Transition to chaos for two-frequency systems

2014 Many biperiodic flows can be modelled by maps of a circle to itself. We decompose the boundary of topological chaos in the space of C1 circle maps of degree 1 into four subsets, and describe the typical routes to topological chaos. Tome 45 N~ 15 1~ AOUT 1984 LE JOURNAL DE PHYSIQUE LETTRES J. Physique Lett. 45 (1984) L-741 L-746 ler AOUT 1984, Classification Physics Abstracts 05.40 In many experiments on chaotic dynamics, both physical (e.g. [1] and references therein) and computational [2-4], one sees a sequence of behaviour on varying a parameter, which can be described in the phase space as follows : flows : ATTRACTING 1 ATTRACTING 2 ATTRACTING 3 « CHAOS » EQUILIBRIUM -+ PERIODIC ORBIT ~ TWO-TORUS -~ (return)maps : ATTRACTING 2 ATTRACTING 3 « CHAOS » FIXED POINT -+ CIRCLE 0 Our interest is in transition 3 in cases for which the dynamics on the attracting circle can be described by an orientation preserving circle homeomorphism (as when transitions 1 and 2 are « Hopf » bifurcations). Although this has been greatly studied [2-7], one is still far from a complete classification of the possible routes. Maps of a circle to itself have been proposed [8], and widely used [3, 7, 9-13], to model many systems exhibiting this transition. (*) Permanent address : Mathematics Institute, University of Warwick, Coventry, CV4 7AL, England. (**) Permanent address : Physique de la Matiere Condensee, LA 190, Parc Valrose, 06034 Nice Cedex, France. Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019840045015074100 L-742 JOURNAL DE PHYSIQUE LETTRES Remark 1. All (return) maps having a contracting ring not containing two disjoint ones, are expected to reduce in some sense to circle maps. Here we say that A = M is a contracting ring for a map f : M -+ M, if A is topologically a circle cross a codimension one closed disk, and is mapped strictly inside itself. This covers a large class of systems exhibiting transition 3, though it does exclude some cases of interest [14, 15]. In this Letter, we classify all routes to chaos for a class of circle maps. More precisely, we decompose the boundary of topological chaos in the space C1 (1f1) of degree one, C1 (i.e. continuously differentiable) circle maps into four well-characterized subsets, and describe the typical ways it can be crossed. First we need to define topological chaos and the concepts of lifts, degree, rotation number and frequency locking for circle maps. We say a map f has topological chaos if, given any accuracy of observation, there exist a strictly increasing sequence of integers ni and a real number h > 0 such that the number of distinguishable orbit segments of length ni grows asymptotically like exp(ni h). The largest such h is called the topological entropy hcr) [16]. Although topological chaos does not necessarily imply complicated asymptotic behaviour of most orbits, it is a prerequisite. Given a continuous map f of the circle T 1 = R/Z to itsel~ it is useful to consider its lifts to continuous maps F of the line R to itself, i.e. those maps such that /o jr = 7r o F, where 7r(x) = ~(mod 1) is the canonical projection from R to lr 1 . Different lifts differ only by an integer constant. F(x + 1) F(x) is an integer constant, known as the degree off We consider only degree one circle maps, since this is the case that arises by deforming orientation preserving homeomorph ms.1 Given a lift F of a circle map, we say a point x has rotation number p(x) = lim (Fn(x) x) n-oo n if the limit exists. Given F, the set of all rotation numbers is always a single number or a closed interval [6, 17, 18], which we call its rotation interval p(F), and is translated by an integer if one chooses a different lift of the same circle map. If p(F) is a single rational co, we say F is frequency locked with ratio c~. In the course of this Letter we will define various sets of circle maps. Their relative disposition is illustrated in figure 1, which we recommend the reader keep continually in view. Remark 2. Because there are two main ingredients in their dynamics, namely the nonlinearity and the rotation number, we represent spaces of circle maps by two dimensional pictures, As a guide, think of two parameter families like Firstly, we denote by Lw the set of frequency locked maps with ratio co, and define the OJ-Arnold tongue Aw to be the set of all maps F such that OJ E p (F ). We extend the definitions of La> and Aw to (u irrational, though then the names are somewhat inappropriate. For any c~ E R, Lw is strictly contained in Aw, but both notions coincide for monotone maps, as illustrated in figure 1. The main properties of these sets are described in [19] ( 1 ). Remark 3. For c~ irrational the possibility of F E Lw but non-monotone, which is somewhat pathological, can be eliminated by considering only analytic maps or by restricting C 1 (T 1 ) slightly to a certain class Al (lfl) [ 18,19]. Next, we define Zp/q to be the subset of Ap/q consisting of those maps for which all periodic orbits of rotation number p/q have period of the form q.2", for some n (all fractions are supposed to be written in lowest terms). Now we are ready to describe the boundary of topological chaos in C1 (If 1 ). (1 ) See also Boyland [20] for an independent treatment of these notions in a subclass of C1 (~ 1 ). L-743 TRANSITION TO CHAOS FOR TWO-FREQUENCY SYSTEMS Fig. 1. A two-dimensional representation of the space A 1 (T 1) (see Remark 3~ illustrating various relevant subsets and their boundaries. We show four sample paths whose characteristics are illustrated in figure 2. The figure is also intended to indicate : the opening of irrational tongues at the boundary of non-invertibility in class A 1 (T 11 the fact that all boundary points of L~~~ are accumulation points of other tongues, and the complex structure of part D pill of the boundary of chaos, which will be described in [26] in terms of kneading theory for families which look like F~. Remark 4. It is good to impose as weak smoothness conditions as possible, since the induced circle map for the contracting ring need not be smooth. Some smoothness, however, is necessary for the transition problem to make sense, since any continuous circle map can be approximated by continuous circle maps with positive topological entropy [33]. Theorem [19]. The boundary of topological chaos in C1 (lf 1 ) is the union of the following four subsets : L-744 JOURNAL DE PHYSIQUE LETTRES Remark 5. It is convenient furthermore to decompose S into Sp q U. S~~q, the sign indicating whether larger or smaller rotation numbers may be found on crossing that part of the boundary; similarly for Hp/q and 1~. The relative disposition of all these sets is shown in figure 1. To each of the four parts of the boundary corresponds a characteristic route to chaos for « neat » crossings. Different routes yield different evolution of the rotation interval; all give positive topological entropy. For generic one parameter families, one sees only the following routes, and p(F) and h(j) evolve as indicated. We denote by c the difference in parameter from the critical case, and the C represent constants, C1, C2 > 0, C3 C4. Route Sn 9. The transition is marked by a Saddle-node which destroys the last orbits with rotation number p/q, and one obtains a non-trivial rotation interval p(F) > ~/?(S~) or M(S~) : Route H~. The transition * 3 marked by creation of a Heteroclinic loop in the sense of [22], giving a non-trivial rotation inte ’ with plq as left or right endpoint, respectively. This involves either a heteroclinic tangency in the sense of [23] (as for the family F~,,n [20]), or a saddle-node. In the first case, the other endpoint ofp(FJ evolves like : and in the second case, like : Route Dp~9 : Accumulation of period Doubling bifurcations with periods q.2" : p(F) remains lQE2 equal to {p/q }, and h( f~) oc E’~ga, b ~ 4.6692 (provided the crossing occurs in the stable manifold of the known quadratic fixed point [24, 25] of the doubling operator). Remark 6. Route I does not appear in our list because it is not stable to perturbation in one parameter families. It is stable, however, in two parameter families. Also Dp/q has special points, where different scaling is to be expected, which cannot be avoided in non-trivial two parameter families. Further details and justifications of all these results will be given in references [19, 26]. Routes S and D have already been recognized in experiments [1]. Route H is new, but need not have immediate consequences for experiments since in general, crossing Hp/q does not remove the stable periodic orbits present in Lp~9 f 1 Zp~. However, the crossing creates other attracting sets, some of which are typically chaotic. These may make themselves evident on jogging the system or by hysteresis [ 19, 27]. Both p(F) and A(/) have implications for experimentally measurable quantities e.g. the rotation interval of a point x [28] : and the Lyapunov exponent ~(/) (when well defined : see e.g. [29] and references therein). One has