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Characterizing the basins with the most entangled boundaries

Published online by Cambridge University Press:  20 June 2003

HELENA E. NUSSE  and
JAMES A. YORKE
HELENA E. NUSSE
Affiliation:
Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA (e-mail: [email protected]) University of Groningen, Department of Econometrics, PO Box 800, NL-9700 AV Groningen, The Netherlands.
JAMES A. YORKE
Affiliation:
Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA (e-mail: [email protected]) Department of Mathematics and Department of Physics, University of Maryland, College Park, MD 20742, USA (e-mail: [email protected])
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Abstract

In dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is non-empty. The purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. If a two-dimensional basin has a basin cell (a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well-chosen periodic orbit), then the basin consists of a central body (the basin cell) and a finite number of channels attached to it and the basin boundary is fractal. We prove the following surprising property for certain fractal basin boundaries: a basin of attraction B has a basin cell if and only if every diverging path in basin B has the entire basin boundary \partial\bar{B} as its limit set. The latter property reflects a complete entangled basin and its boundary.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 23 , Issue 3 , June 2003 , pp. 895 - 906
Copyright
2003 Cambridge University Press

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