Ergodic Theory and Dynamical Systems



Characterizing the basins with the most entangled boundaries


HELENA E. NUSSE a1p1 and JAMES A. YORKE a1a2
a1 Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742, USA (e-mail: h.e.nusse@eco.rug.nl)
a2 Department of Mathematics and Department of Physics, University of Maryland, College Park, MD 20742, USA (e-mail: yorke2@ipst.umd.edu)

Article author query
nusse h   [Google Scholar] 
yorke j   [Google Scholar] 
 

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.

(Received January 29 2002)
(Revised September 5 2002)


Correspondence:
p1 University of Groningen, Department of Econometrics, PO Box 800, NL-9700 AV Groningen, The Netherlands.