Geometry and topology of escape. I. Epistrophes.
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.
Mitchell
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Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795, USA. kevinm@physics.wm.edu
Handley
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Tighe
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Delos
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Knudson
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SK
D017709:Fractals; D006859:Hydrogen / Q000737:chemistry; D008433:Mathematics; D008954:Models, Biological; D008962:Models, Theoretical; D017711:Nonlinear Dynamics; D013997:Time Factors
D006859:Hydrogen