Fractal boundary for the existence of invariant circles for area-preserving maps: Observations and renormalisation explanation

Physica D 35 (1989) 318-334 North-Holland, Amsterdam

FRACTAL BOUNDARY FOR THE EXISTENCE OF INVARIANT CIRCLES FOR AREA-PRF~Q~RVING MAPS: OBSERVATIONS AND RENORMALISATION EXPLANATION J.A. KETOJA and R.S. MacKAY Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Cooentry CV4 7AL, UK Received 2 November 1988 Accepted 19 December 1988 Communicated by R.H.G. Helleman

Breakup of a golden invariant circle is studied in an extended version of the standard map with two parameters. The critical line in parameter space turns out to have a Cantor set of cusps which can be organized in a self-similar tree. A theoretical explanation is conjectured in terms of a horseshoe for a renormalisation operator. The results of some other researchers on similar systems are shown to fit this explanation as well.

1. Introduction Much of the study of Hamiltonian dynamics has concentrated on exploring phase space struc. tures of area-preserving maps (see articles in [15], for example). These maps appear not only as natural discretized versions of Hamiltonian motions with two degrees of freedom but also when solving one-dimensional variational problems. A typical picture of the phase space of an area-preserving map is the following. One finds many hyperbolic and elhptic periodic orbits. The elliptic ones are surrounded by invariant circles (equiv. alent to KAM tori in continuous time), on which the orbits are quasiperiodic. These "islands" of regular motion then lie in a chaotic "sea" where the longtime behaviour of orbits is very comphcated. An important quantity characterising invariant circles is the rotation number (or winding number). It describes the average speed at which an orbit goes around an invariant circle. It is often useful

to expand this as a continued fraction ~=a0+

1 a 1+

a2+

...

(1.1) The invariant circles act as barriers to transport. This means that a chaotic trajectory cannot wander freely around the phase space. When the values of parameters are changed, it sometimes happens that an invariant circle breaks up and one is left only with a partial barrier formed by an invariant Cantor set called a cantorus [16]. Because transport is an important issue hu many physical problems, a lot of ef~brt has been put into studying the breakup. The breakup is o~ten found to possess asymptotic self-similarity and universality. Especially interesting is the breakup of an invariant circle whose rotation number is the inverse golden mean, ~' = (V~"- 1)/2, whose continued fraction expansion is [0,1~] (1~ =

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J,A. Ketoja and R.S. MacKay~ Fractal boundaryfor the existence of invariant circles

1,1,1... ) - mainly for two reasons. Firstly, there is numerical and theoretical evidence [4, 22, 3, 13, 17] that the frequencies whose continued fraction expansion ends in 1~ (called noble numbers) are the locally most robust ones: if an invariant circle of some non-noble frequency exists, then there is also a noble invariant circle close by. Secondly, the simplicity of the continued fraction expansion of {; makes it the simplest case to develop renormalisation theories [8, 23, 9] for breakup. Greene and MacKay [9] developed an exact renormalisation scheme in which the renormalisation operator has a fixed point in the space of maps, correaponding to a map with a critical noble invariant circle. In this paper we study breakup of an invafiant circle with rotation number ~" in an extended version of the standard map, previously studied k,a [5], which has two parameters. We determine the critical line which separates the suberitica~ parameter region, where an invariant circle of rotation number ~ exists, and the supercritical one, where there is no invariant circle. Axel and Aubry [1] made a similar study in a~nother system which can be shown to be equivMent to an area-preserving map with two parameters: the Frenkel-Kontorova model with two sublattices- a system which simulates incommensurate structures in an electric field. They found that the critical line was not differentiable at an infinite set of eusps which appeared to form a Cantor set. Work on quasiperiodic Schrtdinger operators [20, 21] gives similar results and suggests that this phenomenon is genetic in a certain class of systems with two parameters. There may also be some connection with Wilbrink's results [24] on another family of standard-like mappings where the critical line is found to have infinitely many pieces, which join in a non-analytic case, however. We show that the critk a~ fiae b~s a Can~or ~e~ of cusps Mso in the extended standard map, and organize the cusps in a self-similar tree. Using the action representation we are able to give a plausible explanation for the self-similarity. We develop a renormalisation approach based on half-cycle actions which were first introduced by Kadanoff

319

[8] in order to explain breakup in the one-paran,eter standard map. For the Cantor set of cusps we suggest an explanation in terms of a horseshoe for the renormalisation operator. There are ninny aspects of this problem which are still not clear to us, however, so this paper should be seen as just a first step.

2. Extended standard map The standard map and its extended version have the same form:

=y. +f(x.),

(2.1)

In the extended version, the function f ( x ) depends on two parameters kx and k~: f(x)=

kl -'~s

k2 ~(2¢rx) - ~-~ sin(4~rx).

(2.2)

The standard map is realised as the case k 2 = 0. One can think of the extended version as the result of taking into account also the second Fourier expansion term of an odd function f ( x ) with period one. Since f ( x ) is periodic, one can identify x and x + 1 and think of the map as acting on a cylinder. The rotation number of a trajectory is defined by w= lira Xu--Xo N-~oo N

(2.3)

if the limit exist% Periodic orbits X,+Q = x,, + P have rational rotation numbers P/Q. The value of the function f ( x ) remains the same if one changes the sigri of k~ and shifts x by 1:~ " ~c~e~efe~c, ~ ;~ ~s afl~cient to st~dy the map in the parameter region k 1 ~ 0. Greene et al. [51 determined the critical line in the first qaadrant kl >_O, k~ >_0 for the invariant circle of rotation number ~'. They found a smooth curve connecting the critical points on the k~ and k 2 axes. k~ = 0.97164 and k~ = 0.80472 respectively. The criticat

J.A. Ketoja and R.& MacKay~ Fractal boundary for the existence of inoariant circles

320

line had a cusp at the latter point where the map belonged to a different universality class from the one seen everywhere else along the curve. When k~ = 0, the map has an additional symmetry given by the shift of x by ½, which means that the invariant circle has effectively the rotation number 2~. MacKay [10] showed that it should result in a period-three cycle in the scaling, which Greene and Mao [6] were then able to relate to a periodthree cycle in function space of the renormalisation operator. They found that the unstable manifold of the three-cycle is at least two-dimensional whereas at the normal fixed point the renormalisation operator has only one unstable eigendirection. If k~ = 0, shifting x by ¼ and changing the sign of k 2 does not affect the value of f(x). This implies that there is a critical point at k~ = 0, k2 = - k ~ as well. In section 3 we will determine the critical line in the parameter region k~ > 0, k 2 _< 0 and show how it will connect this critical point with the curve found in the first quadrant. The critical line will turn out to have a Cantor set of cusps in the same way as the critical line of the model studied by Axel and Aubry [1]. In explaining our numerical observations we will take advantage of the fact that our map is "reversible", i.e. it can be represented as the composition of I2I 1 of two involution operators (i.e. operators whose squares are the identity) I x and 12. In our case we can take

For each line a, one can find symmetry coordinates ~v,,, ua) such that u~ is left invariant by the corresponding involution operator while t,~ is flipped in sign:

Vo=X,

Uo-'-Y+ f(x.) 2 ' l+y

Vl=X--

2" '

Ul=Y'

(2.7) 1

v2=x-5,

u 2 = y + f(

Y

V3 = X - "~,

u3 = Y .

Here x is defined modulo unity. Instead of thinking in terms of the original map, it is often much more fruitful to work in the action representation. One writes an area-preserving map F: (x, y) --, (x', y') using a generating function

r( x, x'):

y,

0r(x, x') =

~X'

'

(2.8)

0r(x, x') Y=

bx

One can think of these relations as the rt.sult of looking for a stationary value of the free energy (or action) Er(x,,x,+l).

(2.9)

i

I,: ( x , y ) ~ ( - x , y + f ( x ) ) , I2: ( x , y ) ~ ( y - x , y ) .

(2.4)

If x is considered to be defined modulo rarity, there are two symmetry lines which remain invariant under 11: lineO: line 2:

x=0, , x=3,

(2.5)

and two symmetry fines corresponding to/'2: line i: line 3:

x= (1+y)/2, x =y/2.

(2,6)

An area-preserving map F always has a generating function if it satisfies the twist condition ax'/Sy > 0. The extended standard map is generated by the action

r(x, x')=

x): + g(x),

(2.1o)

where g(x)= fff(t)dt. In order to study orbits with tugh periods, we will need generating functions for maps which result from composing the extended standard map several times with itself. If two maps 7' and U have generating functions r

J,A. Ketoja and R.S. MacKay/Fractal boundaryfor the existence of invariant circles

and ~,, the composition TU is given by the generating function

l,*~'(x,x")=~,(x,x')+~'(x',x"),

(2.11)

where x'(x, x") is chosen to make the sum stationary with respect to variations in x'. If both T and U satisfy the twist condition, a stationary value x' always exists, but it need not be unique, different choices giving TU in different regions of the phase space. It is often useful to write a generating function in symmetry coordinates v~, o~ or even in the form where one mixes the symmetry coordinates of two different lines together [8]. If the motion of eq. (2.1) is represented in the symmetry coordinates (v o, Uo), it turns out that it is generated by the action • oo( Oo, o;) = ½(o; - o0) +

+ g(o;)).

(2.12) In order to find the corresponding generating functions in other symmetry coordinates, we first define c,,¢(o,, v,~) to be the generating function for the coordinate transformation (v~, u~) --, (va, u¢). It exists for all other (~, r ) except for the pairs (0,2), (2,0), (!,3), and (3,1). An arbitrary %~(v,, v~) can be written as a composition of ¢0o and generating functions f,,~ the coordinate changes. For example, ~'22= ~'20° ~'02 where ~'2o= c21 °(c10 ° too) and ~'02= (~'00° Col)° q2" Here the order of taking the compositions is important because c2t o qo and co~ o q2 do not exist. The linear stability of a periodic orbit x o, x~.... , x 0 = x 0 + P is determined by the residue

2-Tr(M) R =

4

'

where M is the Jacobian matrix of the Q times iterated map at x~: M = Q-'[I[..[ i=o

1

if(x/) l+f'(xj)

]

(2.14)

321

For a stable elliptic cycle, 0 < R < 1, whereas the residue of an unstable hyperbolic orbit is negative or greater than unity. It can be shown that a non-degenerate minimum of the action or a "physically stable configuration" is always a hyperbolic orbit with negative residue [141.

3. Seif-similad~ Breakup of an invariant circle is most easily tracked numerically by studying the sequence of orbits whose rotation numbers are truncations of the continued fraction expansion of ~" [4]. The truncation after the nth entry gives an orbit with rotation number ~'. = F./F.+ 1, where F. is a Fibonacci number: e . + , = r. + v . _ , ;

Fo=0.

= 1.

(3.1)

This orbit is the best approximation to an invariant circle (or a cantorus) among those periodic orbits whose periods are F,÷ t or less. At all parameter values, there are at least two orbits of a given rotation number ~',: one with a non-negative residue and another with a non-positive residue (see [11], for example), ltd. the usual critical case. there are precisely two such orbits and their residues have the limits [9]

R ~+--,0.250088...,

(32)

R,~ -~ - 0.255426 .... In the subcritical case (existence of an analytic invariant circle), R ---,0, whereas in the supercriticat case (existence of a hyperbolic cantorus), R _ oo for all well-ordered orbits cf rotation number ~'~ as n tends to infinity [12]. This suggests that the subcritical parameter r e , o n can be estimated as the part S~ of parameter space where the nonpesitb.,e residues ,~ orbits o? ro~afi~m ' "~ lie between 0 and some negative constant r. The estimate should become more and more accurate with increasing n. In order to guarantee rapid convergence in the parameter region where the breakup corresponds to the usual critical case, we choose the value -0.25 for the constant r.

J.A. Ketoja and R.S. MacKay/ Fractal boundaryfor the existence of invariant circles

322 i

(a)

k~

2

•

kl

"

",I

M' '

o

k~

k2

2~

- 1

(b)

-~

(e)

2 ~ ,

k~

k:z

2~

(d)

(:)

k~

k~ ~

x

Ol

~ , ,

-0.9

"

$7

k2

-0.4

Fig. 1. Successive estimates of the subcdtical parameter region. Mature cusps are denoted by the symbol M and immature ones by the symbol I. In (d) and (e) the parameter reg/on inside the small box Js similar to the total picture three levels earlier. (D gives the blowup of this region for n = 7 which is similar to the total picture for n -- 4.

In fig. 1 we present results of our numerical study for the extended standard map. A combination of the gradient method [1] and the Newton method is used to find, for every rotation number ~',, several hyperbofc orbits in the parameter plane. Most of them appear at too lugh absolute values of the parameters to be relevant to breakup of the invadan~ circle. The solutions we are interested in are tho~e which are crucial in determining the re#on 5.~. The boundary of S, is built up from the curves of the constant residue -0.25 for solutions which are relevant to breakup. These curves are shown in ~'he figure. Each S, is a connected set of parameter v~lues around the origin of the parameter plane. A crossing of two different constant

residue curves gives rise to a point where the boundary of S,, is not smooth. Following the terminology of Axel and Aubry [1], we call these points "cusps" although they become real cusps (zero angle) only in the limit n ~ ~ . At the boundary of S,,, the ground state (having the least action among the orbits with rotation number ~',,) is usually given by the orbit with the most negative residue. Only in the vicinity of a cusp, where two differ~n,t orbits have almost the same residue, does t; & rule fail. Both S~ and $3 have only one cusp in the parameter region k 1 > 0, k z < 0 whereas S4 has two cusps there. One could say that the cusp in $2 is just shifted to S 3, whereas the one in S3 splits into two cusps ha $4. We call a cusp in S~ mature if it splits into two cusps in Sn+ 1 and immature otherwise. In fig. 1 we denote mature cusps by the symbol M and immature ones by the symbol I. We observe that the cusp pattern of a general S, seems to be generated by the following rules: 1) An immature cusp in S, becomes a mature one in S~+ r 2) A mature cusp in S, splits into an immature and a mature one in S,+ I. One ends up with the self-similar tree of fig. 2 of successive layers of cusp patterns. Note that it consists of a main tree and a new "'sapling" on every third layer of the tree. In this paper, we will ~..M~ M

I --M( \I

-"-I --M --/~*'~"I --M

I I o

Fig. 2. The cusps arranged in a self-similar tree. The arrow points at the cusp with the symbol sequence [MMIMMM, Besides the main tree, there is a new "sapling" on every third laver.

J.A. Ketoja and R.S. MacKay / Fractal boundaryfor the existence of invariant circles

concentrate on the main tree. Let the number of branches in its nth layer (which corresponds to rotation number ~',+ l) be B~. From the third layer on, the tree can be viewed as a union of two parts which are both similar to the original tree. This allows one to write B,, as the sum of the numbers of branches in these two parts: Bn = B , - I + B,,_2, with B~ = B 2 ---1. Therefore, the number of branches is equal to the Fibonacci number: Each branch of the main tree can be represented as a sequence of symbols M and I. For example, in fig. 2 the arrow points at the cusp with the symbol sequence IMMIMMM. The fact that an immature cusp can be followed only by a mature one is most naturally taken into account if one drops the first I out of a sequence and rewrites it using symbols 0---MI and 1 = M. So the new representation for the cusp IMMIMMM would be 10111. In this new representation every sequence of O's and l's corresponds to some real cusp. The representation is also unique. The saplings have exactly the same self-similar ~tructure as the main tree. Everything seen in the parameter regiou k x > 0, k 2 < 0 is repeated in a smaller box around the k2-ards three levels later as shown in fig. l(d, e and f). We do not completely understand this structure yet, ~hough it is clear that it is related to the copy at k~ = 0, k 2 = - k ~ of the three-cycle found by Greene and Mao [6]. The type of perturbations occurring near k:~ = - k~ appear to be quite different, however, from those near k~ = +k~. In fact, we found that the scaling near k 2 = - k ~ is given by eigenvalues of about

323

4.25 and 12.66, instead of 4.25 and 10.7 as found by Greene and Mao near k 2 = +k~ (see table I), Thus it appears that there is a third unstable direction from their three-cycle, with eigenvalue 12.66. In Section 6 we will show that the above selfsimilarity is not a property of the extended standard map only but for example the results of Axel and Aubry [1] can be viewed in this way, too.

4. Why do we have mature and imma~,m'ecusps? In order to explain fig. 2, one should understand why the cusps are divided into mature and immature ones. The generating function formalism can be used not only to derive a plausible explanation but also to shed some light on the structure of the orbits which are involved in defining the border of S,. This information is later used in constructing the renormalisation theory for a breakup, First we will show how one can build up the action W~, L(V,~,v~) corresponding to a configuration which in K steps increases x from v, to v~ + L in symmetry coordinates. It is sufficient to show how W~O'L(Vo, V;) iS constructed, because W~'L(v,, V~) can be written as a composition of Woo x' L(V0, V;) and the generating functions for co, ordinate changes. The symmetry coordinates were defined by :~dng x modulo unity in eq (2.7). Therefore

wg,°(Vo,.;)=

(4.1)

is the action for a step from vo to v[~, whereas Table I The parameter values enrrognnndino lt~ th~ firS{ i m m ~ [ u r . ° cusp of the new sapling in S,, and the resulting estimates of the unstable eigenvalue.~ 81, 8~, n

k1

k2

81

$2

2 5 8 11 14

1.6606707435 0.1413838786 0.0109939225 0.0008671539 0.0000684938

-0.78528765 -0.82837257 -0.80938378 -0.80584665 -0.80498666

11.746 12.860 12.678 12.660

-0.8219 5.0761 ~.1543 4.2900

wg'

1( v o ,

= %.( vn, v; + ! )

(4.2)

corresponds to a step which increases x from v0 to v[~ + 1. W,~" L can be expressed as the composition ,L

Wdo, , O

" : o . . . o U : l " ""

(43)

K

{o,1},

E

L.

324

J.A. Ketoja and R.S. MacKay / Fractal boundaryfor the existence of invariant circles

We are interested in the ground state with rotation number ~',. It appears that in the parameter region k2 < 0 periodic ground states are always symmetric, i.e. invariant under 11 or I v On the other hand, one can easily show that every symmetric periodic orbit must go through two symmetry lines [2]. So the ground state can be found, if one considers periodic configurations xo, x l , . . - , xr,+~ = Xo + F,, such that two of the points lie on a given pair of symmetry lines and the action is made stationary with respect to variations in the other points. If the symmetry lines involved are t~ and /3, the corresponding action is denoted by W~, ( = W~,,a). It tams out to be useful to represent W ~ as the sum of two half-cycle actions:

t,'& = w~(o, o) + w;.(o, o),

(4.4)

where

where ot~ is chosen to make the sum an extremum. If Iohl is small, one can estimate the action b y W..~+I(O.O) = W..",(O.O) + W;r-l(O.O)

and similarly for Wv'~+l(0,0). Thus, the approximation for the action of the full configuration is

wg*.l= w:,(o,o) + wg-~(o,o) + w.7,(o,o ) + w~.(o,o) = w~a + w ~ 1.

w5.*1 = w.3(o,o) + w~-l(o,o) + w,7,-1(o,o)

+ w;a(o,o ) -

(4.5)

The integers K, and L. are determined by ~', as in [2]. We now give a heuristic argument for determining which sy~nmetry lines will be involved in ground states of successive rotation numbers ~',. Suppose that the ground state of rotation number ~'. passes through the lines ~ and fl so that

w & = w.'~(0,0)+ w;.(0,0)

(4.1o)

Similar estimates for the configurations which use other pairs of symmetry lines would give bigger values for the action. For example, the action of a configuration which has two points on the lines a and 8 could be approximated in the following way:

w2~, ( o,, q~ ) = wS., ~. ( o., q, ) ,

w;°(o~, o;) = wCF+,- K.:.(~o-~.)(~., ~;).

(4.9)

(4.6)

is the least action one finds. Suppose that the ground state with rotation number ~',_x uses the lines fl and 3', so has the action

wanBa + Wf~3 1 .

(4.11)

The result is bigger than W"+l,,r. because W~r~1 is smaller than W ~ x. After repeating the same for the other configurations one could see that the orbit which goes through the lines a and y is the probable ground state. In fig. 3 we sketch parameter regions A and B where the ground states for n = 2 and n = 3 are both deep minima. In the region A, the half-cycle actions of the ground states are n=2:

W~ ( F3/2 steps) W~ ( F3/2 steps),

n=3:

Wo~ ( ( F a + 1)/2 steos)

(4.12)

W3~ (( F 4 - 1)/2 steps), (4.7) and in the re,icon B Then our claim is that the ground state with rotation number ~'.~_~probably uses lines ~ and 3'. We write the corresponding half-cycle action in the form

n=2:

Wl~ ( F3/2 steps), n=3:

w~+ o, ~(0,o) = w;;~(0, ~,k) + ~ ; - ~(~, 0),

(4.8)

W32 ( F3/2 steps) W3 ( ( F 4 + 1)/2 steps) W~ (( i4 - 1)/2 steps).

(4.13)

J,A. Ketoja and R.S. MacKay/ Fractalbounda~,for the existence of mvariant {qrcles

3

325

structed by mixing the building blocks of A and

B: Wo]= W3 ° W3] = W ~ ° W 3 ((Fs + 1 ) / 2 steps), (4.16)

W~o= W&o W~ = W~o W~ (( Fs - 1 )/2 steps).

kl

1 -2

k2

0

Fig. 3. The parameter regions A and E where one can find well-defined ground states for both n = 2 and n ?= 3. The solid curve gives the border of S3 and the dashed c~rve that of Sz. In the region C both solutions which correspond to the ground states in A and B for n = 3 are hyperbolic. Here the actions are assumed tc~ be evaluated at the point (0, 0). In the way ~hown above, one can u:e these half-cycle actions as building blocks to construct the half-cycle actions of the ground states for n = 4. For example in the region A one can write

w~ = ¢vg o wg ((F, + 1)/Z steps),

(4.t4)

W~= W,i° Wol ((+'~ - 1)/2 steps), and in the region B

W ~ = W 3 o W ~ ((F5 + 1 ) / 2 steps),

(4.15)

W~ = W3] o W~2 (( F 5 - 1 ) / 2 steps), which are the same as in t~-,~ region A. In the vicinity of the cusp in fig. 3, however, the m i n i m a in the symmetry classes of (4.12), (4 13} have almost the same depth. So the above argum e n t will not work. Consider the parameter regm~ C where the continuations of ~he ground states from A and B for n = 3 are both hyperbolic. Numerically one notice~ that neither of the above combinations gives the ground state in C for n = 4. Yet, there is a third possibility which is con-

It turns out that this configuration has the least action in the region C. Because for n = 4 there are three hyperbolic solutions involved instead of the previous two, the cusp in S3 splits into two cusps in S4 and it is therefore mature. We will analyse both cusps in $4 in the same way as we studied the one in S 3. Because there is a cusp between the regions C and B, we write similar action tables for C and B when n = 3 and n = 4 as we did for A and B when n = 2 and n = 3. In the region C the half-cycle actions for n = 3 and n = 4 are n=4:

W4 ( ( F s + 1 ) / 2 steps)

w(o (( ~ - 1)/2 steps), n=3:

{4.17)

w 3 (( F 4 + 1)/2 steps) W~3 ( ( F a - 1 ) / 2 steps}.

For n =- 3 we choose here the actions of the previous region A because one finds nume'Acally that C concentrates on the region where the solution corresponding to actions of A with n = 3 is hyperbolic, The actions for B are now n = 4:

g4/'~ ( ( F 5 - 1 )/2 steps)

n=3:

Wa43(( Fs + 1)/2 steps), ~ ((F4- 1)/2 steps)

(4.18)

ve?~ (( r4 + I)/2 st~ps) As for the symmetry lines, these new tables for C and B are exactly the same as the previous tables (4.12) and (4.13) for A and B if one interchanges i and 2 (fffis point was made clearer by writing the actions in the order abvce). Therefore, we expect our new cusp in S4 to be also mature and to be

326

J.A. Ketojaand R.S. MacKay/Fractal boundaryfor the existenceof invariant circles

followed by the same cusp-splitting pattern as the cusp in S 3. Next we analyse the other cusp in $4 which lies in between the regions A and C. For that purpose one needs a new action table for the region A: n=4:

W~ ( ( F 5 - 1)/2 steps) W2~ ((F5 + 1)/2 steps),

n=3:

(4.19)

W33o (( F4 - 1)/2 steps) W~ ( ( F 4 + 1)/2 steps).

Mixing the building blocks of A and C gives the actions Wo~, W25o, W3~, and W~ but all these can be found also by composing the actions of A or C alone. Therefore the number of possible ground states remains two and the cusp does not split. In other words, it is immature. Finally one has to analyse the cusp in $5 which follows the immature cusp in $4. The half-cycle actions for n = 4 and n = 5 in the region A are n=5:

W5 (F6/2 steps)

W~ (F6/2steps), n=4:

(4.20)

W~3 ( ( F s + 1)/2 steps) W~2 ( ( F 5 - 1)/2 steps),

and in the region C n=5:

W~ ( F6/2 steps)

(r6/2 steps), n=4:

malisation procedure which will be given in the next section will be based on the half-cycle actions. We have already laid down the foundations of the theory by showing how these actions are involved in forming mature and immature cusps. From what has been written above, it might appear that there is a multitude of qualitatively different solutions which are crucial in determining the boundary of a general S.. A closer inspection reveals that there are in fact only two distinct solutions which are relevant to breakup (the correspor,ding half-cycle actions are given in an appendix). By looking at the half-cycle actions for n = 2, n = 3, and n = 4 one notices that two soluliens never go through the same symmetry line if their constant residue curves form a cusp. From the self-similarity it follows that this is true for an arbitrary n. So if one fixes the pair of symmetry lines for one solution, there is only one possibility for the pair of any neighbouring solution. By induction, there are on!y two possible pairs of symmetry lines for all solutions involved in determining.the boundary of S,. Numerically we found that near the boundary of Sn the solution is uniquely determined if one specifies the pair of symmetry lines it meets, the number of steps between these lines, and the average length proceeded in x during one half-cycle. The residues of our two qualitatively different solutions decrease and increase along the boundary of S, in such a synchronized fashion which makes the existence of several cusps possible.

(4.21)

W~ ( ( F s + 1)/2 steps)

5. Renonn~.lisafion

W(o ( ( F 5 - 1)/2 steps). Af*er interchanging 0 and 2 the symmetry lines involved are equivalent to the ones in the tables (4.12) and (4.13) of A and B for n = 2 and n = 3. Thus, we expect the cusp to be mature and to be followed by the same cusp-spfitting pattern as the other mature cusps already found. In conclusion, we have given a heuristic explanation for the self-sin~larity of fig. 2. The renor-

If the invafiant circle breaks up at parameter values do not correspond to a cusp point, the breakup can be explained in terms of the standard renormalisation theory [9]. In tiffs approach one considers high iterates of a mep on small length scales around the dominant line which is usually defined to be the symmetry line on which there is a point of an elliptic orbit for every rational rotation number. In the extended standard map

J,A, Ketoja and R.S. MacKay~ Fractal boundaryfor the existence of invariant circles

one cannot necessarily find such a line at all parameter values, and therefore we relax the deftnition considerably. We define the dominant line to be the symmetry line on which there is a point of an elliptic orbit for every rotation number ~,, with n large enough (n >_.N). Greene and Mao [6] noted that the cusp in the critical line of the extended standard map at k~ 0, k 2 = k~ is combined with a change in the dominant symmetry. In the region k 1 > 0, k2 > 0 the dominant line is 0. Because changing the sign of k I and shifting x by ½ leaves the map invariant, the dominant line in the region kl < 0, k 2 >__0 must be 2. In other words, the dominant symmetry is different along the two halves of the full cusp. This feature can be attached to the cusps in the region k 2 < 0 as well. Assume that at some point of the parameter plane, away from all cusps, the ground state with rotation number ~', (n > N ) uses one of the following pairs of lines depending on the value of n: aft ( n ( m o d 3 ) = 0 ) , fly ( n ( m o d 3 ) = l ) , or ya (n(mod3) --- 2). Here one has taken into account that the ground states with rotation numbers ~'~-1 and ~'~ use a common symmetry line so that one can compose the corresponding half-cycle actions to form the ground state with rotation number ~,+ v Then there is an elliptic orbit which becomes the ground state beyond the nearest cusp and goes through the lines yS, 8a, fl~ respectivdy. Because there is an elliptic orbit through the line 8 for every rotation number ~'n (n >_N), 6 is the dominant tine. But ~ cannot be the dominant iine beyorM the nearest cusp, because there an orbit through the line ~ is always hyperbolic. In other words, every cusp is combined with a change in the dominant symmetry. This must br, taken into acco~ant when one uses the standard renorrnalisation technique. In order to explain a breakup which takes place at one of the cusp points, or to give reasons for the existence of the Cantor set o~ cusps, we need a new theory. In the following, we develop a renormalisadon approach to the breakup of golden invariant circles for reversible area-preserving maps, capable of explaining this. A new feature in

327

our approach is that the renormalisation operators act on a triplet of actions instead of acting on a pair of actions as in the standard theory. In section 3 we represented a cusp as a sequence of symbols 0 and 1. Assume that there are two cusps for which the end of the symbol sequence is the same. They are both preceded with almost the same cusp-splitting pattern, and if one considered the scaling in the pattern, it would probably turn out to be very similar. One could explain this by showing that the half-cycle actions of the solutions which form these two cusps are similar. For that purpose, one needs renormalisation operators To and T1 such that the actions corresponding to a cusp a_ ka _ , + ~... a_ ~ (a~ {0,1}) can be generated by the eperator T,_T~_2... T,_, acting on the half-cycle actions of the first immature cusp in S2. it is most convenient to write the renormalisation operators in the form

r0=IoM,

(5.1)

TI = M, where the operato: M generates the actions of a mature and the operator I those of an immature cusp from the half-cycle actions of the cusps which precede them in the cusp-splitting tree. Let us consider a mature cusp in S. which splits into a mature and an immature one in S. + 1. There are ground states with rotation numbers ~',_ ~ and ~'. on both sides of the cusp in S,. On one side the ground states correspond to the half-cycle actions ff/~- 1 and W~v, which contain a con~x~on symmetry line fl so that one can compose them to construct the action of the ground state with rotation number ~n+ ~- On the other side of the cusp, the ground state with rotation number gn cannot touch the lines ]3 or 7. In other words, the hagcycle action for that solution must be Ws'~. From the actions { WJ~- 1 Why, Wd'~} one can construct the corresponding triplel of actions for the mature and immature cusp in S,,÷~. In the case of the mature cusp in S~+ ~, the half-cycle actions on one side of the cusp are ~.~ and W~+ 1 = g~,~-1 o g ~ , and on the other side the ground state corresponds

J.A. Ketoja and R.S. MacKay/Fractal boundaryfor the existence of invariant circles

328

to the action Wa"~+1= W ~ o W~#-1. Again one would like to be able to compose these actions in order to write down the triplet of the mature cusp in S,,+2. The actions W~v and W~ +l cannot be composed directly, but Kadanoff [8] showed for the standard map that the operation '/', defined by the equation

(5.1)

~-"(x, x,) = ~-(- ~', - ~ )

can be used to turn the order of symmetry lines around:

(w;,) (o, v,)= w;;(o, v,). n

(5.3)

'p

In the appendix, Kadanoff's argument is carried out for the extended version. So we can write the triplet of the mature cusp in S,+ 1 as {W~:, (W~+I) *, Wn~+l}. For the immature cusp, the actions on one side of the cusp are Wd~ and W~ + t = Wd~ * W~-x, so for the other side there remains only one possible pair of symmetry lines a and y, which results in the action W"+l=,,v W ~ - l ° W~ • The actions can be composed to generate the ones for the mature cusp in So+ z only if the triplet is written in the form ((Wfl,) ~', W ~ +1 tW"÷~)q'}. ,t. oty In conclusion, we have shown that if the operators M and I are defined by the relations

(5.4)

I(v,~','q} = {~,r/ov,(vo,r)~},

one generates the half-cycle actions of a mature cusp by the M operation on the actions of the preceding cusp in the cusp-splitting tree:

'

'

fly '

ya

•

(5.5) The half-cycle actions of an immature cusp follow if one operates by I on the actions of the preced-

ing mature cusp: I{ W ~ - ' , W;~,, W,~}

= { w:,, w~ +,, w.+~,o }.

(5.6) The triplet for the first immature cusp is { W2x3,W3~, W0~} where W~3 is chosen, because it is the only action which gives the triplet the right form when the other two actions are fixed. The triplet of any other cusp can be generated from {W~3, W3], W0~} by using the operators T1 and TO in the order given by the corresponding symbol sequence. For the Cantor set of cusps we suggest an explanation in terms of a horseshoe (see [7], for example, for a definition and properties) for the renormalisation operator T which acts like the operator Ts in one part of the space of triplets and like Ts in another part; these operators are defined by the equations Tos _- i s . M s, TxS= MS ,

(5.7)

where M s and I s are compositions of the operators M and I with appropriate scaling operators B M and Bt: M s = BM o M o BM1, I s = B t ° I ,, BF ~.

(5.8)

MacKay and van Zeijts [18] proposed a similar explanation for the period-doubling tree in two parameter families of bimodal maps of the interval. It is straightforward to apply their ideas to the present case. One represents each triplet of the horseshoe by a bi-infinite sequence of symbols P(A-,A+); A - = . . . a_2a_x, A + = a o a x . . . ( a , ~ {0,1}). The operation by T on this hyperbolic invariant subset is equivalent to the shift on the set of symbol sequences:

rge(A-,0A ÷) = e(A-0, A +), TIse(A-,1A') -- e(A-1, A +).

(5.9)

The horseshoe has 2-dimensional unstable mani-

J.A. Ketoja and R.S. MacKay / Fractal bounda~.,for the existence of invariant circle~

folds V~(A-) and codimension.2 stable manifolds VS(A +) intersecting at P(A-, A+). There are infinitely many periodic points which have the form A---

. . . A N A N A N,

A+--ANANA N ....

329

The firq model we consider is the variation of the one-timensional discrete Frenkel-Kontorova model studied previously by Axel and Aub~ [1]. The free-energy of the system is given by eq. (2.9) with

(Sa0) x,+ 1) = ½(x,+, - .,,,)2 + r(1 - cos(.,,,)) where AN=ata2...a N (a~e{0,1}). We expect the least period to be 3, and in general the period to be N, if rood(N,3)= 0, and 3N otherwise. The operator T has period-three cycles which are not contained in the horseshoe. One corresponds to the usual critical fixed point, viewed near a subdominant symmetry line in addition to the dominant line [23]. It is a three-cycle for T s. If one estimated a periodic point by the triplet { g ~ - x, W~v, W~ }, the actions W~a-t and W~, would correspond to the half-cycle actions of hyperbolic orbits of rotation numbers ~._ x and ~', as before, but Wn". would this time be the action of an elliptic orbit with rotation number ~, (refer to the beginning of this section). Another period-three cycle for Ts, not included in the horseshoe, is the one found by Greene and Mao [6]. For the bimodal period-doubting problem, MacKay and van Zeijts incorporated the usual Feigenbaum fixed point into the renormalisation horseshoe picture. It remains to be seen how these extra three-cycles for our renormafisation operator can be incorporated into the picture with a horseshoe.

6. Universality While the above renormatisation theory is formulated for maps of standard map form, we expect that similar behaviour to that of the extended standard map ca~ be found in a relatively large class of two-parameter faraJfies of reversible areapreserving maps. We discuss two other examples where the critical fine has already been found to have a Cantor set of cusps, which probably belong to the same universality class as the extended standard map.

+(-1)'Ex,,

(6.1)

where the parameter E simulates an electric field which breaks a symmetry between two equivalent sublattices in certain incommensurate structures. If one looks for the ground state, one is led to a variational problem which can be turned into an area-preserving map:

Y,.t --Y. + K sin (x.) + ( - 1 ) " E . x.+a = x. + y. +

(6.2)

K sin(x,,) + ( - 1)"e.

The change of variable

zi = xi+ ( - 1 4) iE

(6.3)

transforms the action into the form

(6.4) There are a number of symmetries which reduce the problem considerably. Firstly, one can change the sign of K and shift z by ,rr without affecting the free energy. Secondly, changing E into - E is equivalent to exchanging the role of the even and odd sublattices. Thirdly, changing E into 4~r- E is equivalent to shifting z by ~ ~nd exchanging the role of the even and odd sublatrices. Therefore, it is sufficient to study the model in the parameter region K >_0, 0 __ E < 2~r. Next one would like to find out the effective rotation number of a solution in different regions of the parameter space. In the most general context, rotation number can be defined for commuting pairs of maps (U, T) [9]. For example, if a

J.A. Ketoja and R.S. MacKay/ Fracml boundaryfor the existence of invariant circles

330

point is invariant trader UeT Q, it is said to have rotation number P/Q. In the case of the extended standard map, T is given by eq. (2A) and U is the map (x, y ) ~ (x - 1, y) when k 1 > 0 and the map (x, y ) ~ (x - 1/2, y) when k 1 = 0. If the map of eq. (6.2) is denoted by F~ when n is even and by F_ E when n is odd, the commuting pair for this model in the region K>_0, 0 < E < 2 c r is (R, F _ r F E ) where the map R is defined by the equation Yn+ ~ ~'~Y n '

x,,+l = x~ - 2¢r,

(6.5)

and the other member of the pair is taken to be F_ EFt, because with this choice one gets rid of the explicit dependence on n of the map and is likely to find generic behaviour. The definition of rotation number in terms of this commuting pair results in an equivalent definition of rotation number (or (in)commensurability ratio) ~ the limit

1

~0 = ~

.

~m

x2N-x o N

(6.6)

But when E = 0 (K >__0) the commuting pair can be taken to be (R, Fo). So an orbit wbdch has rotation number P / Q and is invariant under R~'(F_eFe) Q when E > 0 , is invariant under R eFffO when E - 0 and therefore has on this axis effectively rotation number P/(2Q). When E = 2~, the map F_eF ~ can be written also in the form R - IFo2 so the same orbit is invariant under R(e-°-)Fff c2 and thus has effectively rotation number ( P - Q)/(2Q) on this line. In summary, if rotation number ~o is defined as the limi~ (6.6) in the parameter region K > 0, 0 < E < 2~r, the effective rotation number is ~o/2 at E = 0 and ( # - l j / 2 at E = 2¢r, Axel and Aubry [1] estimated the subcritical parameter region for an invadant circle with rotation number ~" in the same way as wa~ done in case of the extended standard map in gection 3. Each S~ had two groups of cusps in the region K > 0, 0 < E < 2~r: one r~ear the ax~s E = 0 and

another near the line E = 2~r. If their results are compared with ours, it seems that the cusps in both groups can be represented by the same selfsimilar tree of fig. 2 as the cusps in the extended standard map. In their case, the rotation number which corresponds to the appearance of the first immature cusp of the tree differs in the two groups. For the group near the line g = 0 it is ~3 and for the group near the line g = 2~r it is ~'4- The fact that the formation of cusps is one level retarded near g = 0 and two levels retarded near g--2~r from that in the extended standard map can be understood by looking at effective rotation numbers. For the extended standard map, the effective rotation number of an invariant golden circle at k 1 = 0 is 2~" which has continued fraction expansion [1,4oo]. The first convergents of 2~', i.e. the successive truncations of its continued fraction expansion, are 1/1, 5/4, 21/17, . . . . On the other hand, the invariant circle is approximated by orbits which have effective rotation numbers 2~n (0/1, 2/1, 1/1, 4/3, 6/5, 5/4, 1 6 / 1 3 , . . . ) at k 1 = 0. Every third of these is a convergent of 2L Therefore it is not surprising that there is a threecycle for the renormalisation operator at k: = 0 and in the vicinity of that line one sees a formation of cusps beginning at rotation number ~2 = 1/2 whose effective value is the first convergent of 2L In the Frenkel-Kontorova model the invariant golden circle has effective rotation number ~/2 = [0, 3,400] (coLvergents: 0/1, 1/3, 4 / 1 3 . . . . ) at E = 0. Again every third of the effective rotation numbers L / 2 (0/1, 1/2, 1/4, 1/3, 3/10, 5/16, 4/13 .... ) of the approximating orbits is a convergent of ~'/2. The formation of cusps begins at rotation number ~3 = 2/3 whose effective value 1/3 is the truncation of ~'/2 after the "non-universal" part (i.e. the part preceding the infinite sequence 4~) of its continued fraction expansion. At E = 2~r the effective rotation number of the ingariant golden circle is (~" - 1)/2 - [0, - 5 , - 4 J for which the convergents are 0/1, - 1 / 5 , - 4 / 2 1 , . . . . The effective rotation numbers for

J.A. Ketoja and R.S. MacKay/Fractal bounda~ for the existence of invariant circles

the approximating orbits are - 1 / 2 , 0/1, - 1 / 4 , - 1 / 6 , - 1/5, - 3/16, - 5/26, - 4/21 . . . . . This time the trunacation after the non-universal part of ( ~ ' - 1 ) / 2 gives rotation number - 1 / 5 . Because it is the effective value of ~4 = 3/5, we expect the formation of cusps to begin at this rotation number. Axel and Aubry conjectured that the total number of cusps for S, in the parameter region 0 < E < 4¢t is the Fibonacci number F,+ x and that F,_ 1 of them are centered around the line E = 2~r whereas the other F, lie near the lines E = 0 and E - 2~. This picture seems to be correct. In order to draw this conclusion one has to take into account four groups of cusps, each of which contains the main tree and saplings (with F, cusps on their n th level), and the cusps formed at the lines E = 0 and E = 2~r (there is a cusp at E --- 0 when n(mod3)=l,2 and a cusp at E = 2 ~ r when n(mod 3) = 0, 2). The second case which probably belongs to the same universality class is a system studied by Schellnhuber and Urbschat [21]. They study a two-dimensional map which results when one tries to calculate the eigenstates of a certain quasiperiodic SchrSdinger operator. In their model the subcfitical parameter region corresponds to an extended state and the supercfitical to a localized state. These two regions are separated by a fractal curve, very similar to the critical curve in the extended standard map. Unfortunately their paper does not contain enough information to conclude whether the self-similarity in the cusp-pattern is exactly the same as in our case or not. These examples suggest the following general principle: If a two-parameter family of reversible area-preserving ~ a p s has a line in the parameter space where the map has Z 2 symmetry, one is likely to see a Cantor set of cusps in the critical fine for breakup of the golden invariant circle. The formation of cusps follows the self-similar tree of fig. 2 and begins at rotation number ~, whose effective value corresponds to the truacation after the "non-universal" part of the conthmed fraction expansion of the effective value of L

331

7. Discuss|on In section 3 numerical results were presented suggesting that in the parameter region k~ > 0, k 2 < 0, the boundary of the existence of a golden invariant circle for the extended standard map has a Cantor set of cusps. In section 3 we developed a renormalisation approach in order to explain tiffs. The cusps were found to be arranged in a simple tree and the underlying self-similarity was explained by using the action representation. One could represent each cusp, and the corresponding triplet of actions, by a sequence of two symbols. This led to a renormalisation conjecture in terms of a horseshoe for the renormalisation operator in the space of triplets. We did not make any attempts to prove our conjectures. They could be tested by first solving for periodic points of the renormalisation operator and corresponding eigenvalues and then comparing the results with the scaling of cus!0s in the parameter space [18]. Both parts of this test are difficult but not impossible. In order to approximate a periodic point of the operator one should represent three actions by power series in two variables. The requirement of perio2icity would lead to a large system of nonlinear equations for the coefficients. On the other hand, finding the precise position of a cusp on the parameter plane would require a three-dimensional Newton method. Tiffs would not take too much computer time if cne had a good initial guess. Our renormalisation horseshoe seems to have quite a large universality class. In section 6 we discussed the Frenkel-Kontorova model with two sublattices, where the cusps seem to form the same tree-structure as those in the extended standard map, and ScheUnhuber and Urbschat's model which display similar behaviour. Another case which may be related to this universality class, though it is not so clear, is Wilbrink's [24] family. Wilbrink studies breakup in a map which, in addition to the "'nonlinearity" parameter, contains a parameter which can be used to "tune" the map from the standard map to a piecewiseqinear case

332

J.A. Ketoja and R.S. MaeKay/ Fractal boundary for the existence of invariant circles

C triangle" map). The subcritical parameter region in that two-parameter system resembles very much the one in the extended standard map. He observes cusps which result from stability exchanges between two orLk~ of the same rotation number (refer to section 4). Everything written about these exchanges goes word by word to the extended standard map. An elliptic orbit changes to hyperboric, emitting two elliptic orbits which do not go through the symmetry lines. They will move towards the other hyperbolic orbit and change that to an elliptic orbit. He notices also that stability exchanges result in variation in the dominant symmetry in the same way as in the extended standard map. The cusps, however, are all located on the line corresponding to the non-analytic case which is outside the space of systems that we consider. Satija and Wilbrink [19] conjecture that the phase diagram exhibits asymptotic self-similarity when approaching the non-analytic case, and support this with some computations. While the theory is limited at present to maps of the standard map form, we expect it to generalise to all reversible area-preserving maps. Furthermore, by analogy wi'h the case of areapreserving period-doubling we do not expect any new unstable directions if non-reversible maps are allowed. So we expect the universality class to include many non-reversible maps, too. One feature of the critical line in the extended standard map that remains to be explained is why the perturbations in the immediate neighbourhoods of the points k: = 0, k 2 -- + k~ and k 1 = 0, k 2 = - - k ~ appear to be restricted to different subspaces. The latter point is equivalent to Greene and Mao's [6] three-cycle but gives birth to a new sapling every tlfird renormafisation. We found that the three-cycle has at least one more unstable direction in addition to the two they found. Also the second eigenvalue that they found is still not understood. Besides the actual breakup, there are also a number of ether things which require more careful consideration. For example, the structure of a cantorus after breakup could vary in different

regions of parameter space. At a cusp there are more than two orbits with the same rotation number - a fact that might lead to an invariant two-hole Cantor set instead of the ordinary one-hole set. It is clear that this has to happen on kl = 0 on passing through k 2 - 5: k~., for example, by the 12 symmetry. It would be interesting to study formation of a second orbit of gaps in a eantorus.

Acknowledgements J. Ketoja is grateful to the Academy of Finland and the Emil Aaltcnen Foundation for their financial support. The computations were made on equipment granted by the UK Science and Engineering Research Council.

Appendix Half-cycle actions As was explained in section 4, there are two significant locally m:nirrfising states with rotation number ~'., which interchange being the ground state as one passes through a cusp in the extended standard map. One is given by the half-cycle aetions n ( m o ~ 3) = 0:

W0~ ((F.+ 1 + 1 ) / 2 steps) W~ (( F,,+:- 1 ) / 2 steps),

n (rood 3) = 1 :

W~ (( F.+: + 1 ) / 2 steps) Wl~ ((F.+ 1 - 1 ) / 2 steps),

n ( m o d 3 ) = 2:

(A.1)

W~ ( F.+:/2 steps) W~ ( F. + 1/2 steps).

and the other corresponds to the actions n (rood 3) = 0:

Wz] (( F,,+I + 1 ) / 2 steps) Wl~ (( F.+ a - 1 ) / 2 steps),

~ (rood 3) = ~ :

W~ ((F.+: + 1 ) / 2 steps)

Wf2 (( F.+ 1 - 1 ) / 2 steps),

(A.2)

J.A. Ketoja and R.S. MacKay / Fractal bounda~, for the existence of invariant circles

n ( m o d 3 ) = 2:

W3] (F.+1/2 steps)

333

which increases x by unity. The half-cycle actions are

W~ ( F,+ x/2 steps).

Wo'~(o,o')=~o,(O,o') = ( ~ ' - o + ~)~ +~g(' ~), These tables can be easily proved assuming that the half-cycle actions of the ground state with rotation number ~'.+1 can be always expressed as the compositions

Wlo(o, v ') = q0(o, o' + i) = ( o ' - ~ + ~)~- + -~g~~'), W~3(v,v')=',h3(v,o'+ 1)

(A.5)

= ( o ' - o+ ~)~ + " g ( ~ + ~),

w : z I = y:~ . w ~ - l, (A.3)

w~(,,,v') = ~(o,

w;+ ~ = w ; ( ' o w~'gt ct

o')

= ( ~ ' - o + ~)~ + ~g(o' + ~).

of the actions of ground states with rotation numbers ~,, and ~ - t - By comparing (A.1) and (A.2) with the tables (4.12) and (4.13) one first notices that the half-cycle actions are correct for n = 2 and n = 3. After that one has to check that the compositions of half-cycle actions for n (rood 3) = 2 and n(mod 3 ) = 0, as in eq. (A.3), give those of n (mod 3) = 1, and that similar results hold for the compositions of actions vdth n ( m o d 3 ) = 0 and n ( m o d 3 ) = 1 and with n ( m o d 3 ) = 1 and n ( m o d 3 ) = 2. Then by induction the tables (A.1) and (A.2) are correct for all n. Essentially the same tables are contained in Kadanoff's early work on the standard map [8]. According to eq. (A.3), one can compose an arbitrary half-cycle action if one knows the actions for n - - 0 and n = 1. The case n = 0 corresponds to the motion in which the full cycle consists of one step and the total progress in x is zero. This kind of motion is generated by the half-cycle actions

w ° ( o , o9 = ~o~(O, o9 = ( o ' - o) ~ + ~g(o), w ° ( ~ , o3 = ~o('O, ~') = ( o ' - ~)~ + ~g(o9,

wo(o, o')= ~,(o, ~')= (~'- ~)~ + ~g(o+ ~), w~(o, o')= ~,~(v, ~')= (o'- o) ~ + ~g(o' + ~). (A.4) When n = 1 the full cycle consists of one step

Finally we will show that the same symmetry property which Kadanoff [8] found for the halfcycle actions of the standard map is obeyed by those O~ the extended version as well. Take the definition of the operator 'P" as in eq. (5.2): Ct'(x, x') = I-(- x', - x ) . Then 'r

o

~)'/" ~ p ~ o T ~.

(A.6)

From eqs. (A.4) and (A.5) we see that the symmetry property of eq. (5.3), (W~"v)~'(v,v') = W~( v, v'). holds for n = 0 and n = l (g(-x)=g(x), g ( - x + 1/2) = g(x + 1/2)). Assume that it holds for n - i and n. From eqs. (A.3) and (A.6) it is easy to show that the same property is true for n + 1, and therefore for an arbitrary n.

References [1] F. Axel and S. Aubry, Polaxisation and transition by breaking of analyticity in a one-dimensional model for incommensurate structures in an electric field, 3. Phys. A 20 (1987) 4873-4889. [2] R. DeVoge!aere~ On the structure of symmetric perodic solutions of conservative systems, with applications, in: Contributions to the Theo~ of Nonlinear Oscillations, S. Lefschetz, ed., vol. IV (Princeton Univ. Press, Princelom N J, 1958), pp. 53-84. [3] D.F. Escande, M.S. MohamedBenkadda and F. Doveil, Threshold of global stochasticity and universality in Hamiltonian systems, Phys. Lett. A 101 {1984) 309-313. [4] 3.,4. Greene, A method for determining a stochastic transition, J. Math. Phys. 20 (1979) 1183-1201, [5] J.M. Greene, H. johannesson, B. Schaub and H. SuhL Scaling anomaly at the critical transition of an incommen-

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