Chaos: An Interdiscipliary Journal of Nonlinear Science
American Institute of Physics
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, K. A.; Handley, J. P.; Tighe, B.; Delos, John B.; and Knudson, Stephen, Geometry and Topology of Escape. I. Epistrophes (2003). Chaos: An Interdiscipliary Journal of Nonlinear Science, 13(3), 880-891.