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Full-Text Articles in Physical Sciences and Mathematics
Geometry And Topology Of Escape. I. Epistrophes, K. A. Mitchell, J. P. Handley, B. Tighe, John B. Delos, Stephen Knudson
Geometry And Topology Of Escape. I. Epistrophes, K. A. Mitchell, J. P. Handley, B. Tighe, John B. Delos, Stephen Knudson
Arts & Sciences Articles
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 …
Geometry And Topology Of Escape. Ii. Homotopic Lobe Dynamics, K. A. Mitchell, J. P. Handley, John B. Delos, Stephen Knudson
Geometry And Topology Of Escape. Ii. Homotopic Lobe Dynamics, K. A. Mitchell, J. P. Handley, John B. Delos, Stephen Knudson
Arts & Sciences Articles
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 end point 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 …