitly for arbitrary (xo,yo), the sequence of inverse transformations,

if it converges, yields then a general explicit solution of (5). Such an occurrence is of

course quite rare for unrestricted (xo,yo), but locally, i.e. for (xo,yo) sufficiently

close to (0,0), it may occasionally take place.

-11-

Practically the method of successive transformations is only successful when the do-minating terms happen to be of a sufficiently low order, because in general the complexityas well as the notational bulk of the successive transformations increases very rapidly.The "order" of a dominant term can be defined in various ways. In this paper, because ofthe continuity and differentiability assumptions already made, it will be defined as thedegree In of a term in a Taylor series. For a two-dimensional Taylor series the term oforder m is actually a sum

. I: J: I I = m2.; 045.94.; ,HJ (32)with constant coefficients aij' Such a sum is called an "mth order form".

5.2 Local analysis. The Cigala transfonnation

Let us now describe briefly the method of successive transformations as worked out byCigala”). Suppose that the recurrence (5) has an exceptional case at (0,0). If not, it isalways possible to transform a point (x,y) to (0,0) by means of a linear transformation.As a second preparatory step, the recurrence to be studied is transformed into a complex-conjugate form by means of

. ‘ . -1hum . ramm an“? ,pumwrb . as)In.the new variables the recurrence takes the form

_ on

Mn“ = aw.“ 4‘ Z 13' (“n.vn)- F

' ‘fVVIH : E‘ - 1T“ + §=m 0} (KV‘ IV")

where

& - -é) . . “3—1 v1

Paw”) = «Z7, (1 C“ ‘ (35)i 3 * 1 2,..1

Qjflkfif) = é(1)C51'M' .‘V

The asterisk represents the complex conjugate, and the binomial coefficients Q) areintroduced for convenience. They permit simplifications in some later algebra. Theconstants Cj Y are completely defined by Eqs. (33) and the original recurrence.

The successive transformations permitting dominant terms to be identified are of thetype

I; = M. + ‘fmo‘u‘d ”I = M“ + 4"" (”h”)

Irma-1r) = ZN— (?)°

-12-

the arbitrary (free) constants ocY to be so chosen that in the new variables 5, n the

recurrence (34) contains as few terms as possible in the corresponding mth order forms.

This condition implies the following values for OLY

-id ___e—————-— z 0 1....,m, (37>' 1 _ éd(m—zv—1)$ C“’ 1 "||

The transformation (36) will thus succeed only if none of the denominators in Eq. (37)

vanishes, i.e. it will succeed if one does not have

‘f’ = 2-“- _%l_ , P, 1/ = integers .(3.8.)-

The exceptional case (38) is therefore defined as a case where the removal of all mth order

terms cannot be attained. In other words, if Eq. (38) holds, then not all OLY in Eqs. (36)

can be chosen according to Eq. (37) , and the recurrence (34) will contain some mt order

terms in the new variables 5, n. The constants any inconsistent with Eq. (37) can be chosen

arbitrarily (for example, set to unity).

Let the transformed recurrence be

§. Jig“ MM.) ”am-”W'asmtiW =50? w Wm" .w 5— w 2.. H ,

where Uj , V. are jth order forms. The "canonical" forms Um’ Vm define the dominant terms.

arm"

The advantage of Eqs. (39) compared tovthe original recurrence is twofold:

(39)

i) If the critical case is such that one or more invariant curves traverse the point (0,0),

then at least in principle it is possible to determine the slope of these invariant

curves at (0,0) from the form of Um and VIn alone.

ii) If no invariant curve traverses (0,0), then (39) is analytically more convenient than

the original recurrence if one wishes to determine the qualitative shape of the regular

invariant curves near (0,0) .

In the case (i) the angles between the invariant curves can be found by transforming

qs. (39) into polar coordinates

i 9 43g = y.e , W‘f'e . (40)

Since €.n = p2, one readily finds

a. I!5%.“ = g: + Am(6n.‘f). 5.. +---- (41)and with a bit of patience

9n 1 = 6n + Q1?" I e" ' (P) ' (42)

As a matter of convenience, the dependence of Am and 9m on the "unperturbed rotation angle"

4) has been indicated explicitly. The condition for the existence of invariant angles 6 is

obviously - * (43)

6M1 = e" + ‘f

-13..

The elimination of en+1 between Eqs. (42) and (43) leads to an algebraic equation for the

unknown real constants en = é.

Once the angles éj’ j = 1, 2,... are known, en = éj defines an invariant segment

emanating from p = 0. The relative position of successive consequents on each invariant

segment is given near 9 = 0 by Eq. (41), i.e. one knows then whether these consequents

converge to or diverge from p = 0. Once one is in possession of this knowledge, there re-

mains only the straightforward but tedious task of transforming the angles 5. into slopes

expressed in the variables x,y, and then computing the invariant curves by means of Eq. (2m.

Let us now apply the Cigala method to the critical cases ¢ = Zn/S and o = 2n/4 at (0,0)

of the recurrence (22) with a quadratic non-linearity. In both cases

WY= I" .'P-.»=1LW =1£¢mf. (44)

The use of Eqs. (33) leads to the Cigala starting recurrence (34),(35) with

C10 = '17; [(1 +2mqv)~imtr + i(1+ mflfi - amen {:3 if-

Cu = HM? - i

-14..

The three constants on. are well defined if q: = 21T/4, and all quadratic terms can thereforebe eliminated. If ¢ = 21r/3 the constant 0L2 is not finite, and one quadratic term turns outto be dominating. The resulting canonical form is

gnu = eL‘r.§n *' C21”: *"I

. ,‘ I. (48)v + .u -

V‘VH‘ 5 Que . V'n + Cu. in

In order to determine the possible existence of invariant angles, the transformation topolar coordinates is carried out:

. ' so) 4 .“(w-5M C” _e|-(‘f* .. ] 3:“ +--S)“: = f: *‘ [ Caz - e + n (49)

-' + 59..+ + MZE1 + C“? 3-s-S’n1‘ - (50)

emu = 6" S,"

For lpn]

-15..

For larger Ixnl or lynl it is necessary to take into acCount higher order terms inEqs. (55). This means that from the cubic terms in (55) it is not possible to determinewhether the continuations of the hyperbolic segments given by Eq. (56) join smoothly in the

finite part of the phase plane or not.

The fourth-order forms to be added to Eqs. (55) arels)

mm = :‘(z-n)“ +2(5‘vw‘)[i(§+n‘-(§vw)‘]

' (57)L. a” 1. 1, -

" .- I'

\L. (E M3 = ‘-‘;—(€ 4:) +2(; a. Wm) +(s 0)]With the use of Eqs. (57) the polar form of Eqs. (55) becomes

1 1I 0' o.

o

fm4q = f“ + ( 4M 46“) f“ 4P

(58)

0.. m - 2(1~4m49»)3’«9mFrom Eqs. (58) it is possible to conclude that the continuations of four hyperbolic segments

join as shown in Fig. 19. In fact, (58) implies

f: +(16Wl‘eu3f: I'".II2.fh+4

eu&4 =

(59)

6h —- 3(1- 6064603),» *"H

From Eqs. (59) it is easily seen that close to the origin pn is stationary for en = 0, 1r/2,

n and 3n/2.

The local analysis of Eqs. (22) with a cubic non-linearity does not disclose anything

qualitatively new. The exceptional case :19 = 21r/3 appears to be trivial because there are

no quadratic terms to be eliminated from Eqs. (22). The case (1) = 217/4 has cubic dominant

terms, but they are such that no invariant angles é exist. Similarly to Eq. (56) one finds

near the origin the approximate solution , p so ,

‘* " + Mam ‘) =C°nst- -2

-16-

It was foundls) that this centre also turns into a three—line saddle. The corresponding

slopes in x,y coordinates are p1,2 = 12 and p3 = im.

The main difference between the local properties of cycles and of the fixed point (0,0)

of Eqs. (22) is that the former may possess critical as well as exceptional cases. The

starting point for a study of critical cases on a cycle is normally the explicit form of

the recurrence (15), which, as pointed out before, is extremely hard to obtain, even with

the help of present algebraic computer programs. By applying suitable non-linear transfor—

mations to the properly prepared recurrence (15) it is again possible to define canonical

forms and dominant terms.

A partial classification of dominant terms was carried out by Birkhoffz), but most of

the solutions are still lacking. This is unfortunate because solutions of critical cases

appear to be the key to a profound understanding of large-amplitude behaviour of non—linear

dynamic systems.

As an illustration of the present limitations of local analysis, consider the critical

case A = -l occurring during the transition centre-saddle of type 3. Such a transition is

frequent for cycles of the recurrence (22). Since cycles of order k = 2 do not occur in

this recurrence, the simplest possible case is k = 3. For the quadratic non-linearity one

has then 1 = -1 for u = -%, the coordinates of one point of the cycle being (-b%,0). The

first terms of the iterated recurrence (15), with (-5§,0) moved to (0,0), are

1. [f +jlxt+....1,“ +613“ —21n‘jn a "H

t{Ia-r“;

rmAlthough the quadratic terms appear to be dominating, the qualitative nature of the solu-

tion of Eqs. (62) is still unknown. Other methods of studying critical cases are therefore

indispensable.' One such method is contained in the bifurcation theory of Poincarél).

(62)"3" + 4x” .123: .451t -491:+...

5.3 Bifurcations

Consider a family of functions G(x,B), depending continuously on the real parameter S

in a given interval 81 < B < 82. A.va1ue So is said to be a bifurcation value of G(x,B) if

B; < Bo < 82 and if G(x,B) undergoes a.qualitative change at B = 80, i.e. if the qualitative

properties of G(x,Bo) are different from the qualitative properties of G(x, Bo + A8),

[A8]

-17-

The first bifurcation encountered in the study of Eqs. (22) happens to be the transi-

tion centre—saddle of type 3, the bifurcation value being A = -1 (cf. Tables 1 to 9).

While the bifurcation solution is still unknown, in principle it is possible to find the

non-bifurcated solution just "before” and just "after" the bifurcation, i.e. for a slightly

weaker and a slightly stronger non-linearity. The properties of the bifurcation solution

can then be roughly estimated by means of a continuity argument. If there were no

practical limitations on the analytical or numerical operations, the choice of the specific

cycle to be studied in connection with the 1 = -l bifurcation would be a matter of indif—

ference. But since there exist several practical constraints, this choice is best made by

experience. Extensive data on cycles, such as those shown in Tables 1-9, are of course

quite helpful.

Consider Eqs. (22) with the quadratic non-linearity. For the cycle 4/1 the data are

II II-0.103 x-0.104 x

0 ¢ = 2.9883512 = -1.22558 ,

0.532268 y0.533703 y43

k

(63)

the eigenslopes in the latter case being p1 = ~0.0545, p2 = -p1. The bifurcation A = -1

thus takes place in the interval -0.104 < u s -O.103. Since the saddle of type 3 has real

eigenslopes it is possible to compute by means of Eq. (20) the invariant curves passing

through one point of this cycle. A representative part of the invariant curves traversing

the point given in Eqs. (63) is shown in Fig. 21. The corresponding saddles 4/1 of type 1

are far removed from the invariant curve loops of Fig. 21. The closest points of the cycle

saddle 4/1 are

u = -0.104 X = 0.164754 y = 10.211856 11 = 2.33845 . (64)

Thus a.natura1 question is whether there exist points of another cycle inside the main loops

of the invariant curves of Fig. 21. The answer turns out to be in the affirmative: for

u = -0.104 two points of the cycle centres 8/2 with 6 = 0.583397 are located at

(x = 0.519141, y = 0) and (x = 0.546543, y = 0), respectively.

For intermediate values of u one finds for the cycle 4/1

0 = -0.1036 x = 0.533130 y = 0 12 = —l.l3268u = —0.1o34 x = 0.532844 y = 0 12 = -1.05210 (65)u = -0.1032 x = 0.532556 y = 0 ¢ = 3.03975

and for the cycle 8/2

= _" " x = 0.524416 y = 0} =11 0.1036 . {x = 0.541193 y = 0 :6 0.354186= _ x = 0.529369 y = 0} = 6611 0.1034 {x = 0_536210 y = 0 4 0.143779 C 3

u = -0.1032 x = none y = none .

From Fig. 21 and the data in Eqs. (65) and (66) it appears that the centres 8/2 approach the

saddles 4/1 of type 3 as u approaches no from below. Since for u > no the centres 8/2 cease

to exist, it is natural to conclude that they merge with the saddles 4/1 of type 3 as

u + Uo- This conclusion is confirmed by the behaviour of the invariant curves, whose loops

get flatter as u + no. For n = -0.1036 the eigenslopes are p1.2 = $0.0334 and for

' 18 :J

u = -0.1034 they are already p1 2 = $0.0136. Using the terminology of Poincaré one can,

therefore say that as u traverses no in the direction of increasing non-linearity, the

centres 4/1 bifurcate at u = No into saddles 4/1 of type 3 and into centres 8/2.

In order to verify whether the preceding conclusion holds also for the previouslyestablished equivalence between the strength of the non-linearity 6 and the distance 5 ofa cycle from the origin, the singular invariant curves defined by the saddle of type 1-saddle of type 3 configuration were computed for the rather weakly non-linear case u = 0.707,k = 17, r = 1. These invariant curves are shown in Figs. 22 and 23. Inside the first loops

of Fig. 23 were found two points of the cycle centres 34/2. For a moderate non-linearity,

each“island'of the cycle 9/1 with u = 0.5 was found to contain in its interior two pointsof the cycle centres 18/2 (Fig. 16).

The bifurcation A = —1 of the cycle 3/1, described by Eqs. (62), produces the centres6/2, one point of the bifurcated cycle being

u = -0.506 x = -0.332005 y = -0.036424 ¢ = 1.72444 . (67)

As the strength of the non-linearity increases, the rotation angle ¢ of the bifurcatedcentres grows, and may attain in its turn the bifurcation value ¢ = n (A = -1). For thecentres 6/2 this happens in the interval

{u11

If the iterated recurrence (15) has a critical case A = -1 at a point (x,y) for k = k0,then it will have a critical case A = +1 at the same point for k = 2kg. But it is wellknownz) that there exist also critical cases A = +1 which do not arise in this way.Consider, for example, Eqs. (22)-with the quadratic non-linearity and the cycles 3/1. Thecoordinates of one point of these cycles can be found explicitly in terms of u

2.74582-1.60884 .

-0.510-0.511

-0.046945 ¢-0.049216 A2

K II -0. 331126 y68

-O.330906 y ( )II II II II

2xziit+n=o 3‘0: (69)1 — r - 2(1- r) r

This equation yields

_ _ x = -0.300054 y = 0 45 1.43177 701‘ ' 0‘45 {x = -0.079256 y = 0 A1 = 1.33221 . ( )It is however obvious from the form of Eq. (69) that the centres and saddles (70) will merge.

This happens for

H=v°=1t'rrfi',

where Eq. (69) has a double root

=-.l:c z :_+L!:- x -o.zo7107 ., g = 0 , A = *7 - (71)

For n > no the roots of Eq. (69) are complex, i.e. the cycles 3/1 no longer exist..

-19-

By straightforward but practically tedious algebraic calculations it is possible to

find the iterated recurrence at the double root (71) in canonical form, i.e. the double

root moved to (0,0) and only dominant terms retained

1M5 = In ; flu ‘ 6"“) x: '(72)

.4“, = t3" rah-Hz: +2(1+.)x..t3.. -(1-;Lo)g:+---

The invariant curves of the critical case (72) cannot be expressed in the form (20). Itwill be shown later (Part 2) that the invariant curves passing through the points given by

Eq. (71) can be described by an asymptotically convergent series in fractional powers of its

argument, the first term of which is

.3“ = :2 13-9a . (73)

The invariant curves therefore form a cusp at the double point (71). Longer segments of

the invariant curve defined by Eq. (73) are shown in Fig. 24. These segments do not join

smoothly, but form homoclinic points.

It is obvious that the cycle 3/1 has also something to do with the exceptional case

¢ = 2n/3, because contrary to what happens to the saddles n/r in Tables 1 to 4, the saddles

3/1 move inwards as the strength 6 of the non-linearity increases. More closely to Mo = -%

one finds

u = -0.49 x = —0.013709 y = 0 11 = 1.06158u = -0.5 x = 0 y = 0 X; = +1 and ¢ = Zfl/S (74)

u = -0.51 x = 0.012995 y = 0 X1 = 1.06201u = -0.55 x = 0.059493 y = 0 Al = 1.35216 .

The exceptional case ¢ = 2n/3 at (0,0) is therefore produced by the coalescence of a centre

with three saddles of type 1 of the cycle 3/1. The invariant curves corresponding to the

crossing of this bifurcation are shown in Figs. 25 to 27. It should be noted that just as

in the case of the cusp, from which these invariant curves have evolved, the invariant

curves emanating from the saddles 3/1 of type 1 do not join smoothly. The loops on the

interior invariant curves (those closer to the origin) are much less pronounced than those

on the external ones. The iterates on the latter diverge very rapidly.

The exceptional case ¢ = 2n/3 occurring on the centres 3/1, cf. Eqs. (61), is a bifur-

cation involving the saddles 9/3 of type 1. For u = -0.47870 one such saddle is at

(x = -0.322341, y = 0) with 11 = 1.00263 and the eigenslopes p1,2 = 1.973.

Since the correlation of the cycles 3/1 with the exceptional case ¢ = 2n(1/3) at (0,0)

turned out to be successful, it is worth while to examine the relationship between the cycles

4/1 and the exceptional case ¢ = 2w(1/4). The locations of the points of the cycles 4/1can be expressed analytically in explicit form. For Eqs. (22) with a = 2 the saddles oftype 1 are at

1 a. _+V(’Z) r= m(r*—W—Zr)» :1— 437%,: (75)

-20-

whereas the centres, and after the 1 = -1 bifurcation, the saddles of type 3 at

_—__—-1 ‘I— a

: 2(1-m("r 14% 'ZA) ' 3 ' 0 ,(76)

Aw—rF-‘a?Both cycles exist only for u < 0 and both merge with (0,0) when u + 0. The rate of approach

of (0,0) is, however, unequal. Sample locations are

{u = —0 0001 x = —0.084329 y = 0 ¢ = 0.009548‘40 = ~0.001 x = -0.150566 y = 0 ¢ = 0.054121

' = _ x = -0.225650 y = 0 ¢ = 0.1839111“ 0'005 {x = -0.047326 y = 0.049813 11 = 1.04091 (77)I“ = _0 1 x = -0.267843 y = 0 ¢ = 0.313781k ' x = —0.065235 y = 0.070185 x, = 1.0837 .

The bifurcation of the exceptional case 0 = 20/4 therefore releases an island structure 4/1

when u = no = 0 is crossed in the sense of increasing non-linearity. The difference of the

rate at which centres and saddles 4/1 leave the vicinity of (0,0), illustrated.in Eqs. (77),

explains the deformation of the regular invariant curves at u = U0 = 0, because in order to

preserve continuity the shapes of these invariant curves (Figs. 19 and 20) must ”preadjust"

t0 the shape of the singular invariant curves traversing the 4/1 saddles, which are about

to appear.

All higher exceptional cases of Eqs. (22) at (0,0) examined so far show the same

behaviour: the island structure n/r, n > 4, is released at pa = cos 2w(r/n) and exists

only for u < Ho- The exceptional cases on centres of a cycle have the same behaviour when

the exceptional case is such that no invariant curves pass through a bifurcating centre of

the cycle.

The bifurcations described above allow an explanation of the origin of all cycles in

Tables 1, 2, 3, and 5, and of most cycles in Table 4. For example, the cycles 10/2, 15/3,

20/4, and 25/5 all result from bifurcations of the centres 5/1: 10/2 at A = -1, 15/3 at

¢ = 2n/3, 20/4 at ¢ = 20/4, and 25/5 at ¢ = Zfi/S. The reason for the non-uniqueness of the

type 1 saddles 13/2, 17/3, and 19/3 is still unknown. The existence of surplus saddles of

type 1 implies that still other bifurcations take place when the non-linearity is suffi-

ciently strong or the distance 5 of the cycle from (0,0) sufficiently large.

These still unknown strongly non—linear bifurcations are probably responsible for the

faster rate of growth of the stochastic instability region than one would expect on the

basis of the value of [AI of the corresponding saddles, especially in the interval

-1 < u < 0. As an illustration consider the type 3 saddles 3/1 for u = -9/10:

x = - 31—8 (1 + 1/161) y = 0 12 a: —29.73 . (78)

If the value of x is rounded off to eleven significant digits, and the computations are

carried out with 15 digits, the first 14 consequents of (78), computed by means of (22),

-21-

are as shown in Table 10a. For the 20th consequent one has lxl > 101°°. Following the

same procedure for the type 3 saddles 4/1 at u = —9/20

x = §%.(9 _ ¢fi281) y = 0 l2 x -54.47 , (79)

but rounding off the value of x to nine significant digits, one finds the first 14 conse-

quents of Table 10b. At the 20th consequent the value of [XI is still below one. For the

saddles 4/1 both the rounding error and llzl are larger, but the divergence of consequents

is much weaker. I

This more pronounced stochasticity is the reason.why the cycle 3/1 was not chosen

initially for a detailed study of the A = —1 bifurcation.

Consider now the exceptional cases at (0,0) for (22) with a cubic non-linearity. The

first case of interest is o = 2n/3. There exist centres and saddles 3/1. One centre is at

1 +2“0: fiV—zqfi ' (80)

= .3_p_' =0 1x “1”“ , ‘1, , (8)

and one saddle at

where z is a real root of

1 .. 23'5 + (1’1‘331 + ("1‘33 + ? -— O .

(8)

Both cycles exist only for u < —% and coalesce with (0,0) as u + —%. For formulae (80)

this conclusion is obvious, for Eqs. (81), (82) it can be arrived at by examining some well-

known algebraic inequalities, but it is simplest to inspect the following numerical results

= —0.5001 x = 0.009428 y = 0 AI = 1.00001

= -0.501 x = 0.029804 y = 0 A1 = 1.00018 (81')

u = —0.55 x = 0.207352 y = 0 A1 = 1.06457 .

One point of the centres 4/1 is at

__ ' -; LL

x ’ Ti? ' ‘1 1—r ' (83)

whereas one point of the saddles is at

= 413—- : 84x ”14* , *3 0 C)

The island structure 4/1 exists only for u < U0 = O, and is generated at (0,0) when

¢ = 2n/4.

_ 22 -

N0 explicit expression in terms of elementary functions is known for the locationsof the saddles 5/1. One point of the centres 5/1 and centres 5/2 is at

(3' : LL , (85)x=0 .

where z is a real root of

m" - “M; Mr”); '1 ‘ 0- (86)The centres 5/1 exist only for u < no = cos (Zn/5) c 0.309, and they merge with (0,0) asu + no. The same property holds for the centres 5/2, except that no = cos (4w/5) z —0.809.

One saddle 6/1 is at

:0 = 1'1 871 , Eli/717% , C)and one centre, and later a bifurcated saddle, at

x=—3-:-L 21:0 (88)

where

33 ~ (14th}; +0414} -* ‘15 = 0 - (89)

1Both cycles exist only for u < no.= cos (Zn/6) = % and merge with (0,0) as u + i. For(88),(89) this is easy to see from

u = 0.499 x = 0.051588 y = 0 ¢ = 0.000358u = 0.4999 x = 0.016328 y = 0 ¢ = 0.000011 (88')u = % x = 0 y = 0

From the accumulated analytical and numerical data it is possible to conclude that inthe case of the cubic non-linearity the island structures n/r arise also from a bifurcationof (0,0) at ¢ = 2n(r/n). This conclusion holds also when (0,0) is replaced by a centre ofa cycle, provided of course the centre does not turn into a singular point traversed by(real) invariant curves. For instance, the type 3 saddle 18/3 in Table 9 is produced bythe bifurcation of the centre 6/1 when the rotation angle of the latter attains ¢ = 2n/3.In contrast to the quadratic non-linearity, the cycles 10/2 and 15/3 do not appear in thattable because the A = -l and ¢ = 2n/3 bifurcations of the centre 5/1 take place foru < 0.125.

Another more fundamental difference with the quadratic case is the fact that theA = -1 bifurcation is not unique. First there exists the already familiar centre n/r-type 3saddle n/r bifurcation, which gives off a cycle centres Zn/Zr. An example is: n = 12,r = 1:

_ 23 _

u = 0.5206 x = 0.933734 y = 0.057360 ¢ = 2.90028 (90){u = 0.5204 x = 0.933918 y = 0.057241 kg = -l.05826 ,

the centres produced being n = 24, r = 2

u = 0.52 x = 0.934716 y = 0.055338 ¢ = 1.03691 . (91)

The second A = -1 bifurcation is of the type centre n/r—centre n/r. An example is thecycle 12/1 ' '

u = 0.5660 x = 0.904441 ’ y = 0.071435 ¢ = 3.120090 = 0.5655 X = 0.904641 y = 0.071390 ¢ = 3.14028 (92)

u = 0.5650 X = 0.904840 y = 0.071346 ¢ = 3.12253 ,

and another the cycle 6/1

p = 0.135 x = 0.743725 y = 0 ¢ = 3.0269

n = 0.130 = 0.746496 y = 0 ¢ = 3.1108 (93)u = 0.125 x = 0.749219 y = 0 ¢ = 3.0860 .

This type of A = -1 bifurcation produces both a centre and a type 1 saddle 2n/2r. In thecase of Eqs. (93) it is a 12/2

x 0.799854 0 0.785630_ = y = 45 =u ‘ 0-125 {x = 0.762687 . y = 0.016234 M = 1-25904 - (94)

Since the eigenvalue 12 of a type 3 saddle may pass through a minimum, for examplefor n = 12, r = 1 one has

u = 0.5127 x = 0.939818 y = 0.053464 12 = -2.87580

u = 0.5126 x = 0.939884 y = 0.053423 Ag = -2.87614u = 0.5125 x = 0.939950 y = 0.053381 12 = -2.87599 ,

a bifurcation of a type 3 saddle into a centre (in the direction of increasing non—linearity) becomes possible in principle. For the cycle 12/1 it actually takes place 1

{u = 0.506 x = 0.943782 y = 0.050947 12 = -1.64124 (95)

u = 0.505 x = 0.944313 y = 0.050610 0 = 2.86028 .

At this A = —l bifurcation, type 1 saddles 24/2 are produced:

u = 0.505 x = 0.943231 y = 0.051506 11 = 2.22857 . (96)

A frequently encountered bifurcation with the cubic non—linearity is of the typecentre—saddle of type 1, occurring when A = +1 (¢ = 0). One example is n.= 12, r = 1

{u = 0.5295 x = 0.917631 y = 0.068195 ¢ = 0.208874 (97)

_u = 0.5290 x = 0.917794 y = 0.068151 11 = 1.37574.

- 24 -.

The points of the cycle (97) form a set of doubly symmetrical pairs, i.e. for each point

(x,y) there exists both a point (-x,y), x # 0, and a point (x,—y), y # 0. Such a symmetry

is to be expected from the general form of Eqs. (22) with a = 3. The bifurcation centre n/r-

type 1 saddle n/r generates two (distinct) cycles of the type centres n/r, with comple-

mentary symmetry and the same eigenvalue. Each newly generated cycle is only symmetrical

in y, i.e. for every point (x,y) there exists a point (x,—y). The missing x—symmetrical

points, required for over-all symmetry, are supplied by the other newly generated cycle,

so that the combined points of the two cycles form a complete set of doubly symmetrical

pairs. The two unsymmetrical cycles 12/1 generated by the l = +1 bifurcation (97) are

defined by

0.907622 y0.928938 y

XX

0.075973 _0.060483} ¢ — 1.69965 . (98)n = 0.525 {

The configuration of this A = +1 bifurcation is shown in the phase plane in Fig. 28.

The first loops of the invariant curves passing through the symmetrical saddles n/r, which

resulted from the bifurcated symmetrical centres n/r, wind around the newly generated

unsymmetrical centres n/r. .This configuration is in principle identical with that shown

to exist for the centre n/r—type 3 saddle n/r bifurcation with a quadratic non-linearity

(cf. Fig. 23), except that the role of the cycle centres Zn/Zr is now taken over by the

two unsymmetrical cycles centres n/r.

The bifurcations described above are sufficient to explain the presence of all cycles

listed in Tables 6 to 9. Whether other bifurcations exist for a very strong non-linearity

could not yet be ascertained.

A stochastic phenomenon similar to that illustrated in Table 10 for a — 2 appears

still to exist, but it is considerably weaker. A.possib1e explanation might be the rather

slow bifurcation rate of the centres 3/1 and 4/1. In fact, no bifurcation into a saddle

has taken place yet for the values of u used in Table 10:

u = —9/10 centre 3/1 x = 0 y = 7%? ¢ = 1.59480 , (98)

3 3= —9 = ———— = ———— = 1.80411 . 100u /20 centre 4/1 x ¢§§ y V7§ ¢ C )

The centre 4/1 has, however, become a saddle at u = -9/10. In order to illustrate the

resulting stochastic divergence at u = -9/10, the first 25 consequents of the two saddles

4/1

3x = ——- y = 0 11 = 69.8961Vl§ (101)

3x = ——— = ———- l = 118.113 .as y «1’9 1

are shown in Table 11. The initial values xo,yb have been rounded off to 11 significant

digits in both cases. The consequents of the bifurcated saddle have grown by a factor of

105, whereas the consequents of the normal saddle have grown by a factor of 101°, in spite

of the smaller eigenvalue of the latter.

Summarizing the results of all known bifurcations of Eqs. (22) it is posSible to

affirm that Birkhoff's picture (Fig. 1) applies to a sufficiently small neighbourhood of

-25-

centres (fixed points or cycles) not only for d) 75 21r(r/n), but also for (b = 21T(r/n), pro-

vided the exceptional case does not turn the centre into a singular point traversed by real

invariant curves. Except possibly for a very strong non-linearity, the island structure

n/r released by the crossing of the exceptional case is extremely flat, the angles between

the tangents to the invariant curves at the saddles n/r are extremely small (cf. Table 5).

Because of this flatness, a newly generated island structure is almost indistinguishable. : . . .from a continuous family of regular invarlant curves.

For a strong non—linearity the' neighbourhood of a centre to which Birkhoff‘s picture

applies may become so small that it is almost negligible. The size of this neighbourhood

can be readily estimated from a table of cycles like the Tables 1 to 4 and 6 to 8. The

borderline appears to be attained when a non—bifurcated saddle of type 1 reaches the

eigenvalue A1 z 2.

TRANSVERSE MOTION IN THE FORM A RECURRENCE

A recurrence describing the transverse motion of a particle in an accelerator or

storage ring can be found in a straightforward manner if the assumption is made that the

ring is composed of localized elements only.' Consider a transverse reference plane at some

point of the ring, and let yn, 5,“, Zn in designate the position and velocity of a particle

in this plane after n turns around the ring. If the transfer function of each ring element

is Imown, then in principle the position and velocity of the particle can be determined

, 2 can be expressed in terms of y, yn,211+ 1 n+1after one more turn, i. e. yn+1’ yn++1,

zn, zn. This expression is the required recurrence

‘jnu = G1(‘ju,‘jn ,3u.3u) , linn = G;(‘jn.'j" v3" :3?)

3N1 1' G503“ .‘iu .3" , 3M) , 3mm = Gd‘jm'Qn-S" '3”)

(102)

if all ring elements are linear, the functions Gi’ i = l, 2, 3, 4 in Eqs. (102) are

linear forms, easily obtained by transfer matrix multiplication. The simplest way of

introducing non-linear elements consists in the thin lens approximation. Such an approach

was used for the CERN PSB. The resulting recurrence, assembled in the form of a computer

program, was called the "simulated PSB". Typical distributions of consequents, fomd by

means of this computer program, are shown in Figs. 29 and 30 1°). The presence of an

island structure and the reduction of its thickness by means of a reduction of the strength

of the non-linearity are clearly visible.

Equations (102) simplify considerably if there is only one non-linear ring element,

because the linear part of the ring can then be described by the usual parameters on, B, Y

and the betatron wave number Q 19) . Suppose furthemore that the single non-linear thin

lens is at a location where on = 0, and that the transverse reference plane passes through

the centre of this thin lens. With these assumptions Eqs. (102) simplify into

4. Jib, *(3n.3u3‘3“.1 = on ‘3» "’ b1 can

4» taughgu) + %¥(flnu,3nu)

(1M1 = ”h ‘1“ + “1"!“

3m = M« Win ~ tawny)in“ = -b;, 3n + 01

3" "241%(‘301 , 3n)1' 17:30q ,Smi) I

(103)

-26..

where

b=045u2n1M , $4 = (Lt—[0,],as (104)a" = w 217%,. , bl = M away!— I w;

= a} " [as] I

and f (y,z), g(y,z) are the vertical and horizontal transfer functions of the non-linearlens, respectively. The recurrence (103) is obviously conservative for arbitrary f, g.

If the non-linear force acting on the particle is a gradient of a potential, then fand g are not mutually independent. For example, for magnetic sextupoles and oCtupoles 2»one has

+(%,3) =1 *1(%1_31) ’ $01.3)= — 2k;u} , k1 = const (10

5)

and

a 7' L ' 3 k; = cons“W = ML; *533) . 311.31 = 443% 3), 1:006)respectively.

The theory of fourth-order recurrences such as (102) is not yet fully worked out, butsome of its features are similar to those of second-order recurrences. There exist, for

example, one-dimensional solutions (fixed points and cycles), which can be classified firstaccording to the nature of their eigenvalues. By analogy with Eqs. (5), a fixed point or apoint of a cycle of Eqs. (102) is called a centre if its eigenvalues are of the form

3‘ ’ 211?;A... = e “" , x... - e - amWhen a generalized Cigala. transformation is applied to Eqs. (102), in order to identify

and isolate the dominant terms, it is found that an exceptional case occurs when“)

- 21! A- n - 4 " 4("‘11 - mu) T1 ” ("4* 241% Y [a (108), _ = 2-" 1

(W111 " ”2.23% "' (n12 ' “115%- ‘f‘ ’

where mi., ni. , hi’ i, j = 1, 2 are integers (or zero). For the particular recurrence(103), (104) one has '

Y1 = 21'1" . 7* = “‘P (109)

and Eqs. (108) are found to be satisfied when

m," + ma;z = h , (1.10).m, n, h being integers (or zero). The exceptional cases (108) thus describe the resonancesof accelerator theory.

From an inspection of Eqs. (103), (105) it is obvious that a particular solution of(103), (105) is given by (yn, 9n, 2n = 0, in '= 0), where yn, yn is the general solution ofthe second—order recurrence

gum 41‘3“ " his?“ * 12““ *1}:,(_111)

a». ' z‘b13'1 {'k‘ju + f‘iigkgflgz *jikz'svm

-27-

The same conclusion holds for Eqs. (103), (106), except that yn, yn is the general solution

of, 511.1 = www * ‘2'“? (112)

3l 3 L kv .L L + 3 M1+ a

1nd ‘ "h'ju" ‘13" 2 ' sfiu z a

Let us now show that Eqs. (111) and (112) contain only one independent parameter.

Dropping the indices in the constants a, b, k, consider first Eqs. (111) and let

, 2v“ = b‘jn + a?“ + Az- bk I3" . (113)

Inserting Eqs. (113) into (111) and rearranging yields

aan“ = 13’“ I VVH-‘l = '43“ + zafima 4' bk 5n“ . (114)

With the scaled variables

in = (HO-1» ., 17., = (hm-"3‘ (115)Eqs. (114) reduce to

_ _ _ - - _.zfirm = '0’“ , ”mm = ~‘3n 1' za'jnu "’ 'jm. , (116)

which contains only one parameter.

The same result holds for Eqs. (112). The substitution

5. 1. bigv" = by" + 0-3.. ‘ z ‘3“ (117)leads to

- '5 118)-

Z a. * bk In (‘2"1’4 = 1r" ' VMH a: \2“ 4 (301“ V?“ I

and the scaling

E“ = 153.3" , {7n = V“! ”I" (119)

to__ .. 3

-

- , ._ _ “ . 1201mm = ‘0'“ 1 “mm ~ ‘1“ + 2 a 3““ + an“ ( )

Consider now the recurrence (22) , examined in detail in Section 3-5, but to avoid

ambiguities of notation, let yr1 = nn

I: 3 ’75 +F(xH-OA)'

1““ = 4h + F(Xn\ ,““1 M

(121)

4

H1) = '11, +(1'H1 , at“ Z, 5

The ' substitution

u.“ g m + H1“) ‘ ’ '(122)

-28-

reduces Eqs. (121) to the diagonal format

.. + x .xh*‘ = “'M ’ “H‘W - ’36“ *' 2."- xfl.‘ “4‘4

(123)

From a comparison of Eqs. (121) and (116) , (120) one finds in both cases the parameter

equivalence

a,= cmZ-mv: f‘ ' —1

-29-

ested in the solution of Eqs. (128), (129) for points not too far from (0,0), then a

locally equivalent recurrence can be found either by expanding the elliptic functions in

power series or, more conveniently, directly from Eqs. (128), (129) (Ref. 22). Let us

follow the second method.

As usual it is convenient to reduce first the number of parameters by means of the

scaling

‘i‘

. 1 .

1. 3 I it : '5 t ’ (130)w

a»: ll 8

which leads to

52m + 1(2) + 6(1). x’m = o ,

+1 “T < Q‘t < (“+3.2YT 1% “=0'1‘l’... . (131)

'1 . (n«r-‘-i\'\"

S“ =-z:+4ze-{-—25£T+208-2—:-+

914 = “5-3; 14%.; _ 201%?»

9n = —6—Z—:+66%‘.—6lz-%’+-

60-5 = -é—2—-,s+66-?9-:+-

R30 = ‘64'4-g-E-Z‘S—E-E +208—g—Z-yn-

Ru =-3-2—: +24%—zo?—g Haas—£31,...

Kn =-5—59.:+66—g-: -6l2—§-}+-

R03 = 73-: +66%: —£Iz—3—;+--

Contrary to the localized octupole case (112), the recurrence (132) contains all termsin the third-order forms. The coefficients cij and di' are not independent becauseEq. (131) is conservative. The point (0,0) of (132) is a centre with the rotation angle

q» = wT . (133)The exceptional cases of Eqs. (132) can be analysed in the same way as the exceptional casesof (22). Since Eqs. (132) contain no quadratic terms, the case (1) = 217/3 is not essentiallydifferent from a centre. Applying the Cigala transformation to the exceptional case(b = 211/4, for which

4T = 31 . (134)wit is found that (0,0) is traversed by four invariant lines, i.e. it is a four-line saddle.Contrary to the localized octupole case (112) , the recurrence (132) possesses an unstableq = 2-, resonance. Such a resonance cannot be compensated by localized octupoles.

The properties of the recurrence equivalent to

2(t) + w2z(t) + k26(t)z2(t) = 0 . (135)

have not yet been fully examined, and it is not yet known whether "thick lens" sextupolesproduce a qualitatively new phenomenon.

Note: All computations have been carried out with at least 15 digits and then rounded off .

Acknowl edgment

The numerical computations have been carried out by J .K. Trickett, who has also mademany useful contributions during the analysis of critical and exceptional cases.

._ 31 -

Tab1e 1

Distribution of cycles, F(x) = 0x + (1 - u)x2 u = 0.8

Centres Saddles

k T X Y ¢ X Y 41 P1 P2

10 1 0.250525 0 0.00014 0.241022 0.036586 1.00014 -1.900 -1.893

11 1 0.541472 0.049656 0.0039 0.553787 0 1.0040 130.3

12 1 0.701273 0 0.0108 0.690907 0.042711 1.0109 -2.138 -2.018

13 1 0.784273 0.033838 0.0199 0.792378 0 1.0201 46.02

14 1 0.852882 0 0.0313 0.846716 0.025958 1.0318 -2.278 —2.017

15 1 0.889927 0.019591 0.0460 0.894553 0 1.0471 24.26

16 1 0.923875 0 0.0653 0.920430 0.014648 1.0674 -2.442 -1.953

17 1 0.942232 0.010886 0.0910 0.944784 0 1.0053 13.39

18 1 0.959826 0 0.1255 0.957941 0.008058 1.1338 1.823 2.746

19 1 0.969319 0.005948 0.1728 0.970709 0 1.1882 7.381

P2 = ‘P1 if Y =

Table 2

Distribution of cycles, F(x) = ux + (1 - 11)::2 u = 0.6

Centres Saddles

k r X y ¢ x y 11 P1 P2

1 0.270826 0.078992 0.0202 0.291725 0 1.023 12.35

1 0.636402 0 0.2323 0.612900 0.094901 1.259 -4.303 -1.402

1 0.768053 0.071259 0.5364 0.783907 0 1.686 2.570

10 1 0.866626 0 0.9792 0.855053 0.049575 2.475 7.515 —1.170

11 1 0.908344 0.033302 1.7514 0.915504 0 4.101 81.400

Centre-saddle Abifurcation 2

12 1 0.946208 ‘0 -3.618 0.941152 0.022154 7.729 1.014 -1.600

13 1 0.962248 0.014531 -12.25 0.965337 0 16.18 1.050

14 1 0.977719 0 -32.38 0.975562 0.009536 36.18 1.067 -0.9718

15 1 0.984249 0.006201 —80.22 0.985561 0 83.73 0.9816

16 1 0.990688 0 —194.0 0.989776 0.004048 196.9 0.9857 -0.9712

17 1 0.993399 0.002623 -464.9 0.993953 0 466.1 0.9751

18 1 0.996095 0 -1110 0.995711 0.001708 1107 0.9768 -0.9752

19 1 0.997229 0.001105 —2644 0.997462 0 2632 0.9768

Centres 6

15 2 0.503853 0.050169 0.0127 0.509777 0 1.013 101.62

17 0.717118 0.041262 0.1367 0.723500 0 1.146 16.41

19 0.827394 0.029627 0.5191 0.827460 0 1.664 5.995

P2 = 'P1 if y = 0

-32-o

uz

m“a.

n«a

H.H.Nm

HqH

Hoo.H

m.HoHco

IIIIIIlllllll1

lllllllllll|

EHown

6

---w~w-mmmqm-.m

Hoo.o

mags

ommoé

IIIIIIIIIIIII

IIIIIIIIIIIIIIIIIIIIII|IIIIllllllllll|n

illu

A.

owevmo

6

newNrm

é

EBmg

.0

a.onH:

Lu2

3:0

0

mam.H

New.H

on.5.»

mwm.w

ill|llllII|IIlIlIJIIIIIIIIII|III

Own4.

andm

mmpé

oo

mmoé

ruin

---:}:r------------

|IIIIIIIIIII|‘llllllllllll

«3%wwmé

ommawn

.o

mo.mn-'l-llllll-‘Il-IIL

omrén

NwN¢No

.o

«.mmm

nod

mamammd

wemam

.o

NN

3S

NK:oflm

onsfin333mm

u.8

“c

cosFor

.0

5Rmmwd

«can.o

5.8.o

cvm.wwo.oHmc

Heed

omww

ord

mumm

ite

Nm

mHmH

$4$

3o

mH

JHI

NoH.H-

mH—.Hn

mmo.HI

omo.Hu

Hwo.Hn

IIIIIIIIIIIIIIIIII1

IIIII|IIIIIIII

HHvN.H

mmm.m‘

Willlla

luu

llllllulIllln

xl||llllil

omm.N

Illllllllllllu

lllllllllll|

wrwcoo

.o

oo

3%moo

.o

o

m;

mHod

0

wwwwmod

o

bwmoao

.o

o

Illllllllllll1

||llll|lllll

IIIIIIIIIIIIIIllllllllllll

1---mwmgwmmmw-.Hmw

rmmd

mic;mmmd

com55.0

3;wand

«mmmmad

mrom

ad

mumN5

.o

:5m3.

.0

mmo«S

.o

3A...2.-

5mom:

--::mmm:--:--amn:-----N.§“-._----.m..§...--

-----.m.m..._.m...--:---s..a..:.....%...m....

36

.7

ooooo

mmmooo

.oas

mam.o

3%mmmé

mmmmao.o

fivmmmm.o

mmmawa.o

ramcwa.o

HHwfirm.o

Hawmwm.o

Hpmwmm.o

momHow.o.

H —r4

llllllllllllllIIIIIIIIIIIIIlllllllllllllj

.H

--wmm

mm

mm

dm--.-----------‘-------.------:

HH

--Hmm

mwm

m---1------------------

lllllllllllll

llllllllllll‘llllllilllllli

HH

--Hmm

mwwm

dm--l------1------------.------:

IIIIIIIIIIIIIIlllllllllll|1

|IIIII.IIIIII:

HH

wmoans

.0H

m:M:pH.3mHHH-am

..:....-

NHHH3

«AEggnog—

£3333mm

..9

3c

mNHN.Hw

wd

oomm

mm;.0

GEmam

.o

HllIIIIIIIIIIIllIIlIIIIIIIIrlllllllllllll:

«mmHNH

.oH

«uv&

han

$45

v"h

2

883mm

hwufiw

o

m.on

1aa

:Hg

+x:

umxvm

.mmHuxuwe

defipahumHm

mmHa

-33-

-----Ammwumu--.

.-----wwwuw----

:V.

3

a;

an-

------mwm.fim

mvé

m3.—

awe;

mom

é

SN

.“

mam

.“

can.«i

m:

4

Sum;

3mA

39.4

New

.“

Sm

;

mam

;

---.mmm..H8m

.“

com.“

3pm.0

amné

ah.>

:14“

NNN.

we;

mbmmm5.

new

«mm

m.:N

andam

p.

mammm

mmmm

wmwm

vowmam

2;.

Oman _2cflmNawmH~~

mmaowHgmH

-------4mfimm---------m

qmw

m-----------..,----.-----------

---mm

qmm

--.---------------------------

——

.IIIIIIINIu

IHN

INIIIIIIIIIIIIIIIIII

prod

on»ma_.o

.mnnm_u.o

Illllllu

lllrll

:3.936

oaux

pom.c.-----m----------mmm.mmmqm--

o«fig

boo.oe

:wpmwd

a5m.

«1.x.o

om

gmfiw

é

ampruoé

we.mmmd

cam21.0

9.3m

ad

-

mmmomo.o

naa~«v.o

-----Nw------.--¢wwmm

mm

mm

w--o

”#0Nvm.o

0man

Nwwd

pmommm

.o

o«mo

owed

.0

«mmw

and

-----#w-‘--------mmmgwmwmm--o

«wefind

0nbw

pawé

0:0:

0as.

mé

om

;B

ad

mmv$

86

mnwowmd

05mm

mead

mamvm

od3:.

$5.0

oNo"

Nmmd

.ommwwwm--

omoo

cum.c

«3.N2

.c

run}-..unntnuu|unuuuu|=|unuu

33

.31

:}-..

o.z~oA9v

ova;

.-----mmmmn--.----m”mmmu--

I .. .. IMHuumnHHxu .

mw¢n.o

IIIIIIII|IIIIrlIIIIIIIIIlI.

a .mI

-.---mdmmmu--:3

4

mwmfi.o

:bowl

floam-~H+H-

«.mmw-m.#n.-om

dmu

mumJ—u

5ocd

--~mmnmwwdm----Mmmmwmmdw----wmm-mmmqm--

o

m5

35.0

new2

w .0

gorem

d

mmocmo

.o

a;woo...

o

--mmmwmwmmm-1-----~m

-----.

omwSod

o

um»N

R6

cancan

.0

mar$3.6

mamemmd

moom

rwd

wmam

mvd

.36~

36

2:.nerd

woa05.0

3L.and

mamSad

«mawand

omachad

2wmmmd

Onowand

aweand

I:«and

IIIIIIrIIIIII:

mMN

fl..1

.Cum..1.

a.a _a a

na —

mr d

m--MM---.

m--NM---.

mm:

mm~

m«a

vL«A

Ax

2.65

han

mwfiuvam3

.63

mi

m3.3

.82

3c

mNH.on

a«Na:

IHg

+.x:u

nxvm.moHuxu

wesoapznfinpm

fin

¢mFk

_ 34 _.

ou

kMa

Hm:u

«A

3838o;

0$

35

5266

o2

.53

6-

§5

33m80;

o323.0

256o

$25.0-mm

2.

32.”.08;

oEMSN6

08.0o

83

5.?

an3

5::8o;

o$

53

5805

o3

35

6..

a3

mm”;8o;

o£

33

5266

o3

33

6-

em3

5380:

o$

88

6086

o3

3%

.?2

E.

.8380;

o8

58

6806

a8

32

6-

S2.

«new:5

;o

8:3

6:56

o8

3%

.?S

S

3:N

8;e.

82.36~86

o8

23

.?E

3>53

25;o

“agate30.0

o5

38

.?E

av.2.”

moo;‘

.o$

32

6256

o«3256.

E5.

film«

a;

o«8936

Seao

$8

56

.2

3

ma:

m2;

o2

83

5£55

0$

32

.?2

33.5

28

;.o

28

36

93.6o

Nmmwmma-2

33.3

go;o

22

35

356.o

28

36

-.2

S3

;“3

:;o

88

35

926o

“.33

”?a

g2.2

«an:o

33

35

$3

.o

3.2

3.?

m2

:56a

:o

Saw

s;936

o2

32

6.

p3

39mm

a;0

23

3a

N5

6.

a3

35

.0.

m3

03

;gm

.“o

mmvmmvém

2;9.

22

2.0

-m

a9

:;flag.

o.8

33

6mmo.~

05

52

.6-

45

385

gmé

o£83

.0Ems?

oSam

mie-

m2

53.0

$9.:0

$1.5m.o

3N.:..

o8

33

5-

Nm

.504m

82:o

$33.0$

22

-o

3:2

5-

Hm

‘oV

a:4.

hx

«x:3

A?m

xa

x

mm

qoafim

m9?

momagma

- 35 _

Table 6

Distribution of cycles, F(x) = ux + (1 - u)x3 u = 0.8

Centres Saddles

k r x Y ¢ x y A: p: pz

10 1 0.244854 0 0.00034 0.233105 0.044088 1.0003 1.840 -1.826

12 1 0.637054 +0.075703 0.0289 ' 0.655575 0 1.0293 49.85

14 1 0.802387 0 0.0807 0.788431 0.059665 1.0841 -2.339 -2.030

16 1 0.870022 0.042294 0.1525 0.879650 0 1.1647 18.89

18 1 0.924653 0 0.2553 0.918194 0.028817 1.2904 -2.749 -1.999

11 1 0.516110 0.037849 0.00003 0.520909 0 1.00003 33095

13 1 0.737916 0.034151 0.00069 0.741984 0 1.0007 2766

15 1 0.843619 0.024989 0.0032 0.846531 0 1.0032 812.1

17 1 0.903045 0.017232 0.0099 0.904969 0 1.0100 312.0

19 1 0.938721 0.011578 0.0262 0.940079 0 1.0265 129.2

P2 = -p1 if y = 0

-36-

mhcucwo1

35

93

a03

had-n,

ulllllllb

lllll

H:

IIIIIIIIIIIIIIII

o.58

v.mm

om

aHU§m

illllllllllll:llllllllllllllll

H:

-------mHmNH--HmooH

luxulumxu-uJu

o

Sm

muum

9.:US$50...“

u0.52

OH

>go

o

.3920d.---HmmmmHHNm-

chmmmw

.0

Has.0

Hmooé

---wmmgwwmmm-4Illllllllllla

.new

HmHN.o

nx

hmngHo

o.0

MH“A:

u«m

nuunuuuuuunuujN

aH

NpH

mu2

3:0

0

oa«.Hmmu.H”mm.“mum.N

-------mmmum----Nom.H

IIIIIU

IIIIIIIL

IIIII

IIIIII|'Il|

HV .mmam

Hdraw

-----HHqmm-“$

.m

------WNNWHW---------mmqmm--

-------~mmwum-w

ho;

oonxuuunm

unuunu

nunnuumunnuux

luuunnmxuuauu

o

mmm.m9

o

cowHam

.0

mamEb

.0

ampm

wmd

mmHown

.0

o:mrm

.0

m5.wwa

.oH

illiuu

on

uu

n-

..u..!...--.i---

1llllllllllll

wHH‘«mm

.o

mom0

5.0

mi.

map.o

Hedw-

llllllllllllI;

:5as

.oIIIIIIIIII

oroHam

0

HmH

HmH

HrH

HrH

HpH

HmH

NAw

HU

Udm

Ip350

coflmousHHmH

mo.~

-----mM

www-mm

>0.0

mampao.c

---mmmwwwmum--2%

:6.o

vmwrmm

.o

vnew

.0

marHap-o

H2

HHH

lIlIIIl

Ha

1--.N.NN-..N.-.:------Hmm”mu---

NumH

NIX

:

------m..mN.-.:--:NNN..N-

-------N.N.N..N.----Hmmmmmwdm--------m

------

mbmQ

:.o

o

5::5

.o

r;m

.oru

n---in

...-i-

Illllllllllll

0mmova

.o

mmm«om

.o

mwmour

.0

wmm0mm

.o

NEVH-

mo.mmn

newmac

.oHwo

mum.o

mmmwmod

wowrwa

.o

HwH

H2

z2%

amI

.3“c

:oHHmOHEHmH

mmHm.H----m

mm

$m--

NNNm.ommwm.o

rlllllllllllln

IIIIIIIIIIIII

rlllllllllllln

IIIIIIIIIIIII

o

---mmm.mmmum--a

mmmmmH

..o

mamawa-o

mHmmwmd

HmvHHS-c

mNHmmmd

rlllllllllllll.

rllllll|llllll.

H.HuH

HNH

HoH

lllllllllllll,

Hw

MmrA

%x

mum

Na

x

mEumm2

35

0

06

":mam:

IHg

+x:

nHxvm

.moHuxu

moQ

OHuaHum

HQ

namH

_ 37 _

23

auwwm

upsfinan

AS:

IIIIIXlIIIx

335mmfivcoam

otoo505a?

mannaE

gamma--------w

$..w-:---

.8.2

-------m..wmm-.----co—

cm

-

-

---

-«5.4

-

moooco.H.-----...,..mo...m:.-----.mmm..m--------.n.3...fi.-.

booooofl

E

--§.mm.m...m-. N—.5250m:

2532

.33mfluudm

3:he

08%“

3oz

Inuuuunmuluvnn

ram«mm

.o

wmm£6

.0

newHE.

.0

$6356

OH

%

idmmmmmumt-----m$m..m--Eiflmmum:

«m50

IIIIIII

Secs.3

HOO

Hx

m2.3m

.0

-

-.mmm._._.w...o.

-

{4%.

“We:

-

5Nammo:

mgnmm

.o

IIIIlllllllllnllllltlv

lllg

.Hmfifio

oao

munE:

u“A.

----m--:-m

ml;

----m--.:m

::N

EN

SN

2m

M:

Ar:3

5O

rma.H

Jami:

mm

n.~

mam

;

Sim

m.rmmmw

mmm

b.NQMN

ruin

:

mam

a”

ownmvm

.o

2m8a

.0----P.m.m

wwwfli---.o.ew.mwm..m--

m5»vmm

.o

-:-.p...a.m.fi..:N

ova-

rIIIIIIIIIIII.

:MmmMmmum:Nun

wuwé

un

ulllm

ulu

lu

o

«A5

35

.33

3w

EU

dmuqw

«.54.

:23

.man

.Hrlllllllllllll

:0;

ram:w

.o

wmrmom.

.0

3mm.0

move.0

mwv1.0

.6

canmmoé

Emas

.0

mmmmmm

.o

93

52

3.

mm

mam

;

E:g:Nm

moé-

adocm

admom

.-::.

-38-

Table 9

Distribution of cycles, F(x) = LIX + (1 - 11)}:3 11 = 0.125

Centres and bifurcated saddles Saddles

k r x y ¢ or A x y A1 P1 P2

5 1 0 0.499598 ¢ = 0.2693 0.506708 1.3059 8.862

7 l 0 0.719507 A2 = -33.64 0.872819 38.938 2.116

9 1 0.900774 0 A1 = 1398 0.962584 1368.4 2.271

9 1 0.976489 0 A1 = 678.811 1 0.902004 0 A1 = 3900011 2 0 0.617970 ¢ = 0.7666 0.606154 2.0807 2.744

13 2 0 0.674263 A2 = -251.1 0.863099 320.45 2.139

15 2 0.875734 0 A1 = '4999 0.894221 9990.5 2.150

15 2 0.972147 0 A1 = 263.317 3 0 0.643999 A2 = - -2.95219 3 0 0.660568 A2 = -2667 0.862151 0 3490.7 2.144

6 1 0.749219 0 ¢ = 3.0860 0 0.654654 11.511 -0.2950

8 1 0.895574 0 A1 = 272.210 1 0.980539 0 A1 = 7353 0.960505 0 9477.6 2.255

12 1 0.902042 0 A1 = 2071312 2 0.799854 0 ¢ = 0.7856 0.762687 0.016234 1.2590 0.0053 -6.091

14 2 0 0.682554 A1 = 1499.616 2 0.977099 0.005240 A2 = -2227116 3 0.584123 0 A1 = 2.49618 3 0.633322 0 A2 = -40.2020 3 0.865464 0 A1 = 11012

p2 = -p1 if y = 0

-39-

Tab1e 10

Quadratic non-linearity. Stochastic instability, growth rates of consequents

(a) Type 3 saddle 3/1 (b) Type 3 saddle 4/1u = -9/10, 12 = -29.73 H = -9/20, 12 = -54.47

n xn Yh Kn Yn

0 -0.360228 0 -0.461915 0

1 0.570757 I 0.465497 0.517241 0.617087

2 0.570756 -0.465476 0.772260 0.000000

3 -0.360196 0.000081 - 0.517241 -0.617087

4 0.570603 0.465271 -0.461915 0.000000

5 0.570345 '-0.465856 0.517241 V 0.617087

: 6 -0 . 361108 . 0 . 002411 0 . 772260 0. 0000007 0.575168' 0.472012 0.517242 -0.617087

8 0.582915 _ -0.454190 -0.461914 -0.000001

9 -0.333213 -0.072065 0.517239 ‘ 0.61708410 0.438786 0.304118 0.772254 -0.000007

11 0.275024. -0.542295 0.517224 -0.617100

12 -0.646404 1.10063 -0.461946 0.000073

13 2.47629 10.0685 0.517370 0.617254

14 19.4907 701.767 0.772562 0.000411

|x2°| > 101“ lol < 0.55 |y2.[ > 7- 10“

P2“= 'P1 if y = 0

- 40 _

Tab1e 11

Cubic non-linearity. Stochastic instability, growth rate of consequents

(a) TYpe 1 (b) TVpe lbifurcated saddle 4/1 saddle 4/1p = -9/10, 11 = 118.1 0 = -9/10, 11 = 69.90

D X0+2n y0+2n xo+2n yo+2n0 0.688247 0.688247 0.6882471 —0.688247 -0.688247 -0.688247 1.6 x 10'112 0.688247 0.688247 0.688247 -1.3 x 10'103 -0.688247 -0 688247 -0.688247 1.1 x 10-94 0.688247 0.688247 0.688247 -9.4 x 10-95 -0.688247 —0.688248 -0.688247 7.9 x 10-86 0.688245 0.688244 0.688247 —6.6 x 10-77 -0 688272 -O.688285 -0.688245 5.5 x 10—58 0.687973 0.687838 0.688227 -4.6 x 10'59 —0 691214 -0.692720 -0.688079 3.8 x 10‘”

10 0.654994 0.642508 0.686841 —3.2 x 10'311 -0.943167 -1.33218 —0.676519 0.02612 -29.3874 -48192.7 0.592521 —0.18

p2=-p1 if y=0

1)

2)

3)

4)

5)

. 6)7)8)

9)

10)

11)12)

13)

14)15)16)17)

18)

19)20)21)

22)

. —41-

REFERENCES

H. Poincare, Collected papers (Gauthier-Villars, Paris).

G.D. Birkhoff, Collected papers (American Mathematical Society).

A.A. Andronov and L. S. Pontryagin, Doklady Akad. Nauk 14,247 (1937). This theoremcan also be found in any treatise on non-linear dynamic systems [for example, inA.A. Andronov, A.A. Witt and S. IG1aikin, Theory of oscillators (Pergamon Press,NY, 1966].

E. Picard, Legons sur quelquesléquations fonctionnelles (Gauthier-Villars, Paris).

I. Gumowski and C. Mira, submitted for publication, 6th Int. Conf. on Non-LinearOscillations, Poznan, 1972.

S. Lattés, Annali di Matematica (1906), p. 1.

J. Hadamard, Bull. Soc. Math. Fr. (1901), p. 224.

I. Gumowski and C. Mira, Proc. 8th Int. Conf. on High-Energy Accelerators, CERN (1971)(CERN, Geneva, 1971), p. 374.

Ph. Rabinowitz, Numerical methods for non-linear algebraic equations, NumericalAnalysis Acta (Gordon and Breach, NY, 1970).

~L.J. Laslett, E.M. McMillan and J. Moser, Long-term stability for particle orbits,Courant Institute, New York University Report NYO-1480-101, July 1968.

I. Gumowski and C. Mira, CR Acad. Sci. Paris, 174, ser. A, 1271 (1972).

B.V. Chirikov, "Research concerning the theory of non-linear resonance andstochasticit ", Novosibirsk preprint No. 267 (1969); CERN Trans. 71-40 (1971).

AM. Liapunov, Annales de la Faculté des Sciences de Toulouse, Vol. 9 (1907).French version of a paper published in Russian in 1892; reprinted in French byPrinceton University Press in 1947 (Monograph 17 of the Annals of MathematicalStudies).

T. Levi-Civita, Annali di Matematica (1901), p. 221.

A.R. Cigala, Annali di Matematica (1904), p. 67.C. Mira, private conununication (1970).

V.N.Me1ekhin, Zh. Eksper. Teor. Fiz. 61,1319 (1971), and Proc. atth Int. Conf. onHigh-Energy Accelerators, CERN (1971) (CERN, Geneva, 1971), p. 377.

C. Bovetl, G. Guignard, K.H. Reich and K. Schindl, Proc. 8tth Int. Conf. on HighEnergy Accelerators, CERN (1971) (CERN, Geneva, 1971), p. 380.

ED. Courant and H.S. Snyder, Ann. Phys. §, 1 (1958).

C. Mira, CR Acad. Sci. Paris 273, ser. A, 1727 (1971).

E.A. Crosbie, T.K. Khoe and R.J. Lari, IEEE Trans. Nuclear Sci. NS-18 (1971), p. 1077(Proc. Chicago Accelerator Conference, 1971).

L. Pun, Thesis, Université Paul Sabatier de Toulouse, June 1971.

_:__E_igur‘e.4 _Phase plance picture according to Birkhoff

The center 0 is first surrounded by a family of regular invariant curves, thenby one or more island structures, defined by the closed singular invariantcurves F1, F2 passing through saddles of type 1, and finally by one or moreinstability rings, defined by the intersecting singular invariant curves emanatingfrom the saddles of type 1 like Si, Si, 35. These intersecting invariant curvesare limited from below and above by the closed regular invariant curves T3 andP», respectively. Inside the instability ring there exist closed regular in-variant curves surrounding centers Ci, C5, 0%.

FABtAIION ““552

41

L;

“330.9.a:

ug

x9:3%

?mmaohu

weacausnwuuwfim

.NNAJIS

+x:

nCam

w.c

n1

"N.mfim

EmNxt

{mNR:

:5+2

#

md|

C>

‘

.33

50

omfiuwmm

xwm.~

u~&

"no.Hvmwfiumo

moaoflusnfluumfia

Nxfinlfiv+

x:u

Axvm.

0.0u

an

m.wa%

(E:m

NB,to

NB—Na—

dL

“wuaw

uo

mawwmmx

No.Nu

H&"Ao.Hvmmfiuhu

monewuanwuum

«nNNAJIHV

+x:

nAxvm

m.ou

1"

......... Il:m

>u:oufimaum>afi

Haanwwm

£9

EE

EN5

3”2.:

Nacx

.or

aa

”no-amdfl

wa

«oa

”/34, ,.

..,

/ /’..

/a

XXX

A””v

_\‘.

oo

oq.m

amusmoo

waumwmx

H&"AO.HV

.mmaomomo

somuanm

uuwfln

.mflé

ua

um

.mNxflalfiv

+x:u

Axvm

ll}cx

.3.x

momxo/

a;3:38.

.9x

33:5awax

mowuxo

kn«83592500

9.82

3550F

ag

in

mmmfiwm£963»

mfimmamam

inoEfin

gfi

an:

_\\v

.V

\IIa

o9..

\\

~\

\m

1/o

\~

.fi/

~\\J

.vU/

r\\_

.fi/r,,

RD

»):“.7

1 ,4.mo

/o

he2%

a”Add.“

mo«Eon

oficflooumum

sho

“Eon

omfifioofiom

Ci

:N—C:

:9

..w

*a

0..ad

_0.0

{oNd

0N61

.VduQOI

>ta

ttrLz

I...

__

.3_

.l

\x1c

lI

lC

Ct

a».;

x‘

+,

,cxx

\x

vx

o,

x,.

x

..x

ov.0!

ax

a,no

u1

yrg

.e.x

x.

xa

xu

xx

.,.

1,

Rx

xNo:

xi

\.

,.,

ia

~r.

iL

xx

I

x:

x..

ir;

.n

Xm

0]x

..r

x‘

K,

.,i

:x

.x

x__

xe

. xa.

._.»x

:7x

x“n

x

mdl

knmoo

C>mfiwvmm

x

wm.~n

HKHAO.HV.m

wauhumo

coaunfimuum

fim

$3

-:+

x:u

3mm6

ua

nm

.wEA

{0—Cm

N2.—:0

.::yo

cx

umufiwoo

mawvmmx

om.mu

HAHAO.HV.mmHu%o

mocoausfiwuum

flcm

xfinlav+

x:u

Axvm0.0

u1

nm

.afikc>

cx22m

Houfiwoo

333mm

x

HK“A

o.H

v.m

mH

u%u

moG

oausnfiuumm

n

mxnllav

+N:

HAxvm

..m

.o,n

1 E."OH

wwflm

map

5.0I

:>

(anJG

N\Z

Jm

me2

asme

mo50

wdmo

«.0no

No8

o

$5

d0

wficwmmxHm.m

n"Kuflohfiv

.mmfiozuwe

aofiusafiuumfloc>

mxfianflv+

x:u

AxvmmNH.o

ua

"HH

.mflmA

omdwho

mfio>50

whomfio

«Nomud

N50F50

obo_

_[Ix

__

__

__

_

INQ

o

£906

lo

.1as

INFG

Ivno

mo.Hu

~&.H\m

mfiuzo«w

allav+

x:u

Axvm0.0

u1

"NH

.wahI

96

c>

m~.Nu

"‘.H\m

ofiumo«“2113

+x:

uAxvm

5m.ou

1u

MH.ofim

.Rd

:>

I8.0

Jmod

-oI9.0

1N8

Iid

Im8

cxbad

.mac

mmoIII/1

“\\\\\\\Nod

_moone

macmba

spa_

_/\

__

._

__

o

1mod

I.56

1mod

Iwed

19d

1mg

I3.0

Nm.~u

”x.H\m

mHvNxfiaufiv

+x:

uAk

nm.eu

au «mnqwmm

,.

C>1

9.0

ow.mu

TA

NxA:-HV+

x:n

Axvg.H\m

m