Geometry and Topology of Escape I: Epistrophes

Details Geometry and Topology of Escape I: Epistrophes Geometry and Topology of Escape I: Epistrophes

2003-07-11

arxiv.org/abs/nlin/0307019v1

Nonlinear Sciences

Chaotic Dynamics

15 pages, 9 figures, to appear in Chaos, first of two papers

Scientific paper

10.1063/1.1598311

We consider a dynamical system given by an area-preserving map on a two-dimensional phase plane and consider a one-dimensional line of initial conditions within this plane. We record the number of iterates it takes a trajectory to escape from a bounded region of the plane as a function along the line of initial conditions, forming an ``escape-time plot''. For a chaotic system, this plot is in general not a smooth function, but rather has many singularities at which the escape time is infinite; these singularities form a complicated fractal set. In this article we prove the existence of regular repeated sequences, called ``epistrophes'', which occur at all levels of resolution within the escape-time plot. (The word ``epistrophe'' comes from rhetoric and means ``a repeated ending following a variable beginning''.) The epistrophes give the escape-time plot a certain self-similarity, called ``epistrophic'' self-similarity, which need not imply either strict or asymptotic self-similarity.

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