The one to two-hole transition for cantori

PHYSlCA

Physica D 71 (1994) 372 389 North-Holland

SSDI: 0167-2789(93)E0300-Z

T h e o n e to t w o - h o l e transition for cantori C. Baesens and R.S. M a c K a y 1 Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Received 2 August 1993 Revised manuscript received 11 October 1993 Accepted 12 October 1993 Communicated by J.D. Meiss

The gaps in a cantorus come in orbits, which we call "holes". In the space of parameters (a, b) for the "two-harmonic" reversible area-preserving twist map family, a b Y' - Y ~-v sin 2wx - ~ sin 4~rx, x' = x + y' (mod 1), application of the idea of the anti-integrable limit establishes that there must be one to two-hole transitions for cantori of all irrational rotation numbers. We have numerically located a curve in parameter space across which a one-hole cantorus of golden rotation number develops a second hole, and we present results on scaling behaviour of several quantities near this interesting transition.

1. Introduction C a n t o r i are the r e m n a n t s of i n v a r i a n t circles for a r e a - p r e s e r v i n g twists m a p s after they have b r o k e n . A l t h o u g h they do n o t s e p a r a t e phase space, the flux from o n e side to the o t h e r is l i m i t e d to pass t h r o u g h so-called " t u r n s t i l e s " in the c a n t o r u s [22]. I n this p a p e r we r e p o r t on n u m e r i c a l results a b o u t a b i f u r c a t i o n in which a c a n t o r u s gains a s e c o n d turnstile. A cantorus for a h o m e o m o r p h i s m f of a m a n i f o l d is a n e m b e d d e d D e n j o y m i n i m a l syst e m (see e.g. [1]). We c o n s i d e r C a area-preserving twist m a p s , that is m a p s f of the cylinder to itself with lift F : (x, y) ~ (x', y ' ) of the p l a n e to itself satisfying

OX~ >0, Oy

detDF=l,

F(x + 1, y) = (x' + 1, y ' ) .

Such m a p s possess a generating function h, such that

y' dx' - y dx = dh(x, x ' ) .

(1.2)

W e s u p p o s e that f is exact, which is e q u i v a l e n t to

h(x + 1 , x ' + 1) : h ( x , x ' ) .

(1.3)

In this c o n t e x t , the rotationally-ordered c a n t o r i play a special role, a n d we will restrict a t t e n t i o n to t h e m . T h e y can b e d e s c r i b e d c o n c r e t e l y as i n v a r i a n t sets with lifts of the f o r m {(X(0), - h , ( X ( O ) , X(O + w ) ) : 0 E N---},

Nuffield Foundation Science Research Fellow 92-93.

(1.1)

(1.4)

for some i r r a t i o n a l n u m b e r o) a n d s o m e increasing f u n c t i o n X (called its hull function), satisfying

0167-2789/94/$07.00 © 1994-Elsevier Science B.V. All rights reserved

c. Baesens, R.S. MacKay / The one to two-hole transition for cantori X(O + 1 ) = X ( O ) + 1, whose right and left limits

exist everywhere, but are not always equal. Here subscript 1 denotes the partial derivative with respect to the first argument, and ~_+ denotes the set of left and right hand limits of real numbers. A gap in a cantorus is a pair of distinct points of the cantorus whose forward and backward orbits converge together. For the cantorus (1.4), they are the images of points 0 - , 0 + for values of 0 at which X has a discontinuity. The gaps of a cantorus come in orbits: if g is a gap then so are f ( g ) , f - ~ ( g ) . We call an orbit of gaps a hole. Aubry calls it a discontinuity class [4]. The number of holes in a cantorus is countable (the easiest proof is that for a Denjoy minimal system the total length of gaps is bounded). The cantorus generated by breaking a circle in the standard map, for which k

h ( x , x ' ) = ½ ( x ' - x ) 2 - 4 - ~ cos 2wx ,

(1.5)

appears numerically to have just one hole, and this can be proved for large parameter, e.g. [25,6]. Examples with two holes have already been observed numerically, e.g. [4], and in systems with 772 symmetry, the number of holes is always even. Aubry and Abramovici [3] introduced the idea of the "anti-integrable limit", which corresponds to generating functions of the form h(x, x ' ) = V(x) .

373

systems for which V(x) has two wells per period, and we proved that there must be many bifurcations of cantori on passing between double-well and single-well regimes. In particular, there must be a bifurcation in which a one-hole cantorus gains a second hole. In the present paper, we demonstrate this phenomenon numerically in the two-harmonic family y' : y + V ' ( x ) ,

(1.7)

x' = x + y ' (rood 1), where V(x) = (a/4~T ~-) COS 2VX + (b/16~r 2) cos 4~rx,

(1.8) which has generating function h(X, X') = l ( X ' -- X) 2 + V ( x ) .

(1.9)

We obtain the cantori as limits of sequences of periodic orbits, and we investigate the scaling behaviour near the transition. The plan of the paper is as follows. In section 2 we describe our numerical procedures. Details of the methods we used for obtaining symmetric periodic orbits and their residues and phonon spectrum are given in the Appendix. In section 3, we describe our results. In section 4, we prove analytic bounds in parameter space on the 1-2 hole transition. We conclude with a discussion of the results in section 5.

(1.6)

An example is the standard map at k - - ~ , after scaling the generating function by k. For systems with generating function close to an anti-integrable limit, they showed that one can obtain strong control over many types of orbit (see also [25]). In particular, it gives one great control over the rotationally-ordered orbits [6]. Using the concept of the anti-integrable limit, in [6] (also [17]), we proved existence of many one and two-hole cantori for families of maps near double-well anti-integrable limits, that is

2. Numerical procedure Given a mapping f of the cylinder to itself, isotopic to the identity, choose a lift F to the plane. For example, for the two-harmonic map (1.7, 1.8), we obtain F by ignoring the "rood 1". Then a cantorus for f has a rotation n u m b e r , given by co = lim (x,, - Xo)/n , for the corresponding orbits of F.

(2.1)

374

C. Baesens, R.S. MacKay / The one to two-hole transition for cantori

We find cantori as limits of sequences of rotationally-ordered periodic orbits whose rotation numbers converge to the desired rotation n u m b e r to. For each periodic orbit there is a smallest pair of integers ( p , q) E 77 x ~ such that

Class 1: q e v e n ,

x0 =0,

Class 2: q e v e n ,

x 0 ~-

Class 3: q odd,

x0 = 0,

Class 4: q o d d ,

x0

--

Xq 2=p/2, X1,

Xq/2+l ~ p

X(q+l)/2 =p

1

for the chosen lift F. We say the periodic orbit has type ( p , q). For rotationally-ordered periodic orbits, p and q are automatically coprime, so its type is completely specified by its rotation number p / q , which we will always write in lowest terms. We chose ,o =

,o0 =½(3

~,/~)

-

=

~,

-2

,

(2.3)

where - 1(1 + X/5)

(2.3a)

is golden ratio, and we chose the sequence of types ( p , , , qm), m >--2, with =

Fm-2,

qm = Fro,

(2.4)

where F m are the Fibonacci numbers defined by F0=0,

F,=I,

Fm+l=Fm+Fm_l.

(2.5)

The two-harmonic family is reversible with respect to the involution R, x' = -x ,

y' = y + V'(x) .

(2.6)

This means that R conjugates f to its inverse. It follows that a periodic orbit is either s y m m e t r i c , meaning that it is sent to its time-reverse by R, or a s y m m e t r i c , in which case it has a distinct p a r t n e r given by applying R to it and reversing time. It will be enough for us to find symmetric periodic orbits. Symmetric periodic orbits can be classified into the following four classes (up to choice of origin of time and integer horizontal translations) (e.g.

[9]):

X(q l),'2.

(2.7) We use the notation " p / q ( c ) " for symmetric periodic orbits of type ( p , q) and class c. To find a symmetric periodic orbit of given class and type, we use a Newton method in the space S of sequences ( x n ~ , . . . , x , 2 ) , where n 1 and n 2 are given by table 1. The action functional W" ~ - - ~ ~ is "defined" by W ( x ) ~-- Z ~X (n+l nE~7

I - Xn

)2 + V ( x , ) .

(2.8)

Orbits of F correspond to critical points of W, that is sequences for which OW

Prn

1

(2.2)

Yn+q = Yn

1).2 '

X(q

2 ,

X(q+l)/2 = p +

x,,+q = xn + P ,

--Xq/2 ,

c)X n

= 0

for alln E77.

(2.9)

Using (2.7), this equation induces an equation on S (details are given in the Appendix). We find the solutions using Newton's m e t h o d on S. We use rotationally ordered periodic orbits. These are ones for which for all n, k, m C 77, x,, < x k + m iff x,,, ~< x k + 1 + m .

(2.10)

To obtain them, we use a result of [6] which states that near an anti-integrable limit, all rotationally ordered orbits are continuations of a class of anti-integrable states which we call w e a k l y rotationally ordered.

In the limit as a---~ +oc, with

Ibl < a ,

there are

Table 1 Lengths of sequences required to determine a symmetric periodic orbit. Class

nt

n2

1 2 3 4

1 1 1 1

q/2 l q/2 (q - 1)/2 (q 1)/2

C. Baesens, R.S. MacKay / The one to two-hole transition f o r cantori

precisely two rotationally-ordered periodic orbits of each primitive rotation type (p, q), and they are given by the following hull functions: Class 1 and 3: x n = ½ + Int(np/q)

if np / q ~ W_,else np / q , Class 2:

x n = ½ + Int((n - ½ ) p / q ) ,

Class 4:

x n = ½ + Int(np/q + 3).

(2.11)

H e r e Int(x) denotes the greatest integer less than or equal to x. Note that the class 1 and 3 orbits have one point per period on a maximum of the anti-integrable potential, whereas all points of class 2 and 4 orbits are in minima. Using [6], it follows that for Ibl < a and a large and positive, the limit set of a sequence of rotationallyordered periodic orbits with p/q---> o~ irrational is a cantorus if only classes 2 and 4 are used, and the cantorus plus a principal homoclinic orbit if class 1 or 3 orbits are used. For la[ < b , b - - - ~ , there are 2q symmetric periodic orbits and 2q asymmetric ones of type (p, q) (see [14]). In this paper we will be interested in the symmetric cantorus (that with = 1 in the notation of [6]). This can be obtained from the limit of periodic orbits with the following hull functions at the anti-integrable limit: Class 1 and 3: x m = ,~(np/q), Class 2:

x, = ~ ( ( n - ½)p/q),

Class 4:

x, = .~(np/q + ½).

(2.12)

Here _w(0) = ~ + Int(0)

i f 0 < {0} < 31,

= 1 - ~: + Int(0 )

if ~1 < { 0 } < 1

=0

otherwise,

(2.13)

where ~ is the unique solution in 0 < ~ < 31 of cos 27r£ = - a / b , and

(2.14)

{0} = 0 - Int(0).

375

(2.15)

The limit as p/q---> oJ irrational of class 2 orbits is a 2-hole cantorus. If class 3 or 4 orbits are used then a principal homoclinic orbit through x = 0, respectively x = 3 , is also obtained. If class 1 orbits are used then both principal homoclinic orbits are obtained in addition to the cantorus. Near the anti-integrable limit, all orbits of "bounded type" (in particular, all rotationallyordered ones) have locally unique continuations [3,25]. They can be continued until D2W becomes degenerate. Periodic orbits can be continued further in the class P of periodic sequences of the same type, namely until D2WIp, the restriction to P, becomes degenerate; this is equivalent to residue 0 [8,20]. To continue symmetric periodic orbits in the space S, it is only D2W[s that needs to remain non-degenerate. We continued orbits along two main types of path in parameter space. The first was from large positive a down a line of constant positive b, starting from an anti-integrable solution of form (2.11). The second was from large positive b along a line of the form a = Kb with 0 < K < 1, starting from an anti-integrable solution of the form (2.12), and this turned out to be more controllable. The steps in parameter space were chosen so that the expected step in sequence space took some pre-assigned value, e.g. 0.01, using supremum norm. For the log-log plots near the 1-2 hole transition, however, we took equal steps in log(a - a c ) , e.g. of size log(I/0.7). We evaluated the determinant of the antisymmetric part D2WA Of D2W for the periodic orbits (defined in Appendix A.2), and found symmetry breakings (det D2WA = 0) for classes 1 and 4 (the ones which have a point on x = ½). We located the symmetry breaking points by using the secant method on det D2WA. We also evalu1 ated the widths of the gaps about x = 0 and 7, and total length of the holes generated by these two gaps, the symmetric and antisymmetric phonon spectra, and the residues of the periodic orbits.

376

C. Baesens, R.S. MacKay / The one to two-hole transition for cantori

3. Results

M o s t of our w o r k was d o n e on the line a = b/4. We found it necessary to w o r k in quadruple precision (34 decimal digits). Figure 1 shows the set {{x,}: 0 ~ n < 144} for the periodic orbit of type 55 / 144(2) as a function of p a r a m e t e r 2 t a n _ ~ / ( a 2 + b2 ) ,n.

(3.1)

along the path a = b/4 >-O. T h e virtues of the choice (3.1) are explained in [6]. N o t e that r = 0 is the integrable limit (a, b) = 0, and r = 1 is the

anti-integrable limit of the path. R e g a r d i n g this as a good a p p r o x i m a t i o n to a cantorus of rotation n u m b e r w0, we see that as r decreases f r o m 1 to 0, first one orbit of gaps shrinks to zero and is annihilated and then the o t h e r orbit of gaps shrinks to zero. T h e second transition is the familiar circle to 1-hole cantorus transition of [16]. T h e first transition is the one of interest here. T o locate the o n e - t o - t w o hole m o r e accurately, we found s y m m e t r y b r e a k i n g bifurcation points (a.... bin) for each type (F,~ 2, Fro) (m >~ 4) (in class 1 for even period, class 4 for o d d period). T h e results are given in table 2, w h e r e

1.0

(

anti-integrable limit

2-hole cantorus 0.8

< 0.6

j/i

i

ii,

i

i

ii

1-2 hole transition

i i!-~,

li

t

I~

I-hole cantoms

<

!i!!!f!!ii!!i! !!!! !/2!j!j!!!!!!!!ii!i,!! if!

circle - cantorus transition

';" 2 2 2 222"" 2"'2JJ2"2;'2"222222222J22J:'22J:I:::::2:::2::::::J2:2ZJ2J2:2:222J222J22222222""'2;22"22222222222J:22::::2:::::::::::::::::::::2::2""

circle

1::::222::2:222~222222.22222::2:2:22~i~i~i~.i~.~i~i~.:::::1::::::::::::::::::;:222222:2222::122:2::::i~i7~.~iiii.~i...~;.~.iii~i~

integrable limit 0.0

0.2

0.4

0.6

0.8

1.0

x Fig. 1. Projection onto the x-axis of a 55/144 (2) orbit, which can be e ~ e d as a good approximation to a cantorus of rotation number w0 = ½(3 - X/5), as a function of parameter r = (2/~r) tan ~Va 2 + b 2, along the line a = xb > 0, with slope x = ¼, in the parameter space of the reversible 2-harmonic family. Note that at the anti-integrable limit all the points are in the two wells x = -+(1/2"rr) cos ~x of the potential.

C. Baesens, R.S. MacKay / The one to two-hole transition for cantori

377

Table 2 S y m m e t r y breaking bifurcations of periodic orbits along the line a = b/4. m

Type

Class

a

2 3 4 5 6 7 8 9 10 11 12 13 14

0/1 1/2 1/3 2/5 3/8 5/13 8/21 13/34 21/55 34/89 55/144 89/233 144/377

4 1 4 4 1 4 4 1 4 4 1 4 4

0.0 0.0 0.339333085826286331286205043062353 0.414545641383064213008038550755724 0.442909558709145996350320601007537 0.429375618404184936414855191983229 0.429439480217106924406719065695338 0.429436703577663087946298253977836 0.429439737991981729620538418494070 0.429439737992501222840647101577907 0.429439737992502941627904255462858 0.429439737992501223898752836653440 0.429439737992501223898752836653609

we also added the analytically calculable cases m = 2 and 3. T h e y seem to converge rapidly to a point (ac, bc) , with

1

a c -- 0.4294397379925012238987528366536. (3.2) It is reasonable to expect the accumulation point of this sequence of symmetry breakings to m a r k the 1 - 2 hole transition for the cantorus of rotation n u m b e r w0. Firstly, by continuity of the p h o n o n gap [7], the accumulation of a sequence of periodic orbits which are undergoing symmetry breaking is not uniformly hyperbolic, and loss of uniform hyperbolicity is a necessary condition for any sort of bifurcation. Secondly, symmetry breaking converts island chains with two rotationally o r d e r e d periodic orbits, one of Morse index 0 and one of Morse index 1, into island chains with two of each. The heuristic picture of Chirikov, G r e e n e etc. for how an invariant circle is b r o k e n is that the elliptic islands around a periodic orbit of Morse index 1 squeeze an orbit of gaps in the circle. Generalising this intuition to island chains with two independent elliptic periodic orbits leads us to expect two orbits of gaps to be created. Repeating the procedure along lines of various slopes 0 < K < 1 gave a curve of 1-2 hole transition points, some of which are shown in fig. 2.

-i

",.

1

Fig. 2. 1-2 hole transition points in the (a,/3) p a r a m e t e r plane, where ce=ar/~a2a2+b2, f l = b r / ~ a Z + b 2, and r = ( 2 / v ) tan ~a ~ / ~

2.

For picture accuracy, it sufficed to find s y m m e t r y breaking of the 5/13(4) orbit, though care was required for r small to take the first s y m m e t r y breaking as r decreases, as the s y m m e t r y breaking curves were found to cross the lines a = Kb twice, the crossings approaching each other as K decreases, and presumably annihilating one another for some small K0. If one wished to go closer to i< = 0, higher period approximations would be required. For K - - 0 , we used G r e e n e

378

C. Baesens, R . S . M a c K a y

/ T h e o n e to t w o - h o l e transition f o r cantori

and Mao's value for breakup for the ~2 symmetric golden circle [12]. Probably, the form of the 1-2 hole transition curve could be derived near the anti-integrable limit, along the lines proposed in ]14]. In section 4 we derive rigorous bounds for it. We investigated the rate of convergence of the symmetry breaking points a m on the line a = b / 4 to a c. Firstly, it is clear from table 2 that there is a period 3 behaviour, reflecting the period 3 cycle (1, 4, 4) of the classes. In fact, the class 1 values of a m are alternately above and below ac, giving period 6 behaviour. We fitted the class 4 results to a scaling law of the form log(A/(a m

-

-

ac) ) ~

Bq~m ,

(3.3)

0

8

--4

34

J, = 0.65-0.7

(3.4)

and A between 1 and 10. The proposed scaling law is perhaps better written as A log - log

a m -- a c B

mu

log y .

(3.5)

Using the value (3.2) for a c we then investigated how some quantities scale near the transition. We write r c for the corresponding value of r (from (3.1)), and t = r -

rc .

(3.6)

The first quantity on which we report is the width of the gap around x = ½, which from fig. 1 seems to scale rather like x/t. Figure 3 shows a log-log plot of this width for 9 approximating class 2 and 3 periodic orbits, against t. We see an asymptotic straight line, with slope close to 0.5, possibly a little more. Thus the behaviour is indeed very close to square-root behaviour. This should be contrasted with the standard circle to 1-hole cantorus transition in which the gap around x = 0 grows like t ° ' 7 l o r so [16]; the exponent there has a renormalisation explana-

'''

55,89 8 144 i

-12

-16

-20 20

and obtained a reasonable, though not wonderful, fit with

,'

13,21

-16

-12

-8

-4

0 log a--ac

Fig. 3. A l o g - l o g p l o t o f t h e w i d t h o f t h e g a p a r o u n d x = a g a i n s t p a r a m e t e r a > ac, a l o n g t h e line a = b / 4 , f o r p e r i o d i c orbits of types 3/8(2), 5/13(3), 8/21(3), 13/34(2), 21/55(3), 34/89(3), 55/144(2). The periods are indicated on the figure, a n d A = 2 . 0 - a c. N o t e t h a t t h e r e s u l t s f o r 5 / 1 3 a n d 8 / 2 1 a r e almost identical and similarly those for 21/55 and 34/89.

tion. Square root behaviour suggests an explanation in terms of the degenerate anti-integrable limit, because the minima of V ( x ) separate like X/I--K as the slope K changes. In fig. 4, we plot the total length of the second hole in the cantorus as a function of parameter. This was computed by summing the horizontal lengths of the gaps in the approximating class 2 periodic orbits, starting from that around x - :1 and iterating in positive and negative time until the smallest gap was reached. It also seems to behave like x/t. This is confirmed in fig. 5, which is a log-log plot of the same data, and appears to limit to a straight line of slope near 0.5. This behaviour should be contrasted with that for the standard circle to cantorus transition, in which the total length of the hole jumps instantaneously to 1 on creation (this can be deduced from the observed hyperbolicity [18] and the result of [19]). Next we describe the behaviour of the phonon

C. Baesens, R.S. MacKay / The one to two-hole transition for cantori

379

gap of the cantorus. The phonon gap G is defined to be G = IID2W 111;1 ifD2W is inverfible in f 2 ,

8/15

= 0, %5 ...,

4/15....... .. . . . . . . . .

~.

3(8................... .............

i I I I

........... J 3 ) 3 4 , 5 5 / 1 4 4 .. "

.

2/15 /" /

/ 0.0

............

j .

0.3

.

0.44

.

.

0.58

.

.

.

0.72

0.86

1.0 a

Fig. 4. A p p r o x i m a t i o n s to the length of the second hole in a cantorus as a function of p a r a m e t e r a along the line a = b/4, using class 2 periodic orbits of types 3/8, 13/34 and 55/144. T h e results for the latter two are almost indistinguishable on this plot, suggesting rapid convergence.

v

0

.~O

3/8

, : , ,

- 4 . . . . . . . . . . . . 1. 3 /. 3 .4 . . . . . .

: ;

55/144

otherwise.

(3.7)

It is the distance from 0 to the spectrum of D 2 W . It takes the same value on all orbits of a cantorus, since every orbit is dense in the cantorus. It can be estimated from periodic orbits, as the phonon gaps for a sequence of periodic orbits converging to a cantorus in Hausdorff topology converge to the phonon gap for the cantorus [7]. The class 2 rotationally-ordered periodic orbits form such a sequence. Note that the other classes of rotationally-ordered periodic orbits converge to the cantorus union one or two homoclinic orbits, so they could be used to estimate the phonon gap for the homoclinics, and hence to detect turnstile bifurcations. For a periodic orbit of period q, the phonon gap can be computed relatively easily. The simplest case is when D2W is positive definite, as is the case for the class 2 orbits (by continuation from the anti-integrable limit); then the phonon gap is the smallest eigenvalue of D2We. This will suffice for the present paper. Note that in the general case, by the F l o q u e t - B l o c h theory, the spectrum of D 2 W consists of bands [Ei, Fi] , i odd, [F/,Ei] , i even, 1 -< i -< q, where E 1 < E 2
-12

~:, = 7/, =-'0n

-16 I

r -20

.

. -16

.

. -12

.

. -8

-4

0

log a-ac T Fig. 5. A l o g - l o g plot of the data in fig. 4 (with A = 2 . 0 -

ac).

forq-
(3.8)

~n+q ~T~n" The spectrum o f D2W1/2 is almost as easy to compute as the spectrum of D2We. Figure 6 shows the phonon gap as a function of parameter for three class 2 periodic orbits. The important thing to notice is that the limit with

. - 20

for 0 -< n (mod 2q) < q ,

380

C. Baesens, R . S . M a c K a y / The one to two-hole transition f o r cantori

1.0 o13

3/8

g g

g~ 13/34

3/g 55/144 13/34 55/144

0.0

•

J/'J

0.1

a

0.7

Fig. 6. Phonon gap for class 2 orbits of types 3/8, 13/34 and 55/144 against parameter a along the path a = b / 4 . The values of phonon gap for the 13/34 and 55/144 orbits are indistinguishable, suggesting rapid convergence to that for the cantorus.

appears to be positive for all parameter values except at the 1-2 hole transition point (and below the circle-cantorus transition). Hence, from [5] we deduce that the cantorus is uniformly hyperbolic except at the transition point. Of course it cannot be uniformly hyperbolic at the transition, because that would violate structural stability. F r o m fig. 6, the phonon gap appears to approach 0 almost linearly on both sides of the 1-2 hole transition. This is confirmed by the log-log plot of p h o n o n gap versus parameter on the 2-hole side in fig. 7, whose slope is very close to 1. Note that the slopes on the two sides of the transition do not appear to be the same in fig. 6, however. Next we describe the behaviour of the Lyapunov exponent of the cantorus. The Lyapunov exponent of the orbit of a point x is defined to be 1

A = l i m -- log[Df~ I

(3.9)

20 2(1

log a-ac -A-Fig. 7. A log-log plot of phonon gap against parameter a > a C for the same orbits as in fig. 6 (with A = 1 . 0 - G).

if the limit exists. For a cantorus, there is a unique invariant measure, hence the limit exists almost everywhere with respect to this measure and takes the same value. This common value is called the Lyapunov exponent of the cantorus. Instead of computing this directly from the definition, it is much more accurate to compute it from approximating periodic orbits. If the cantorus is uniformly hyperbolic then the Lyapunov exponents of any sequence of periodic orbits accumulating onto it converge to the Lyapunov exponent of the cantorus (e.g. [17,10]). Since the phonon gap results above suggest that the cantorus is uniformly hyperbolic except at the transition, this result is adequate for our purposes. Note that Proposition I of [5] leads to the lower bound A>-G2/4+ I ~ ( G 4 ) . Even if the cantorus is not uniformly hyperbolic, its Lyapunov exponent is at least the lim sup of the Lyapunov exponents of any sequence of orbits approaching it [17,10]. Furthermore, [10] proves that there exists a sequence of periodic orbits accumulating on the cantorus, for which

C. Baesens, R.S. MacKay / The one to two-hole transition for cantori

the L y a p u n o v exponents converge to that of the cantorus. Although the periodic orbits they construct need not be rotationally ordered, we strongly suspect that the same result could be achieved with rotationally-ordered periodic orbits. R a t h e r than computing the L y a p u n o v exponents of the periodic orbits, we c o m p u t e d their m e a n residue. The residue R of a periodic orbit of period q for an area-preserving m a p f is defined by R = (2 - Tr D f q ) / 4 .

(3.11)

For comparison, the L y a p u n o v exponent is h = q - t log(2lR I ~_ 1 + 2~/[]RI(IR I -v- 1)1 )

(3.12)

for R > 1 or R < 0, respectively, and for0
2)

(3.14)

for q, R large. Thus A = log/~ + ~ ( q - 1 / z - q )

for/z

>

41/q ,

f o r / z --<4 l/q .

(3.15)

So, for large q , h -- m a x ( l o g / z , 0 ) .

i

i i / 3

l

// 2

21"

..........

0 21

o.o

o.2

0.4

o.;

o.8

r

~.0

Fig. 8. Mean residue tL against parameter r for the periodic orbits 8/21(3), 13/34(2), 21/55(3), 34/89 (3) and 55/144 (2), along the path a = b / 4 > 0 . Note that we attribute the flattening-off of the mean residue for small r to round-off error, as the calculated residues here are of the order of 10 30.

(3.13)

It is clear that

= G(q - I )

/

144 • 89 'ii:: . . .55 . . y .," 34

tx = 14R[ 1/q

A = log/~ + ~ ( q - l R

4

(3.10)

Its mean residue IX is defined by

A=0

381

(3.16)

T h e advantage of using the mean residue rather than A is that it throws away less information. Values of the mean residue less than 1 have a meaning even though they all correspond to L y a p u n o v exponent zero. In particular, it is conjectured that the m e a n residues of good sequences of periodic orbits approaching an analytic invariant circle typically converge to e-a, where 6 is the "analyticity width" for the conjugacy [17]. In fig. 8, we plot the m e a n residue of periodic

orbits of classes 2 and 3 against p a r a m e t e r r. It is larger than 1 for all p a r a m e t e r values near but not equal to r c, corresponding to positive Lyapunov exponent, but there is a narrow spike near re, where the m e a n residue is considerably smaller than for nearby p a r a m e t e r values. This raises the question whether the m e a n residue goes to 1 ( L y a p u n o v exponent to zero) at the transition point. For classes 1 and 4 (not plotted), the residue changes sign, because of symmetry breaking, and hence the m e a n residue goes to 0 near the transition, but in a narrow spike again. Figure 9 shows a plot of log log/z against log log A / t for the class 2 and 3 periodic approximations, for A = 2.0 - a C. We find periodic oscillations about a straight line of slope /3 = 0 . 6 5 , suggesting that h -- log/z -- (log A/t)t3P(log(log A / t ) ) for some periodic function P, with period

(3.17)

382

C. Baesens, R.S. MacKay / The one to two-hole transition for cantori 10

in action AW for the new hole (defined by Mather [24]). One could compute it by taking

o~

A W -- - ~i !ira ~

_ (WF3n-I/F3n÷

1(4)

+

WF~n/F3n+2(4)

- WV3,+,/F3,,~3(2)).

i

(3.21)

i 0.6

I , J I i

...................

8, I3,21

' i i :' :' :,':: . . . . . . . . . . . . . . . . 55 "..~; ............................... 89 ";ii:~, . , , ...................... 34

; 04!

":it;;'-:: ........ "gill; "'.::"'--. "':7:~i['~ 0.2

233 377

":"

144 987 1597

0

0 oo

. 02

. 0.4

I

016

j ~.0

0.8 A log30( 1 log a_-S~c)

Fig. 9. A plot of log log/x against log log[A/(a - at)], w h e r e A = 2.0 - ac, for periodic orbits of the indicated types.

~=o.ss.

(3.18)

N o t e that this is strongly consistent with the relation = 3v log 7 ,

(3.19)

where v is defined in (3.4), and this suggests that there is asymptotic scaling symmetry with period-1 periodicity in log[log(A/t)] ~- ~:

(3.20)

as t--~ 0. T w o further remarks are in order about fig. 8. Firstly notice that the passage of the mean residue through p~ = 1 is almost linear. This is consistent with the exponent ---0.98 from [16,18]. Secondly, note that the m e a n residue appears to a p p r o a c h 0 like v~. This is curious, because for the standard map, G r e e n e showed t h a t / x - r , as r---~

0.

T h e r e are several other quantities which could be of interest to measure. One is the difference

We conjecture that it would grow like ( a - ac) 2, this being the scaling of the height difference between the critical points at the anti-integrable limit. A n o t h e r quantity to measure is the rate of accumulation of the y-coordinates of the class 1 and 4 periodic points on the line x = ½, and also the scaling of the off-diagonal elements of the symmetrised m o n o d r o m y matrix, as was done in [16,18] for the circle-cantorus transition. Finally, it would be good to check that the observed scalings for m e a n residue and phonon gap on the 2-hole side of the transition have counterparts on the 1-hole side.

4. B o u n d s on the 1 - 2 hole transition near the anti-integrable limit

In this section we establish bounds on the 1 - 2 hole transition near the anti-integrable limit. We choose to write the generating function as h(x, x ' ) = V(x) + l t(x' - x) 2 ,

(4.1)

with V(x) = s i n 0

cos 2avx cos 4-rrx 4~r2 +cos0 16,rr2

(4.2)

This agrees with the original generating function up to a scale factor of t. Note the connections t = cot(½-rrr),

K --- tan 0 ,

(4.3)

with the p a r a m e t e r r defined in (3.1), and the slope K used in section 3. P r o p o s i t i o n 1. T h e r e are no bifurcations rotationally-ordered orbits in the regions p a r a m e t e r space (0, t) where

IV'(x)i < t ~ v"(x) > 0 or V"(x) < - 4 t .

of of

(4.4)

C. Baesens, R.S. MacKay / The one to two-hole transition f o r cantori

383

Remark. In particular, for small enough neighb o u r h o o d s U of 0 = ~-~t = 0, by sketching the 4 graphs of V' and V", it can readily be seen that this region includes all points of U with 0 > ~Tr. 1 symmetric 1-hole

Proof. For a stationary state x V'(x,) = tan(x ) ,

(4.5)

where a , ( x ) = x , + l - 2x, + x , _ i •

(4.6)

For a rotationally-ordered state IAn(x)l < 1

(4.7) -±

(e.g. [15]). H e n c e the points xn of all rotationally o r d e r e d states lie in intervals where IV'(x,)l < t. U n d e r the hypotheses of the proposition, this implies that V"(xn)>0 or V"(xn)<-4t. In particular [V"(x,) + 2t[ > 2 t .

(4.8)

Now D2W is symmetric tridiagonal, with diagonal elements V"(x,)+ 2t and off-diagonal elements - t . It follows from (4.8) that D2W is invertible in f~. H e n c e the orbit has a locally unique continuation (see e.g. [3,25,21]) and there is no bifurcation. []

Proposition 2. There is no symmetric 1-hole cantorus for [01 <¼~r, t < ( 1 / V ~ ) sin(J~r - [01).

Proof. For a twist m a p , every cantorus consists of locally minimizing orbits, because it is recurrent and rotationally ordered (this can be proved using ideas in [23], for example). For the a b o v e p a r a m e t e r values a cantorus cannot have a point on x = 0, because imagining varying it to

Fig. 10. Rigorous bounds on the 1-2 hole transition in the (a,/3) parameter plane (defined as in fig. 2).

cantorus cannot have a point on x = ½. So it has gaps around both x = 0 and x = ½. These gaps cannot belong to the same orbit of gaps. For suppose the endpoints of the gap around x = 0 land on the endpoints of the gap around x = ½ + m after n iterations. Then by time-reversal symmetry about x = 0 , the gap around x = - ½ - m lands on that around x = 0 in n iterations. H e n c e the endpoint orbits are periodic of type (2m + 1, 2n). This contradicts irrational rotation number. Hence there are at least two holes in the cantorus. [] Thus we have established that the 1-2 hole transition must approach the anti-integrable limit 0 = Jrr, t = 0 between the lines 0-zrr-~ and t = (1/V~) s i n ( l r r - 101), as sketched in fig. 10.

5. Discussion

X0~

OZW Ox2 - V"(0) + 2t = - s i n 0 - cos 0 + 2t = - X / 2 sin(0 + ¼~r) + 2t < 0 , which contradicts local minimality. Similarly for - V ~ s i n ( ¼ " r r - 0 ) + 2 t < 0 ,

(4.9) the

We have presented numerical results about a highly non-trivial bifurcation, in which a cantorus gains or loses a second hole. One surprise to us is that the L y a p u n o v exponent of the cantorus appears to go to zero at the transition. It is clear that the cantorus cannot

384

C. Baesens. R.S. MacKay / 7"he one to two-hole transition for cantori

remain uniformly hyperbolic at the transition, because this would violate structural stability. It is also clear that the orbit of the point of the cantorus with x - ~ has L y a p u n o v exponent zero at the transition, because taking a limit point of the sequence of gap displacements as a decreases to a,. gives a bounded non-zero tangent orbit. But one might have expected the cantorus to retain a positive Lyapunov exponent (meaning its almost everywhere value). Our results p r o m p t the following conjectures. C o n j e c t u r e I. The L y a p u n o v exponent exists and is the same for every orbit on a cantorus. C o n j e c t u r e 2. The L y a p u n o v exponent of a cantorus is positive if and only if the cantorus is uniformly hyperbolic.

If a cantorus is uniformly hyperbolic then the L y a p u n o v exponent exists for all its orbits and is positive and the same. This follows from unique ergodicity of a cantorus and the fact that in the uniformly hyperbolic case the Lyapunov exponent can be written as the time average of a continuous function, viz. log Dfl ., where u denotes the unstable direction. Furthermore, if the cantorus has L y a p u n o v exponent zero, then the L y a p u n o v exponent exists for all its orbits and is zero, by a general result of I13] for uniquely ergodic systems. So all that remains to be proved is that if a cantorus is not uniformly hyperbolic then its L y a p u n o v exponent is zero. It would be very interesting to investigate the other types of bifurcation of cantori predicted in [6]. In particular, for b < 0 in the two-harmonic family, m a n y annihilations of cantori are expected. Results on the bifurcations of rotationally-ordered periodic orbits for b < 0 are presented in [14]. One important question is how the one to two-hole transition fits into the renormalisation f r a m e w o r k of [16]. We believe that it corresponds to an unstable anti-integrable fixed point,

but from (3.3) its unstable eigenvalue is infinite, so this is not easy to formulate.

Acknowledgements We thank Jim Meiss for his hospitality during a visit of both of us to Boulder in April 1991 when this work was started, and for provision of a computer program which we adapted for initial investigation of this problem. We thank Catherine Wattebot for assistance with the final version. This work was supported by Laboratoire L6on Brillouin (Saclay), the Nuffield Foundation, and the U K Science and Engineering Research Council.

Appendix A. Efficient methods for symmetric periodic orbits A n u m b e r of tricks can be used to reduce the computation time required to find symmetric periodic orbits and compute their residues and phonon spectrum. First we define some notation. Let (A.1)

a, = 2 + V " ( x , , ) .

Denote by tridag[a . . . . . . . , a , ; bin+ 1. . . . .

b ,]

(A.2)

the symmetric tridiagonal matrix with diagonal entries a . . . . . . . . a, and sub- and super-diagonal entries bin+ , . . . . . b,,. N o t e . To increase

accuracy we represent the integer and fractional parts of x,, separately on the computer. A . 1. F i n d i n g s y m m e t r i c p e r i o d i c orbits Class 1. Provided q - > 6 , the equations to be solved are

C. Baesens, R.S. MacKay / The one to two-hole transition for cantori

2xl - x2 + V ' ( x ~ ) = 0 ,

2x.-x.-1-x.+~

Class 3. For q -> 5,

+ V'(x.) = 0

2x~ - x2 + V'(x~) = 0 ,

for2<_n<_q/2-2, 2Xq/2-1 --Xq/2 2 - p / 2

2xn - x n _ l - x , + l + V ' ( x , ) = 0 + V'(xq/2_l)=O

(A.3)

,

which can be written as U~(xl . . . . ,Xq/2 1 ) = 0 ,

for2-
(A.10)

l <-n<-q/2-1.

(A.4) T h e derivative of U is M ( x ) = t r i d a g [ a ~ , . . . , aq/2_ 1; - 1 . . . . .

and M = tridag[a I . . . . , a(q_3)/2 , a(q_l)/2 + 1"~

- 1].

-1 ....

(A.5)

If q = l ,

t h e n x 0 = 0 . If q = 3 ,

3x 1 - p

M ~x = u

with M = a 1 + 1 .

(A.6)

can be solved efficiently for ~x, using tridiagonality of M. We iterate the Newton step (A.7)

x~"~ x - ~x ,

until the step ~x ceases to b e c o m e any smaller (because of roundoff errors). If q = 2 , the solution is x 0 = 0 , x~ = p / 2 . If q = 4, we must solve + V'(xl) = 0,

(A.7a)

for which M = a 1.

(A.11)

, -1].

Let x E ~q/2 1 be a guess for a solution, and evaluate u - - U ( x ) and M ( x ) . Then the equation

2x I - p / 2

385

then

+ V'(xl) = 0,

(A.11a)

Class 4. For q >- 5,

2x I _ _ X 2 - 1 _~_V t ( x I ) = 0 2x n - x n _ 1 - x , + 1 + V ' ( x , ) = 0

f o r 2 - < n - < (q - 3 ) / 2 , 3X(q_l)/2 -- X(q 3)/2 - P - 1 + V t ( x ( q _ l ) / 2 ) = 0 ,

(A.12) and M is given by (A.11) again. If q = 1, then x 0 = ½ . If q = 3 , then 3x I - p

- ~ + V'(xl) = 0,

(A.12a)

Class 2. Provided q-> 4, the equations are

withM=a

3x 1 - x 2 + V ' ( x l ) = 0 , 2x n - x , _ 1 - x , + 1 + V ' ( x n ) = 0

I+1.

A . 2 . Residue

for2<-n<-q/2-1, 3X q/2 -- Xq/2_ 1 -- p ~- V ' ( X q / 2 ) = 0 ,

(A.8)

with

tridag[...,a,,...

M = tridag[a~ + 1, a 2 . . . . , a q / 2 _ l , aq/2 + 1; -1,...,-1].

(A.9)

If q = 2, then the equation is 4xl - p + V'(x~) = 0 ,

with M = a 1 + 2 .

On ~ , D 2 W c a n be represented by the infinite tridiagonal matrix

(A.9a)

; . . . . - 1 . . . . ].

(A.13)

Restricted to the space P of periodic sequences of type ( p , q) it becomes cyclic tridiagonal. By a formula of [8] (generalised by [20]), the residue is given by R = - ¼ d e t D2WIp .

(A.14)

386

C. Baesens, R.S. MacKay / The one to two-hole transition for cantori

The evaluation of the residue can be simplified for symmetric periodic orbits as follows. In classes 1, 3 and 4, we have a n = a~. Define kin = det tridag[a~_ 1. . . . , a 0. . . . , a~ ;

od for calculating the residue of a symmetric periodic orbit, which has an additional advantage. The involution R (2.6), induces an involution R' on sequences ~c = ( ~ , ) of tangent vectors to a symmetric periodic sequence (x,):

-1 .... , -1], D. = det tridag[a~,... , a 0 , . . . ,a n; - 1 , . . . ,

-1]. (A.15)

Then we have the following recurrence relations: D,,+l = a2~+lD, + D,_~ - 2 a , + l H " ,

(A.16)

H,+~ = a , + l D " - H , ,

with initial conditions D 1=0,

D 0 = a 0,

H0=1.

(A.17)

Furthermore,

det D2W] p

=

Hal 2 --

Hq/2_

= D(q 1 ) / 2 - D ( q - 3 ) / 2 -

2 ifqisodd. (A. 18)

In class 2, a_ n = a n ~. Define H , = det tridag[an_

l,

• • •,

a

l,

a

1,

• • •,

a~;

-1 ..... -11, D = dot t r i d a g [ a n , . . . ,a~, a ~ , . . . ,an; - 1 . . . . . -11 .

D 1=-1,

8x'n = - S x l - n -

(A.22)

The space P of sequences of tangent vectors decomposes into the direct sum S O A of symmetric ones S (meaning those for which R'~0c = ~x) and antisymmetric ones A (meaning those for which R ' ~ x = - ~ x ) . For a symmetric periodic sequence, DZw]p commutes with R' and therefore can be decomposed into the direct sum over the symmetric and antisymmetric spaces. Hence det D2W]p = det D 2 W I s • det D2W]A .

(A.23)

As shown in the next section, D2W[s and D2W]A can be represented by symmetric tridiagonal matrices. The determinant M,, of a symmetric tridiagonal matrix tridag[a~ . . . . ,an; b 2 , . . . ,b,~] can be calculated by the recursion mm+l = am+from-

bZ+,Mm -1 ,

(A.24)

(A.19)

Then the same recurrence relations hold, but with initial conditions H0=0,

Class 2:

if q is even,

2

1 -

Classes 1, 3 and 4: ~x'n = - S x n ,

D O= 1,

(A.20)

and we obtain the result

det O 2 W l p = O q/2 - D q/2_ 1 - 2 .

(A.21)

Note. By a shift in the origin of time, classes 3

and 4 can also be brought into the case a_,, = a,, ~ if desired; but then the period is odd and (A.21) must be changed to

det D 2 W I p =- H(q+l)/2 - H ( q _ l ) / 2 - 2.

(A.21a)

There is an alternative equally efficient meth-

from the initial conditions M0=1,

M l = a 1.

(A.25)

Hence the residue is easily calculated using (A.23) and (A.14). The additional advantage that this method brings is that it can distinguish between symmetry breaking and saddle-centre bifurcations. A necessary condition for either is that the residue be zero. But for a saddle-centre of symmetric periodic orbits d e t D 2 W I s = 0 , whereas for a symmetry breaking d e t D 2 W I A = 0 . Thus to search for a symmetry breaking, as we do in this paper, it suffices to look for when det D Z w [ A = 0.

387

C. Baesens, R.S. M a c K a y / The one to two-hole transition f o r cantori

A.3. Phonon spectrum

~C+n=+--vn/V~ ,

l<--n<--q/2--1,

8Xo = O, As explained in section A.2, for a symmetric periodic o r b i t D 2 W l p decomposes into the sum over the symmetric and antisymmetric spaces. So its spectrum decomposes into what we call symmetric and antisymmetric eigenvalues, according as they come from S or A. D Z w is tridiagonal on the symmetric and antisymmetric spaces, as opposed to cyclic tridiagonal on P. There are efficient eigenvalue algorithms for finding the spectrum of tridiagonal matrices. In addition the symmetric and antisymmetric spaces have roughly half the dimension of P. So this decomposition saves a lot of computation time. By S t u r m Liouville theory (e.g. [2]), we expect the lowest eigenvalue to be antisymmetric, so if only the lowest eigenvalue is desired, it suffices to compute the antisymmetric spectrum. We introduce an orthonormal coordinate system (u, v) on P with respect to w h i c h DZWp decomposes into b l o c k s DzWA and DZWs, respectively. The coordinate system depends on the symmetry class of the periodic orbit.

Class 1. The antisymmetric vectors can be written as gx._ =Un/X/2,

~dCq/2 =

(A.29)

0,

so we obtain the eigenvalue equation My = Av but with M = tridag[al,...,

aq/2_l; -1,

~X 0 = U 0 ,

(A.30) (There are no symmetric vectors for q = 2). Note that this is the same as the matrix entering the Newton's method of section A. 1, which is not an accident. The same is true for the symmetric spectrum in classes 2, 3 and 4.

Class 2. The antisymmetric vectors can be written as 8x n = S x l _ . = u . / V ~ ,

l <-n<-q/2,

(A.31)

and so for q -> 4, m = tridag[(a 1 - 1 ) , a 2 , . . . , a q / 2 _ 1, (aq/2- 1) ; -1,...,-11.

(A.32)

For q = 2, M = a 1 - 2.

(A.32a)

~ c n = - S x I n=Vn/X/2, (A.26)

T h e n for q---4, the eigenvalue equation for antisymmetric phonons becomes

l <-n~q/2,

(A.33)

and for q - 4, M = tridag[(a I + 1), a 2 , . . . ,aq/2_ ~, (aq/2 + 1) ; -11.

Mu = Au,

.

The symmetric vectors can be written as

l<--n<--q/2--1,

8Xq/2 = Uq/2 .

. . . , -1]

(A.34)

(A.27) For q = 2,

with

M=al +2. M = tridag[a0,..., -V~, -1,...,

(A.34a)

aq/2;

-1, -V21.

(A.28)

For q = 2, M = tridag[a0, al; - 2 ] . The symmetric vectors have

Classes 3 and 4. The antisymmetric vectors can be written as

~C±n=Un/X/2, (A.28a)

~x 0 = u 0 , and for q --- 3,

l <--n<--(q--1)/2, (A.35)

388

C. Baesens, R.S. MacKay / The one to two-hole transition for cantori

M = t r i d a g [ a 0. . . . -X/2, -1 .....

, a ( q _ 3 ) / 2 , (a(q

~x,,=-~oc l _ n = v , , / N / 2 ,

1)/2 -- 1) ;

-1].

(A.36)

F o r q = 1,

l <-n<--(q-1)/2, (A.41)

~X(q+l)/2 = O ,

and

M=a o -2.

(A.36a)

-1 .....

T h e s y m m e t r i c v e c t o r s can be w r i t t e n as

gx+,=±v,/V~,

M = t r i d a g [ ( a 1 + 1), a 2 . . . . .

a(q_l)/2 ,

-1].

(A.42)

F o r q = 1 o r 3, t h e a b o v e f o r m u l a e for M n e e d suitable modification.

l<_n<_(q-1)/2,

gx 0 = 0 ,

(A.37)

A.4. Gaps in symmetric periodic orbits

a n d for q -> 5, M = t r i d a g [ a 1. . . .

,a(q_3)/2 , (a(q_l)/2 + 1) ;

- 1,...,

- 11.

(A.38)

F o r q = 3, M = a 1 + 1.

(A.38a)

( T h e r e a r e no s y m m e t r i c v e c t o r s for q = 1).

Note. If t h e origin o f t i m e is shifted in classes 3 a n d 4 so t h a t a n = a l _ n , t h e n t h e a n t i s y m m e t r i c v e c t o r s can b e w r i t t e n as ~ x . = ~xl . = u . / V ~

,

l <-n<-(q-1)/2,

T o c o m p u t e t h e gaps in a r o t a t i o n a l l y - o r d e r e d symmetric periodic orbit around the symmetry lines x = 0 a n d x = 1, it is n e c e s s a r y to find w h i c h are t h e n e a r e s t p o i n t s o f t h e o r b i t to t h e s y m m e try lines. F o r a r o t a t i o n a l l y - o r d e r e d p e r i o d i c o r b i t o f t y p e ( p , q), t h e p o i n t s c o m e in the s a m e o r d e r as for t h e r o t a t i o n R p / q : x ~ - ' ~ x + p / q of the circle, w h o s e o r b i t s a r e given by x n = x o + np/q. So this r e d u c e s to a p r o b l e m in D i o p h a n t i n e equations. L e t P / Q be t h e last c o n v e r g e n t of p / q , i . e . , t h e last r a t i o n a l g e n e r a t e d b y t h e c o n t i n u e d f r a c t i o n a l g o r i t h m b e f o r e it t e r m i n a t e s . T h e n

(A.39)

~ X ( q + l ) / 2 : U(q+l)/2 ,

~:= Qp-qP

= ±1.

and M = t r i d a g [ ( a x - 1), a2, . . . , a ( q + l ) / 2 ; -1 .....

(A.40)

-1, -V21.

T h e s y m m e t r i c v e c t o r s can b e w r i t t e n as

L e t xn, ' - m 0 be t h e closest p o i n t o f t h e lifted o r b i t to t h e left of x = 0 ( b u t n o t e q u a l to 0), a n d xn0 ~ - m 0 + be the closest p o i n t to t h e right o f x = 0. S i m i l a r l y , let xnL.2+ - m l / 2 . be the closest p o i n t s of the lifted o r b i t on t h e left a n d right of

Table 3 Solutions of equations for nearest points of symmetric rotationally-ordered periodic orbits to the s y m m e t r y lines (e E {-+} is interpreted here as -+1). Class

Gap s

Equation to solve for (n, m)

Solutions (n, m) (Vk @ 2 )

1 1 2 2 3 3 4

0 12 0 12 0 ,~ 0

4

12

np/q- m = e/q np/q - m = ½ + e/q (n - ½ ) p / q - m = e/2q = 5~ + e/2q (n-½)p/q-m n p / q - m = e/q np/q-m=+_ +e/2q n p / q - rn = - ½ + e/2q np/q - m = e/q

(etrQ + kq, e~rP + kp) ((½q + e)o'Q + kq, (½q + e)o'P + kp) (['2 (P + s)]o'Q + kq, [½(p + e)]o'P + kp) ([½(p + q + e)]~Q + kq, [½(p + q + e)]o-P + kp) (eo'Q + kq, eo'P + kp) ([½(q + e)l~rQ + kq, [½(q + e)]trP + kp) ([½(e - q)]~rQ + kq, [½(e - q)]~rP + kp) (e~rQ + kq, e~rP + kp)

C. Baesens, R.S. MacKay / The one to two-hole transition for cantori

x = 1. T h e n

{0,

t h e e q u a t i o n s t o s o l v e f o r ns~, ms,

}, e ~ { + } ) ,

and their solutions,

d e p i c t e d in t a b l e 3. T h e r e s u l t s d e p e n d

are

on the

s y m m e t r y class o f t h e o r b i t .

References [1] D.K. Arrowsmith and C.M. Place, Introduction to Dynamical Systems (Cambridge Univ. Press, Cambridge, 1990). [2] F.V. Atkinson, Discrete and Continuous Boundary Value Problems (Academic Press, New York, 1964). [3] S. Aubry and G. Abramovici, Chaotic trajectories in the standard map: the concept of anti-integrability, Physica D 43 (1990) 199-219. [4] S. Aubry, J.-P. Gosso, G. Abramovici, J.-L. Raimbault and P. Quemerais, Effective discommensurations in the incommensurate ground-states of the extended FrenkelKontorova models, Physica D 47 (1991) 461-497. [5] S. Aubry, R.S. MacKay and C. Baesens, Equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for Frenkel-Kontorova models, Physica D 56 (1992) 123-134. [6] C. Baesens, R.S. MacKay, Cantori for multi-harmonic maps, Physica D 69 (1993) 59-76. [7] C. Baesens and R.S. MacKay, Continuity of the phonon gap, Phys. Lett. A 183 (1993) 193-195. [8] T. Bountis and R.H.G. Helleman, On the stability of periodic orbits of two-dimensional mappings, J. Math. Phys. 22 (1981) 1867-1877. [9] R. DeVogelaere, On the structure of symmetric periodic solutions of conservative systems with applications, in: Contributions to the Theory of Nonlinear Oscillations, vol. IV, S. Lefschetz, ed. (Princeton Univ. Press, Princeton, NJ, 1958) pp. 53-84. [10] C. Falcolini and R. de la Llave, A rigorous partial justification of Greene's criterion, J. Stat. Phys. 67 (1992) 609-643.

389

[11] J.M. Greene, A method for determining a stochastic transition, J. Math. Phys. 20 (1979) 1183-1201. [12] J.M. Greene and J.-M. Mao, Higher order fixed points of the renormalisation operator for invariant circles, Nonlinearity 3 (1990) 69-78. [131 R. Johnson, Ergodic theory and linear differential equations, J. Diff. Eqn. 28 (1978) 23-34. [14] J. Ketoja and R.S. MacKay, Rotationally-ordered periodic orbits for multi-harmonic maps, Warwick preprint (1993). [15] P. LeCalvez, Les ensembles d'Aubry-Mather d'un diffeomorphisme conservatif de l'anneau deviant la verticale sont en general hyperboliques, CR Acad. Sci. Paris 306 (1988) 51-54. [16] R.S. MacKay, A renormalisation approach to invariant circlcs for area-preserving maps, Physica D 7 (1983) 283-300. [171 R.S. MacKay, Greene's residue criterion, Nonlinearity 5 (1992) 161-188. [18] R.S. MacKay, Renormalisation in area-preserving maps, PhD thesis (1982) (a corrected and annotated version has now been published by World Scientific, Singapore, 1993). [19] R.S. MacKay, Hyperbolic cantori have dimension zero, J. Phys. A 20 (1987) L559-L561. [20] R.S. MacKay and J.D. Meiss, Linear stability of periodic orbits in Lagrangian systems, Phys. Lett. A 98 (1983) 92-94. [2l] R.S. MacKay and J.D. Meiss, Cantori for symplectic maps near the anti-integrable limit, Nonlinearity 5 (1992) 149-160. [22] R.S. MacKay, J.D. Meiss and I.C. Percival, Transport in Hamiltonian systems, Physica D 13 (1984) 55-81. [23] R.S. MacKay and I.C. Percival, Converse KAM: theory and practice, Commun. Math. Phys. 98 (1985) 469-512. [24] J.N. Mather, A criterion for non-existence of invariant circles, Publ. Math. IHES 63 (1986) 153-204. [25] J.P.P. Veerman and F. Tangerman, Intersection properties of invariant manifolds in certain twist maps, Commun. Math. Phys. 139 (1991) 245 265.