Cantori for multiharmonic maps

Physica D 69 (1993) 59-76 North-Holland SDI: 0167-2789(93)E0208-S

Cantori for multiharmonic maps C. B a e s e n s I a n d R . S . M a c K a y 2 Nonlinear Systems Laboratory, Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Received 10 January 1992 Revised manuscript received 29 March 1993 Accepted 29 April 1993 Communicated by C.K.R.T. Jones

We compute all the cantori and their gap and turnstile structures, for area-preserving twist maps near non-degenerate anti-integrable limits with arbitrarily many wells per period. The results imply a rich bifurcation diagram for cantori of families of maps containing degenerate anti-integrable limits. We conjecture the structure of this bifurcation diagram in the case of the two-harmonic family.

1. Introduction Cantori are of fundamental importance to two classes of problem. T h e first is Hamiltonian dynamics. The question here is how invariant tori break as energy and other p a r a m e t e r s are varied, and what happens to the dynamics when they do break. In the simplest context, area-preserving twist maps, A u b r y - M a t h e r theory shows that an invariant circle of irrational rotation n u m b e r breaks by the conjugacy from rotation becoming discontinuous, leaving behind an invariant Cantor set, christened a " c a n t o r u s " by Percival [19]. These allow orbits to leak through. For a large class of m a p s and rotation numbers, the breakup boundary in p a r a m e t e r space for an invariant circle is believed to be smooth, as a result of simple renormalisation conjectures. But for families of m a p s which we call "multiharmonic", the b r e a k u p boundary exhibits a fractal structure Also at: Laboratoire L6on Brillouin, CEN Saclay, 91191 Gif-suroyvette, France. 2Nuffield Foundation Science Research Fellow 92-93.

[10]. We believe that this is because multiharmonic maps can have m a n y cantori of the same rotation n u m b e r (as we shall prove in this p a p e r near the "anti-integrable limit"), and thus the breakup boundary is c o m p o s e d of m a n y pieces, one for each type of cantorus that the circle can leave behind. Thus it is clearly important to understand the range of possible cantori. The second class of problems is variational problems with competing length-scales, such as arise in solid-state physics. The question here is what the physical minimum energy state looks like. In the simplest class of models, called Frenkel-Kontorova models, the minimum energy states are always ordered like a rotation. The rotation n u m b e r depends continuously on parameters, so is often irrational. N e a r to socalled "anti-integrable" limits, each irrational case corresponds to a cantorus for an associated area-preserving twist map. But for multi-well anti-integrable limits there are m a n y cantori of the same rotation number. So which one is selected? In this paper, we give a detailed description of the set of cantori for systems n e a r a non-degen-

0167-2789/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All fights reserved

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C. Baesens, R.S. MacKay / Cantori for multiharmonic maps

crate anti-integrable limit, and more generally of the set of rotationally-ordered orbits. We deduce the existence of many bifurcations of cantori in families containing degenerate anti-integrable limits (the same will be done with yet more detail for the rotationally-ordered periodic orbits [11]). As a result, we make a plausible conjecture about the breakup boundary for invariant circles of 2-harmonic maps. We also make some deductions about the minimum energy cantorus. 1.1. Mathematical context We consider C 1 exact (=zero net flux) areapreserving twist maps f of the cylinder -~ x R with lift F: (x, y) ~ (x', y ' ) to R 2, and generating function h E C2(R 2, R), so y' dx' - y dx = d h ,

(1.1)

and h(x + 1, x' + 1) = h(x, x')

(1.2)

(cf. [4], for example). Their orbits correspond to the stationary states x E Nz of the action functional W(x) = ~

h(x,,x,+l),

(1.3)

nEZ

that is, sequences for which h2(x,_l,x,) + hl(X,,X,+l)=O for all n E 77,

(1.4)

recurrence relation (1.4) R2~, Xn+ 1 = g ( X n _ l , Xn) ,

defines a map

g:

(1.7)

which is often a more useful way of viewing f. We will consider maps for which h is C2-close to an anti-integrable limit [1,20] h(x, x ' ) = V(x) ,

(1.8)

for some function V of period 1. An anti-integrable limit does not generate a map, but (1.3) still defines a well-posed variational problem. For the anti-integrable limit the stationary states are easy to work out. They consist of all mappings from Z to the set E of critical points of V: E = {x E ~: V'(x) = 0 } .

(1.9)

An anti-integrable limit is said to be non-degenerate if V " ( x ) # O for all x E E . As noted by [1,20], many of the stationary states of a nondegenerate anti-integrable limit can be proved to persist on small perturbation, hence giving orbits of the corresponding area-preserving twist map when h is convex. In fact, orbits can be smoothly continued as parameters vary as long as they have "phonon gap", a property which has been shown to be equivalent to uniform hyperbolicity [21. For k, m E 7, define the translations Rkm on states x ~ ~z by (RkmX), = x,+ k + m .

(1.10)

where subscripts 1 and 2 denote partial derivatives with respect to the first and second arguments respectively. We suppose that the tw/st Ox'/Oy is positive, so correspondingly

We give ~z the product topology, that is a sequence of states y J, j E ~, is said to converge to a state x if for all e > 0 and N E N1 there exists J E N such that for all j -> J,

hi2<0.

ly~-x,l
(1.5)

Conversely, functions h satisfying (1.2), (1.5) and for which [hl(X,X')l---~oo

as Ix ' - x [ - - * ~ ,

(1.6)

generate area-preserving twist maps by (1.1). We call such h convex. Under these conditions, the

for In] < N .

(1.11)

A stationary state x is said to be recurrent if there exist sequences of translations Rk,. with k---~ _+oo such that Rk,,X---~x. A subset S of R z is closed if it contains all its limit points (in the product topology). A subset S is invariant under the translations if for each x E S and translation

61

c. Baesens, R.S. MacKay / Cantori for multiharmonic maps Rkm, RkmX E S. A subset S is rotationally ordered

if for any two distinct x, y E S, either x, < y , for all n (written x < y ) , or x, > y , for all n (written x > y ) . For a rotationally ordered set invariant under the translations there is a unique number to such that RkmX < X

if kto + m < 0 ,

RkmX>X

ifkto+m>0,

(1.12)

called its rotation n u m b e r (e.g. [17]). In this paper, a closed rotationally ordered set S of recurrent stationary states, invariant under the translations, with irrational rotation number is called a cantorus if it is disconnected. The reason for the name (due to [19]) is that its quotient by R01 is topologically a Cantor set, whereas if S had been connected its quotient would have been a 1-torus (circle), corresponding to a rotational invariant circle for the map f. For mathematicians, a cantorus corresponds to a Denjoy minimal set for f, embedded so that cyclic order on the Denjoy minimal set corresponds to horizontal order on the cylinder. For area-preserving twist maps, Aubry and Mather proved existence of a cantorus for each irrational rotation number to for which there is no rotational invariant circle (see [4] for a good survey). Near enough an anti-integrable limit there are no rotational invariant circles (this will be justified in Appendix A), consequently there is a cantorus for each to. There can be more than one cantorus for a given to, however. An example is given in Appendix A of [13] near an anti-integrable limit with double well potential, but the idea was already clear to Aubry and Mather (e.g. [18]). Every cantorus has a hull Junction, that is a non-decreasing function X: •--* R with X(O + 1) = X(O) + 1, such that for each stationary state x of the cantorus there exists 0 ~ R -+ (that is the set of limits to real numbers from the right and left) so that x,,(O ) = X(nto + 0 ) .

(1.13)

A gap in a cantorus is a pair of points on the cantorus for which there are no points of the cantorus in between in rotational order. It corresponds to a jump discontinuity in the hull function. For convex h, the gaps in a cantorus come in orbits which we call holes. The number of holes is called its gap structure. If the cantorus is hyperbolic, there is an arc of stable manifold and an arc of unstable manifold spanning each gap. The way they intersect, which is the same for each gap in a given hole, is called its turnstile structure. We will use the following notation. For to ~ R, define the interval

l(to) = (N, N + 1)

(1.14a)

if to lies between the integers N and N + 1, and I ( N ) = ( N - 1, N + 1).

(1.14b)

Also we recall the following result, which is a simple consequence of (1.12). For a rotationally ordered state with rotation number to, down((n - m)to) < x,, - x m < up((n - m)to) , (1.15) where down(x) is the greatest integer less than x and up(x) is the least integer greater than x. 1.2. Plan o f the paper

The aim of this paper is to describe the set of all cantori for maps with generating functions close to non-degenerate anti-integrable limits with an arbitrary number of wells per period. In addition, their gap and turnstile structures are described. We will show that the set of cantori changes qualitatively when the number of wells changes. Hence, families of maps containing a degenerate anti-integrable limit transversally must have many bifurcations of cantori. Furthermore, even when the number of wells does not change, the phase diagram indicating which is the global minimum energy cantorus is complicated, cf. [8]. The form of the bifurcation diagram for cantori

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C. Baesens, R.S. MacKay / Cantori for multiharmonic maps

of a 3-parameter family of two-harmonic maps (6.1) is conjectured, by interpolating between the results obtained here and numerical results of [10] and others on the breakup boundary for invariant circles in this family. The structure of the paper is as follows. In section 2 we establish a one-to-one correspondence between rotationally-ordered orbits near a non-degenerate anti-integrable limit and certain equilibrium states of the anti-integrable limit. In section 3, we describe the set of all cantori of given rotation number, near a nondegenerate anti-integrable limit. In section 4 we count the number of holes for each cantorus, and in section 5 we find the shape of the turnstiles for each cantorus. In section 6 we deduce the existence of many bifurcations of cantori in families containing anti-integrable limits with different numbers of wells, and conjecture a complete bifurcation diagram for the two-harmonic family. In section 7 we discuss which cantorus is the minimum energy one. In section 8 we summarize our results and pose some questions for the future. In Appendix A, we show that there are no rotational invariant circles for all maps with generating function in a neighbourhood of any set of non-trivial anti-integrable limits (meaning that max V(x) - m i n V(x) is bounded away from zero). In Appendix B, we show that rotationallyordered orbits remain rotationally-ordered as long as they can be continued with phonon gap. These two results are necessary for section 6. In Appendix C we show that every hole for a uniformly hyperbolic cantorus has a homoclinic orbit, which is required for section 5.

2. Rotationally ordered states near non-degenerate anti-integrable limits

from an anti-integrable state, near enough the anti-integrable limit? First we show that near an anti-integrable limit, the points of every rotationally ordered stationary state are contained within a set where V'(x) is small (cf. [6,12]).

Lemma 2.1. If a rotationally ordered set S of stationary states for generating function h(x, x') = V(x) + T(x, x')

(2.1)

has rotation number to, then its points are contained in the set where [V'(x)[-< sup{lTz(x- , x) + TI(X , X')[" (X -- X-), (X' -- X) E / ( t o ) } ,

(2.2)

where I(to) is defined in (1.14).

Proof. This is clear from the equation for equilibrium:

T2(x._x, x.) + T,(x., x.+,) + V'(x.) = 0 , plus (1.15).

(2.3) []

L e m m a 2.1 can be strengthened in an important way. For each non-degenerate maximum M of V, choose a closed interval I m about M on which V"(x)< 0. Think of I M as an arc of the circle ~1 = •/Z. Given a subset S of R z, define • -S = {x, mod Z: x E S, n E Z) C T l .

(2.3a)

Lemrna 2.2. For each to E R, and interval Iu as above, there is a C 2 neighbourhood N of the anti-integrable limit V such that given any rotationally ordered set S of stationary states of rotation number to (invariant under the translations) for a convex generating function h ~ N, ItS has at most one point in I M.

Proof. Given a maximum M and interval Ira, let Which stationary states of an anti-integrable limit give rotationally ordered states on continuation to nearby area-preserving twist maps, and does every rotationally ordered state arise

c = min{-V"(x): x ElM) > 0 .

(2.4)

Define N to be the set of generating functions of the form h = V + T with

C. Baesens, R.S. MacKay / Cantorifor multiharmonic maps

IQ[
for(x-x-),(x'-x)~I(to),

(2.5)

where a = T22(x-, x) + Tll(X, x ' ) .

(2.6)

The equation for tangent orbits ~c~ is T21(x-, x) ~c- + ( a + V"(x)) 8x + T12(x,x' ) 8x' = 0 ,

(2.7)

where ~c-, ~c, ~0c' represent three successive values of ~c~, and x - , x, x' the corresponding values of x~. So if h ~ N is convex, x E I M and ~c---0, then at least one of ~c- and ~ ' is negative. In particular, if ~r- > 0 then ~ ' < 0.

(2.8)

Now suppose for a rotationally ordered set S of stationary states, two points l, r of ~rS, with 1 < r, lie in I M. Then l- < r - , by rotational order, For 0 -< t --- 1, define x(t) = (1 - t)l + tr, x - ( t ) = (1 - t)l- + tr- ,

(2.9)

Proof. From Lemma 2.1 and non-degeneracy of the anti-integrable limit, the points of a rotationally ordered stationary state S are contained in vanishingly small neighbourhoods of the critical points. By non-degeneracy, S has phonon gap and so continues uniquely, in particular to the anti-integrable limit, where it becomes an antiintegrable state A. A limit of rotationally ordered states need not be rotationally ordered, but it is at least weakly rotationally ordered. By Lemma 2.2, ~rS has at most one point near any maximum (modulo 1), hence the same is true for ~rA. This completes the proof. [] Next we prove the converse. Theorem 2.4. Given a weakly rotationally ordered anti-integrable state A of a non-degenerate anti-integrable limit such that ~'A has at most one point on each maximum (modulo 1), its continuation for nearby convex generating functions is rotationally-ordered.

(2.10)

and x'(t) = g(x- (t), x(t)) ,

63

(2.11)

where g is defined in (1.7). Differentiating (2.9)-(2.11) with respect to t and using (2.8), we deduce that 1' > r ' which contradicts rotational order. [] Now we are nearly ready to show that every rotationally-ordered state near an anti-integrable limit comes from an anti-integrable state with nice properties. First we make a definition. Let us say a state x is weakly rotationally ordered if for each k, m E Z, either x~ <- (Rk,,X), for all n, or x . >- (RkmX)n for all n. Theorem 2.3. Every rotationally-ordered stationary state S near a non-degenerate anti-integrable limit has a unique continuation to an anti-integrable state A; the latter is weakly rotationally ordered and ~rA has at most one point on each maximum (modulo 1).

Proof. Let x be the unique continuation of the anti-integrable state A for a nearby convex generating function. The points of x remain near the critical points of V corresponding to the sequence A, Suppose y is a translation of x, Rpqx = y ~ x. We have to show that either y > x or y < x. Let B = RpqA. By the hypotheses on A, the only cases we have to worry about are when for some integers m and n (with possibly m =-~ or n = + ~ , but not both), A k = B k, minima, for m < k < n , and without loss of generality, we suppose A m < B , , and A , < B , when m (respectively, n) is finite. Then we have to show that x k < Yk for m < k < n. In the case m, n finite, this is immediate from Aubry's fundamental lemma (for finite minimising segments), which says that the graphs of xk versus k for two minimising segments can cross at most once (e.g. [4]). Although usually proved for globally minimising orbits, the lemma can be adapted to compare nearby locally minimising segments.

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C. Baesens, R.S. MacKay / Cantori for multiharmonic maps

In the case that m or n is infinite, without loss of generality n = +0% x and y must be asymptotic to each other. One way to see this is that D ZW is invertible in ~e2, and hence if the sequence A k B k, k > - m , is in ~¢2 then so is x k - y k . In particular the difference goes to zero as k--->o0. Then Aubry's fundamental lemma applied to semiinfinite minimising segments implies that Xk < Yk, k > m. Again, the lemma is usually proved for globally minimizing orbits, but can be adapted to compare nearby locally minimising segments. This proves the result. [] R e m a r k 1. An alternative proof of Theorem 2.4

can be given by interpolating between x and y by a path of stationary segments ~(t) for which the endpoints move monotonically with t, and showing that the derivative Og/Ot > 0, using Cramer's rule to show that the inverse of a positive definite tridiagonal matrix is a positive operator. R e m a r k 2. We

cannot expect a result like Theorem 2.3 near a degenerate anti-integrable limit, because there is no reason that rotationally-ordered orbits should continue at all. They could, for example, annihilate in pairs before reaching the anti-integrable limit.

3. The set of cantori of given rotation number

From section 2, it is easy to describe the set of cantori of given irrational rotation number to near a given non-degenerate anti-integrable limit. If the anti-integrable limit has just one well ml per period, there is a unique cantorus of rotation number to, and it is given by the hull function X(O) = m~ + [0],

(3.1)

where [0] denotes the greatest integer less than or equal to 0. As explained in the introduction (eq. (1.13)), this means the cantorus is given in sequence space by Mo = {x(O): 0 ~ lI~_+},

(3.2)

where (3.3)

x,(O ) = X(nto + 0 ) ,

and limits from the right (respectively, left) are taken for 0 ~ ~+(respectively, R - ) . If the anti-integrable limit has two wells per period, m l < m 2 ( < m I + 1), then the cantori of rotation number to form an interval in the vague topology (to be explained shortly). For each a ~ [0,1] there is a cantorus M~,.~ of rotation number to given by the hull function

R e m a r k 3. The above theorems have analogues

for some non-rotationally ordered orbits, specifically those of bounded type, i.e., such that IT2(xn_x, xn) + Tl(X,, x~+l) [ - B (called Bbounded). If x is a B-bounded orbit close enough to a non-degenerate anti-integrable limit, then it is the continuation of an anti-integrable B'-bounded state, for B' slightly larger. Conversely, the continuation of a B-bounded antiintegrable state is obviously B'-bounded for slightly larger B'. R e m a r k 4. In the above two theorems, non-

degeneracy of the anti-integrable limit is only required at those critical points which are visited by the given state.

X(O)

[m 1 + [ 0 ] , [ m2+ [01,

{o} = o - [o1.

O<{O}
(3.4) (3.5)

The parameter a specifies the fraction of the natural measure v of the cantorus which lies in the m~-well (mod 7/). Formula (3.4) is illustrated in fig. 1. The vague topology on cantori is defined by saying that two cantori are close if the integrals of continuous functions q~ with respect to the natural measures v~ and v2 on the two cantori (induced by order-preserving semi-conjugacy to rotation) are close. With this topology the mapping a ~-~M~.~ is a homeomorphism of [0,1] onto

C. Baesens, R.S. MacKay / Cantori for multiharmonic maps

x

x=X(0)

1 m

~<

Fig. 1. Hull function X(O) for the cantorus Mo., at a 2-well anti-integrable limit V(x). the set of cantori of rotation number to. These results are essentially already in [18]. Similarly, if the anti-integrable limit has N wells per period, m~ < m 2 < - . • < r a n (
4. Gap structure The gaps of a cantorus come in orbits, which we call holes. The number of holes for a cantorus is countable. For a cantorus near a non-degenerate anti-integrable limit, only finitely many gaps remain at the anti-integrable limit, given by the discontinuities in (3.1), (3.4) and their generalizations for more wells per period; we call them the principal gaps. The number of principal gaps is at most the number of maxima (per period). Each hole must contain at least one principal gap, but it may contain more than one. In the case of a single well anti-integrable limit, the unique cantorus of rotation number to has a single hole. This can be seen from formula (3.1), because there is just one orbit of discontinuities in X(O), viz. 0 = kto (mod 1), k E 77. For two-well anti-integrable limits, however, the n u m b e r of holes depends on a and to. If

65

a = n o - m, for some n, m E Z, then Mo,,~ has a single hole. Otherwise it has two holes. This is because, if a = 0 or 1, formula (3.4) reduces to the single well case (there is only one principal gap), and if a = n o - m for some n, m ~ 7/, n 0, then the orbits of discontinuity 0 = kto (mod 1) and O = kto + a (mod 1), k E Z, coincide (both principal gaps belong to the same hole). Otherwise they are distinct (each principal gap generates its own hole). Similar considerations hold for anti-integrable limits with N > 2 wells, when 1, 2 . . . . , N-hole cantori, will be found, according to oq . . . . , O/n, O.).

One might expect the difference in gap structure to show up in the topology of the set of cantori. It does if one uses the Hausdorff topology instead of the vague topology. Two closed sets are said to be close in Hausdorff topology if each is contained in an e-neighbourhood of the other. For non-degenerate anti-integrable limits with more than one well, and all small enough perturbations, the set of cantori of given rotation n u m b e r to forms something like a Cantor set in Hausdorff topology, with gaps at a = n t o -

m,n, m E Z . To be more precise, consider the two-well case. Then the set of cantori of rotation n u m b e r to, irrational, in Hausdorff topology, is homeomorphic to the set obtained from the interval [0,1] by inserting a gap immediately to the right and left of each point of the form a = nw - m , n, m E Z. The idea of the proof is sketched in fig. 2, where Aubry diagrams (i.e., graphs of x , against n, connected by straight lines) are shown for some orbits of cantori with t~ = to, a just less than to, and a just greater than to.

5. Turnstile structure If a cantorus for an area-preserving twist m a p is uniformly hyperbolic then in phase space there is an arc W S of stable manifold closing each gap and similarly an arc W u of unstable manifold.

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C. Baesens, R.S. MacKay / Cantori f o r multiharmonic maps

xn

Ix= tff-E

xn

uniformly hyperbolic cantorus must contain homoclinic points to the cantorus (see Appendix C), so the arcs of stable and unstable manifold which close a gap must also have an internal intersection. Near the anti-integrable limit, it is possible to compute the approximate shape of these turnstiles. This was done in the case of single-well anti-integrable limits by [20], and it is not hard to generalise, as we shall now do. Points on the stable manifold of a gap are points whose forward orbit converges to those of the endpoints of the gap. This corresponds to forward sequences x ~ R z÷ of stationary action with respect to variations holding x0 fixed and asymptotic to the sequences for the endpoints. If F = [l, r] is a principal gap which does not fall onto any other principal gap under forward iteration, then at the anti-integrable limit the stable manifold of F, in sequence space, is simply the straight line x 0 = (1 - t)l + tr, x, fixed for n > 0 ,

0-t-
(5.1)

Ix=to

If the anti-integrable limit is non-degenerate, then the stable manifold has a locally unique continuation [1,20], as a slightly curved line connecting the slightly displaced endpoints. The unstable manifold is determined similarly, provided F does not land on a principal gap in backwards time. To compare them, we need to project both into phase space, using

xrl

X=Xo,

I x - - to-4-E n F i g . 2. A u b r y d i a g r a m s f o r c a n t o r i in a t w o - w e l l p o t e n t i a l , w i t h ct = to - e, to a n d to + e, e small.

The resulting object we call a turnstile [15]. Note firstly, that up to area-preserving transformation, the turnstile in each gap of a hole is the same, so turnstile structure can be associated with a hole, not just each gap. Secondly, every turnstile for a

y = - h i ( x 0 , xl)

for stable manifold,

y = h z ( x _ l , Xo)

for unstable manifold.

(5.2)

This gives y = - V ' ( x ) + t?(A)

y = t?(;t)

for the stable manifold,

for the unstable manifold,

(5.3)

where ;t is the C2-norm of the perturbation T. Hence if F is the unique principal gap in its hole then we can use (5.3) to deduce the turnstile structure of its hole. If V has a single maximum in F then (5.3) gives a turnstile equivalent up to

67

C. Baesens, R.S. MacKay / Cantori for multiharmonic maps (a) W

W

~

~

1

(b)

Wu

~ r

.I~

,I~

m 0 1 n ~ r 0 w ~

~

r0Jn ~ _ . . . d

mn Ws

Fig. 3. Turnstiles for holes containing a single principal gap: (a) simple turnstile, (b) double turnstile. The arrows indicate whether the corresponding lobe crosses the cantorns up or down.

Fig. 4. Turnstile structure in a gap [l, r] for a 1-hole cantorus with two principal gaps. The names for the homoclinic points indicate which critical points of V their orbits are close to at times 0 and n.

area-preserving transformation to the usual turnstile picture (fig. 3a) which we call a simple turnstile. If V has more than one maximum in F, however, it gives what we call a multi-turnstile; the case of a double turnstile is shown in fig. 3b. If, however, a hole has two or more principal gaps, then the above results (5.3) apply only to the stable manifold of the last principal gap in the hole and the unstable manifold of its first principal gap. T o compare them it is necessary to iterate one or both of these backwards or forwards to a given gap. This gives more complicated turnstiles. For example, suppose a principal gap F0 = [l0, r0] falls on a principal gap Fn = [In, rn] after n iterations, and there is one maximum m0 (respectively, ran) in each of these principal gaps. Then one obtains a turnstile area-preservingly equivalent to figure 4 in the gaps fsF0, for all j. Note that zero net flux (1.2) imposes a constraint on the areas of the lobes of the turnstiles of a cantorus, namely that the sum over holes of the areas of the up lobes must equal the sum of the areas of the down lobes for a cantorus. This requires some interpretation in the case of turnstiles like fig. 4, but the point is that the net area passing through the holes of a cantorus should be zero. The orbits of intersections of W" and W s in a gap are homoclinic to the endpoints of its hole.

We call them principal homoclinic orbits. This is not because they have a special relation to principal gaps but because there can be other type of homoclinic orbit to a cantorus which are not asymptotic to the same orbit of the cantorus in forwards and backwards time. Near a nondegenerate anti-integrable limit, the principal homoclinic orbits can all be obtained by continuation. One simply replaces certain minima in the anti-integrable limit for certain cantorus orbits by appropriate maxima. For example, in the single well case, the unique principal homoclinic orbit is given by choosing 0 = 0 in (3.1) and taking x 0 to be the maximum between m 1 and ml + 1. The principal homoclinic orbits are in fact precisely the continuations of the weakly rotationally ordered anti-integrable limits A such that ~rA has at most one point on each maximum (modulo 1) and at least one point on some maximum.

6. Bifurcations of cantori

Since the set of cantori of given rotation number changes as one passes from a neighbourhood of a two-well anti-integrable limit to a neighbourhood of one with a single well, there must be many bifurcations of cantori. In particu-

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C. Baesens, R.S. MacKay / Cantori for multiharmonic maps

lar in this example, we must lose all but one of them. The ways a cantorus can be lost are limited by the results of Appendices A and B. A cantorus cannot be lost simply by losing rotational order. This follows from the proof of Theorem B.1, where it is shown that some points must merge in order to lose rotational order; but this would violate irrational rotation number. The only way to lose a cantorus is for it to lose phonon gap (equivalently, to lose uniform hyperbolicity). The set of limit points of a sequence of rotationally ordered invariant sets with given rotation number for a sequence of twist maps f~---~f~ is itself a rotationally ordered invariant set of the same rotation number for f~ [17]. In a neighbourhood of any set of non-trivial anti-integrable limits, there are no rotational invariant circles (Theorem A.1), so a cantorus cannot collapse to a circle. Hence we deduce (as already sketched in [13]):

Theorem 6.1. There exist analytic area-preserv-

ing twist maps with cantori which are not uniformly hyperbolic. Two open questions are whether this can happen for the minimum energy cantorus (see next section), and whether examples can be made of cantori which are not uniformly hyperbolic but nevertheless have positive Lyapunov exponent (non-uniformly hyperbolic). Our aim in the rest of this section is firstly to conjecture the typical ways in which a cantorus can lose phonon gap, and secondly to apply these conjectures to guess the bifurcation diagram for cantori for an example family of multiharmonic maps. We conjecture that the typical way that a cantorus loses phonon gap is by collision with a principal homoclinic orbit. Apart from the obvious case where all gaps collapse to give an invariant circle, the main possibilities for collision of a 1-hole or 2-hole cantorus with a principal homoclinic orbit are sketched in fig. 5. We expect routes (a) and (c) of fig. 5 to lead to annihilation of the cantorus. This is because they

12= r2

Cc)

r 2 .... ~ l l ~ r l

1 2 ~

r2

Fig. 5. Conjectured ways a cantorus can lose phonon gap: (a) annihilation of a cantorus with a double turnstile by one of its principal homoclinic orbits, (b) two-to-one hole transition, (c) annihilation of a 2-hole cantorus by a principal homoclinic orbit.

C. Baesens, R.S. MacKay / Cantori for multiharmonic maps

resemble standard fold bifurcations, even though it may not be easy to formalise this mathematically because one has to work with infinite sequences. To be more concrete, the problem in case (a) is to find critical points of oo

Wr(x) = ~, h(xi, xi+l) - h(ri, ri+,)

(6.0)

i~o0

in the space of sequences x satisfying

69

A = 0 consists of anti-integrable limits. For A > 0 , h is convex. The anti-integrable limits are single well for [kl/k21 > 1, and two-well for Ikl/k21
k 1/A = ~b(r) sin 0 , Ir, - x i l < oo . i=

(6.0a)

--oo

T h e sequence 1 is a critical point, in fact a local minimum, otherwise the cantorus would not be rotationally ordered. Its neighbouring homoclinic point corresponds to another critical point, of index 1, which we will call h. As the bifurcation parameter varies, we expect l and h to coalesce into a cubic-like critical point and then annihilate each other. In case (c), we expect the same, using the sequence r 2 for r, and 12 for l. One thing that makes this non-trivial to prove is that if I annihilates, then so must r, because each is in the closure of the other, but r entered into the definition of Wr Case (b) may turn out to be codimension-2 in general, but we believe it is codimension-1 if one restricts to reversible maps. We see no reason in the latter context for the cantorus to annihilate. Instead we expect a two-to-one hole transition. Numerical evidence for this is presented in [3]. Let us consider the implications of our results for the 2-harmonic family l 2 cos (2~rx) h ( x , x ' ) = ½ A ( x ' - x ) 2 + k4,tr k2 cos(4"trx + ¢) + 16,tr2

(6.1)

concentrating on the case ¢ = 0. The case tp = 0 is parametrised by A:kl:k2E RP 2, two-dimensional real projective space (i.e. the set of lines through the origin in R3), because the absolute values of A, k 1 and k 2 do not affect the solutions, only their ratio. The set where

k2/A = ~(r) cos 0 ,

(6.2)

where ~b is a diffeomorphism of [0, 1) onto [0, oo), e.g., ~b(r) = t a n h - lr

r

or

1- r

F

or

Vri-_ r2

r

or

l_rE

or

tan l'trr.

(6.3)

So the boundary represents the anti-integrable limits and the interior the convex generating functions. Our preferred choices for ~(r) are r / ( 1 - r 2) and tan l ~ r , since they give an analytic parametrisation of RP 2, including at k I = k 2 = 0 and at A = 0. The same holds for any odd analytic function ~b(r) with positive derivative on [0, 1), and a simple pole at r = 1, but none of the other choices above have these properties. We have a slight preference for tan ½,trr, as the inverse has a well established succinct notation. Let us specialise to the case to =3' = ½(1 + V5), golden ratio. From the numerical results of [7,10] we believe that the region of existence of a golden circle (the K A M domain) has the form indicated in fig. 6, with a Cantor set of cusps in its lefthand boundary. We conjecture that the regions of existence of golden cantori fit in as shown there. Note that the parameter a for a cantorus is defined by continuation to the appropriate two-well anti-integrable limit. If the set of parameter values to which a cantorus can be continued is not simply connected, the value of a might depend upon the route taken from

70

C. Baesens, R.S. MacKay / Cantori for multiharmonic maps

o

o

o

½

½

o

~

i

o

1

½

o

½

o

0

{

1

½

½

1

1

1

Fig. 6. Conjectured bifurcation diagram for cantori of golden rotation number for the 2-harmonic family with ~ = 0. The shape of the anti-integrable limit potential is shown round the boundary, and some of the types of transition are labelled: a, b, c refer to figs. 5a, 5b, 5c, respectively, and d refers to fig. 7a.

the anti-integrable limit, but we conjecture that this does not happen for ¢ = 0. There must also be bifurcations of turnstile structures. In particular, the a = 0 and 1 cantori of given rotation number should turn into the unique cantorns of that rotation number on crossing the curves marked d in fig. 6 from the lefthand double-well regime into the single-well regime at the top (respectively, bottom), the only change required being in its turnstile structure, which must go from double turnstile (fig. 3b) to simple turnstile (fig. 3a). One way in which we

conjecture this could happen is sketched in fig. 7a. Clearly, a "pitchfork" version is possible too. Furthermore, the turnstile structures for the other one-hole cantori in the region on the left must also bifurcate several times, because on creation from the KAM domain they start with simple turnstiles (numerical observation), but we have proved here that near the anti-integrable limit they have the turnstile structure of fig. 4, which has six more principal homoclinic points. This suggests that there must typically be several bifurcation curves inside each of these tongues to

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C. Baesens, R.S. MacKay / Cantori for multiharmonic maps

(a)

f0)

AI limit

K.AM

Fig. 7. Conjectured turnstile bifurcations: (a) from a double turnstile to a simple one, (b) for a 1-hole cantorus with two principal gaps.

create the required extra h o m o d i n i c points in pairs. A possibility, which we believe is the simplest one, is sketched in fig. 7b. Note that it includes codimension-2 points at which 3 principal homoclinic points interact. Note that the turnstile bifurcation in fig. 7a, when viewed in reverse, creates uncountably

many cantori from the given one. They are cantori of the same rotation number, but which have a small proportion /3 of their points near the locally minimising homoclinic orbit created by the tangency. Their existence can be seen either by a construction like that from the antiintegrable limit, using the Peierls barrier func-

C. Baesens, R.S. MacKay / Cantori for multiharmonic maps

72

tion (cf. [18]), or by constructing a horseshoe for the first return map to a neighbourhood of the homoclinic point and picking out appropriate orbits. What do we expect for ~ # 0? Adding fir to is equivalent to changing the sign of k2, and ~o has no effect if k2 = 0, so we can include ~0 in the parameter space by rotating the disk D about its vertical diameter. Thus we obtain a 3-ball for the full parameter space, though the back half-ball duplicates the front one by reflection in the plane q~--0, so it is enough to consider the half-ball k 2 ~ 0 , 0 -< q~ -----fir. On the hemispherical surface of the half-ball we can compute the boundary between single-well and double-well anti-integrable limits, sketched in fig. 8. In the two-well region of this hemisphere (though not on the whole sphere), there is a continuous choice of labelling for the wells of the double-well antiintegrable limits, so a is likely to be unambiguously defined for a cantorus. The sequence in which cantori is created along the k2-axis is independent of ~0, and somehow the whole Cantor set of cusps boundary of the K A M 0o

ki/k: singlewell

½

J

double well

0

9

~t

-1 single well

Fig. 8. Boundary between 2-well and 1-well anti-integrable limits for q~ # 0.

domain has to shrink to a single point as ~p rotates from 0 to rr, but that is as much as we can say about non-zero ~0 at present.

Remark. If the condition (1.2) is relaxed to h(x + 1 , x ' + 1) =h(x,x') + C

(6.4)

for some constant C, or equivalently, f is allowed to have non-zero net flux • (in fact, qb = C), then other bifurcations of cantori become possible. The turnstile areas of each cantorus must add up (algebraically) to q~. Hence, for example, we can have an analogue of the bifurcations in figs. 5a and 5c for cantori with just one hole and a simple turnstile, where the area of one lobe shrinks to zero and the homoclinic orbit collides with the cantorus, annihilating it, which is our interpretation of the observations of [5].

7. Minimum energy cantorus

An important question in the variational context, though not so much in the dynamical context, is which of the cantori of given rotation number is the one of minimum energy, that is which consists of orbits for which every finite segment has globally minimum energy with respect to all variations fixing its end points. These are the cantori that Aubry and Mather originally studied. They proved existence (in the case there is no invariant circle of the given rotation number) and uniqueness. The complete answer is in general very complicated (cf. [8]). But we can state several selection principles for the minimum energy cantorus, Firstly, if the map is symmetric, in the sense that h ( - x ' , - x ) = h(x, x') then the reflection of a cantorus is another one with the same rotation number. But the minimum energy one is unique, therefore it must be its own reflection. For example, the 2-harmonic family (6.1) is symmetric for ~0 = 0 (after addition of a term of

c. Baesens, R.S. MacKay / Cantorifor multiharmonic maps

73

the form l [ V ( x ' ) - V(x)], which does not change the stationary states nor the value of the energy of any segment apart from end effects). Hence for k 2 > 0 , the minimum energy cantorus has a = ½ (or more precisely, cannot be the continuation of any anti-integrable cantorus with Secondly, for the minimum energy cantorus, detailed balance must hold for the area of turnstile lobes, that is for each hole, the up-going area must equal the down-going area. This is because the difference in action between the orbits of the endpoints of a hole in a minimum energy cantorus is zero (e.g. [17]). In particular, route (c) of fig. 5 is not possible for the minimum energy cantorus. Similarly, for a minimum energy cantorus with a double turnstile, the area of the second lobe cannot exceed that of the first lobe. Otherwise the energy difference between the central principal homoclinic orbit and the endpoints of the hole would be negative. Similar remarks hold for more complicated turnstiles. Thirdly, near an anti-integrable limit whose wells are not the same height, the minimum energy cantorus must be the one which uses only the lower well. Hence we conjecture a diagram like fig. 9 for the minimum energy cantorus of golden rotation number of the 2-harmonic family (with ~ = 0). Define O/me to be the value of O/ for the minimum energy cantorus (assuming we are near enough to the anti-integrable limit that it can be continued back there). Then, for example, the transition from Otme= 0 to Otme> 0 should occur when the second lobe of the double turnstile of the a = 0 cantorus has grown to the same area as its first lobe.

Proposition. The function ame(f ) is a continuous function from C ~ area-preserving maps f near a non-degenerate 2-well anti-integrable limit, to R.

Proof. The minimum energy cantorus is a semicontinuous function from C 1 area-preserving maps to compact subsets of T x R using Haus-

Fig. 9. Conjectured diagram indicating the minimum energy cantorus of rotation number 1/y for the 2-harmonic family as a function of parameters (with ~ = 0). dorff topology [17]. More precisely, define M~,(f) to be the set of recurrent minimising orbits of rotation number to for a map f, and M ' ( f ) to be the set of all minimising orbits of rotation number to. Given a sequence of maps f~--+f in C 1, let K be the set of limit points of Mo,(f/). Then

M (f) c K c M ' ( f ) .

(7.1)

Thus the only jumps that Mo,(f) can make (in Huasdorff topology) are by blowing up some non-recurrent points of M~,'(f) into recurrent points. These non-recurrent points are precisely those principal homoclinic orbits in a turnstile which satisfy the equal area condition: the algebraic area of the lobe formed from stable and unstable manifolds from say the left-hand endpoint of a gap up to the homoclinic point should be zero. Now continuation from the anti-integrable limit gives a homeomorphism from the chosen set of anti-integrable sequences in R z / z to the resulting invariant set in T x R. We already obtained all the possible candidates for

74

C. Baesens, R.S. MacKay / Cantori for multiharmonic maps

M ~ ( f ) , parametrised by t~ E [0,1]. The possible candidates for M'(f) are obtained by choosing the hull function X(O) to be right continuous at one of the discontinuities 0 = 0 or a, and left continuous at the other. It follows that am~ is continuous. [] Note that we also deduce a type of modelocking: ame cannot increase through a value a = r n t o - n instantly, but must keep this value for at least a certain range of parameters. This is because the two equal area conditions, for a to be able to increase and for a to be able to decrease, cannot hold simultaneously. But we expect values of O~megiving 2-hole cantori to be codimension-1 because of the detailed balance condition. One particular question that we raised earlier is whether the minimum energy cantorus can ever fail to be uniformly hyperbolic. We do not expect any examples of this on the left-hand side of fig. 9, because the boundaries of the ame = r n t o - n tongues are defined by the equal area condition, which happens well before the cantorus loses uniform hyperbolicity. On the righthand side, however, if our conjectured picture is correct, the minimum energy cantorus is the a = ½ one right up to the 1-2 hole transition, at which point it loses uniform hyperbolicity. We are not at present able to prove this, however, and can imagine alternative scenarios in which the minimum energy cantorus changes from the a = ½ cantorus to a 3-hole cantorus (not arising from the anti-integrable limit) which then collapses to a 1-hole cantorus with triple turnstile and hence to the simple turnstile cantorus, without ever losing uniform hyperbolicity.

8. Summary and problems for the future We have proved some general results about rotationally ordered states near the anti-integra-

ble limit, and used them to deduce a lot about the set of cantori for such maps. This led to some conjectures about the bifurcation of cantori in families containing degenerate anti-integrable limits, which it would be most interesting to pursue. In particular, what are the typical ways in which cantori can bifurcate? We are studying one case in detail numerically, namely the one-to-two hole transition [3]. Could one also get a handle on this question near degenerate anti-integrable limits? In particular, how do the bifurcation curves approach a degenerate anti-integrable limit? Secondly, what are the typical bifurcations of turnstile structure in a cantorus? Another question which this work raises is whether one can extend renormalisation theory to explain aspects of the bifurcation picture. A renormalisation scheme (initiated in [10]) is under development to explain the cusped boundary of the K A M domain in the 2-harmonic family, but to complete the picture one also needs to make sense of the numerical observation that the non-degenerate anti-integrable limits are attractors under renormalisation, and the degenerate ones are saddles. The ideas of this paper can also be applied to rational rotationally-ordered orbits [11]. Finally, the ideas can also be applied to higher-dimensional symplectic twist maps [14], though there is no known analogue of the crucial concept of rotational order, which played such an important selection role here.

Acknowledgements We thank Jim Meiss for helpful discussions and for his hospitality during a visit of both of us to Boulder in April 1991, where this work was started. C.B. thanks Serge Aubry and the Laboratoire L r o n Brillouin, Saclay for support for the year 1991. Subsequently, this work was

75

C. Baesens, R.S. MacKay / Cantori for multiharmonic maps

supported by the UK Science and Engineering Research Council and the Nuffield Foundation.

state and hence there is no rotational invariant circle. []

Appendix A. Non-existence of invariant circles near non-trivial anti-integrable limits

Appendix B. Persistence of rotational order under continuation

For an area-preserving twist map, every orbit on a rotationally ordered invariant circle has minimal action (e.g. [4]). This enables us to prove non-existence of rotational (i.e., homotopically non-trivial) invariant circles for all area-preserving twist maps in a neighbourhood of any set of anti-integrable limits for which there exists a constant C > 0 such that

Section 2 tells us how to construct all rotationally ordered states near a non-degenerate antiintegrable limit. But do these rotationally ordered states remain rotationally ordered as long as they can be continued? The answer is yes.

AV = max V(x) - min V(x) >- C .

(A.1)

We call such anti-integrable limits non-trivial. No non-degeneracy condition is required on the critical points of V.

Theorem A.I. Given a convex perturbation h(x, x') = V(x) + r(x, x')

(A.2)

of an anti-integrable limit, and points M , m where V is maximum and minimum, respectively, there are no rotational invariant circles of rotation number to if r = sup{ T(x, m) + T(m, x') - T(x, M) - T(M, x'): (x' - M), (M - x) E/(to)} < AV,

(A.3)

Theorem B. 1. A rotationally ordered state of an area-preserving twist map, remains rotationally ordered as long as it can be continued without collision of any of its points and the generating function remains convex. Proof. A rotationally ordered set S for an areapreserving twist map satisfies a Lipschitz condition (e.g. [9]): there exists L > 0 (depending on the generating function and also the rotation number) such that L - l < xn+1-Xm+l <_L --

(B.1)

Xn -- Xm

for any two states x , £ E S , and n,m~2V. So unless two points ( x , , x , + l ) , (£m,~m+l) merge during the continuation, it is impossible for their orbits to change rotational order. []

where I(to) is defined in (1.14).

Proof. By a theorem of Birkhoff (e.g. [9]), a rotational invariant circle is rotationally ordered. Its projection covers the whole x-axis, so in particular, it contains an orbit with x 0 = M. Take the corresponding state x E R e, and consider the effect of displacing x 0 to m. An energy AV is gained from V(x) and the energy required in T to make the displacement is at most ~', using (1.15). Hence if r < AV, x was not a minimum energy

Corollary B.2. A rotationally ordered orbit cannot be lost without losing phonon gap. Proof. Merging of two points of an orbit implies loss of phonon gap. [] Corollary B.3. A non-rotationally ordered orbit cannot become rotationally-ordered under continuation with phonon gap.

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R.S. MacKay, C. Baesens / Cantori for multiharmonic maps

Appendix C. Every hole for a uniformly hyperbolic cantorus has a homoclinic orbit We sketch a proof of this result. Given a hole for a cantorus, with endpoints l, r, define Wr as in eq. (6.0), but restricting the space to X = {x E ~z

li ~ x i ~ ri } .

(C.1)

Then l, r are critical points of W, (otherwise they would not be equilibrium states). If the cantorus is minimum energy then Mather already proved existence of homoclinic orbits in each hole by the minimax principle: ! and r are global minima with the same value of W,, the gradient flow is inwards on aX, and hence there is another critical point in between. The same argument can be used for any 1-hole cantorus (assuming zero net flux), since then l and r still have the same value of Wr. In the general case (N-hole cantorus, N > 1, or non-zero net flux), then l and r need not have the same value of W,, but if we add the condition of uniform hyperbolicity then they are non-degenerate critical points, in fact of index 0, else a contradiction with rotational order can be derived. A variation of the minimax principle can be applied in this context, to deduce existence of another critical point (generically of index 1) in between.

References [1] S. Aubry and G. Abramovici, Chaotic trajectories in the standard map: the concept of anti-integrability, Physica D 43 (1990) 199-219. [2] S. Aubry, R.S. MacKay and C. Baesens, Equivalence of uniform hyperbolicity for symplectic twist maps and pbonon gap for Frenkel-Kontorova models, Physica D 56 (1992) 123-134.

[3] C. Baesens and R.S. MacKay, One to two-hole transition for cantori, preprint (1993). [4] V. Bangert, Mather sets for twist maps and geodesics on tori, in: Dynamics Reported,Vol. 1, eds. U. Kirchgraber and H.O. Walter (Wiley, 1988). [5] S.N. Coppersmith and D.S. Fisher, Threshold behavior of a driven incommensurate harmonic chain, Phys. Rev. A 38 (1988) 6338-6350. ]6] D. Goroff, Hyperbolic sets for twist maps, Erg. Th. Dyn. Sys. 5 (1985) 337-339. [7] J.M. Greene and J.M. Mao, Higher-order fixed points of the renormalisation operator for invariant circles, Nonlinearity 3 (1990) 69-78. [8] R. Griffiths, H.J. Schellnhuber and H. Urbschat, Exactly solvable models for cantorus phase transitions, Phys. Rev. Lett. 65 (1990) 2551-2554. [9] M.R. Herman, Sur les courbes invariantes par les diff~omorphismes de l'anneau pr6servant les aires, Vol. 1, Ast6risque 103-104 (1983). [10] J.A. Ketoja and R.S. MacKay, Fractal boundary for the existence of invariant circles for area-preserving maps: observations and renormalisation explanation, Physica D 35 (1989) 318-334. [11] J.A. Ketoja and R.S. MacKay, Rotationally-ordered periodic orbits for multiharmonic maps, in preparation. [12] P. LeCalvez, Les ensembles d'Aubry-Mather d'un diff6omorphisme conservatif de l'anneau d6viant la verticale sont en g6n6ral hyperboliques, C.R. Acad. Sci. Paris 306 (1988) 51-54. [13] R.S. MacKay, Greene's residue criterion, Nonlinearity 5 (1992) 161-187. [14] R.S. MacKay and J.D. Meiss, Cantori for symplectic maps near the anti-integrable limit, Nonlinearity 5 (1992) 149-160. [15] R.S. MacKay, J.D. Meiss and I.C. Percival, Transport in Hamiltonian systems, Physica D 13 (1984) 55-81. [16] R.S. MacKay and I.C. Percival, Converse KAM: theory and practice, Commun. Math. Phys. 98 (1985) 469-512. [17] J.N. Mather, A criterion for non-existence of invariant circles, Publ. Math. IHES 63 (1986) 153-204. [18] J.N. Mather, More Denjoy minimals sets for areapreserving diffeomorphisms, Comment. Math. Helv. 60 (1985) 508-557. [19] I.C. Percival, Variational principle for invariant tori and cantori, in: Nonlinear dynamics and the beam-beam interaction, eds. M. Month and J.C. Herrera, Am. Inst. Phys. Conf. Proc. 57 (1979) 302-310. [20] J.J.P. Veerman and V. Tangerman, Intersection properties of invariant manifolds in certain twist maps, Commun. Math. Phys. 139 (1991) 245-265.