Bifurcations of basins of attraction from the view point of prime ends

Helena E. Nusse*, James A. Yorke

*Corresponding author for this work

    Research output: Contribution to journalArticleAcademicpeer-review

    6 Citations (Scopus)

    Abstract

    In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is nonempty and the basins often have fractal basin boundaries. The purpose of this paper is to describe the structure and properties of unbounded basins and their boundaries for two-dimensional diffeomorphisms. Frequently, if not always, there is a periodic saddle on the boundary that is accessible from the basin. Caratheodory and many others developed an approach in which an open set (in our case a basin) is compactified using so-called prime end theory. Under the prime end compactification of the basin, boundary points of the basin (prime ends) can be characterized as either type 1, 2, 3, or 4. In all well-known examples, most points are of type 1. Many two-dimensional basins have a basin cell, that is, 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 present a result that says (a basin has a basin cell) if and only if {every prime end that is defined by a chain of unbounded regions (in the basin) is a prime end of type 3 and furthermore all other prime ends are of type 1}. We also prove as a parameter is varied, the basin cell for a basin B is created (or destroyed) if and only if either there is a saddle node bifurcation or the basin B has a prime end that is defined by a chain of unbounded regions and is a prime end of either type 2 or type 4. (c) 2007 Elsevier B.V. All rights reserved.

    Original languageEnglish
    Pages (from-to)2567-2579
    Number of pages13
    JournalTopology and its applications
    Volume154
    Issue number13
    DOIs
    Publication statusPublished - 1-Jul-2007
    EventUS-Polish International Workshop on Geometric Methods in Dynamical Systems - , Germany
    Duration: 9-Jun-200412-Jun-2004

    Keywords

    • PLANE SEPARATING CONTINUA
    • BOUNDARIES
    • REPRESENTATION
    • REGIONS

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