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Basic sets: Sets that determine the dimension of basin boundaries

Part of the Lecture Notes in Mathematics book series (LNM,volume 1342)

Abstract

In this paper we consider the question of how many possible dimensions a basin boundary can have. We conjecture that the number of possible dimension values is at most the number of some well defined asymptotic sets (called basic sets) on the basin boundary. It should be noticed that the dimension of a basic set also has a dynamical meaning. The conjecture will be proved for a class of Axiom A systems (namely two dimensional diffeomorphisms and one dimensional chaotic maps). In addition, we will give numerical evidence for a physical example.

Keywords

  • Hausdorff Dimension
  • Homoclinic Orbit
  • Stable Manifold
  • Basin Boundary
  • Uncertainty Dimension

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Laboratory for Plasma and Fusion Energy Studies.

Departments of Physics and of Electrical Engineering.

Institue for Physical Science and Technology, and Department of Mathematics.

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© 1988 Springer-Verlag

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Grebogi, C., Nusse, H.E., Ott, E., Yorke, J.A. (1988). Basic sets: Sets that determine the dimension of basin boundaries. In: Alexander, J.C. (eds) Dynamical Systems. Lecture Notes in Mathematics, vol 1342. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0082834

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  • DOI: https://doi.org/10.1007/BFb0082834

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