Geometry and topology of escape. II. Homotopic lobe dynamics

Abstract

We continue our study of the fractal structure of escape-time plots for chaotic maps. In the preceding paper, we showed that the escape-time plot contains regular sequences of successive escape segments, called epistrophes, which converge geometrically upon each endpoint of every escape segment. In the present paper, we use topological techniques to: (1) show that there exists a minimal required set of escape segments within the escape-time plot; (2) develop an algorithm which computes this minimal set; (3) show that the minimal set eventually displays a recursive structure governed by an "Epistrophe Start Rule": a new epistrophe is spawned ∆ = D + 1 iterates after the segment to which it converges, where D is the minimum delay time of the complex. Topological methods and symbolic dynamics have long been valuable tools for describing orbits of dynamical systems. For example, if a particle in the plane scatters from three fixed disks, labeled A,B, and C, its orbit can be characterized by a sequence of symbols, such as ...ABA*BCBCA..., giving the sequence of collisions with the disks. The asterisk gives the location of the particle at the present time; as time goes by the asterisk takes one step to the right. In this paper, we describe a new kind of symbolic dynamics, in which the symbol sequence describes the structure of a curve in the plane. The relevant curve is not the trajectory of a particle, but rather an ensemble of initial points in phase space -the line of initial conditions. This line winds around "holes" in the plane in a manner described by the symbol sequence. The dynamical map applied to the line induces a map on the symbol sequence, which is more complicated than a simple shift. We use this symbolic dynamics to derive properties of the epistrophes introduced in the preceding paper. In particular, we use it to obtain a "minimal set of escape segments" and an "epistrophe start rule."

A new topological technique for characterizing homoclinic tangles

“…2, but this is not a requirement for our technique. (In previous work [19,16,2,3], we labeled this point Q 0 instead. )…”

“…This work is the outgrowth of an earlier paper [16], in which we introduced an initial version of homotopic lobe dynamics. That version made use of just one piece of topological information, the 'minimum delay time' D, which we define in Section 5.1.…”

The structure of ionizing electron trajectories for hydrogen in parallel fields

“…In Ref. [7] we proved a general rule, called the Epistrophe Start Rule, for such escape-time plots. This rule states that: (1) there is a minimal set of escape segments, which is determined by the topological structure of an underlying homoclinic tangle;…”

Ray and wave dynamical properties of a spiral-shaped dielectric microcavity

“…Most recently, Chern et al reported that even unidirectional emissions are possible from a truely asymmetric microcavity, a spiral-shaped microcavity [4]. Besides the photonics applications such as optical computing and networking, study of deformed microcavities can also provides invaluable pedagogical insight into cavity quantum electrodynamics [5,6], chaotic transport phenomena [7,8,9], and even the theory of quantum chaos [1].…”