Recognizing the Hyperbolicity: Cone Criterion and Other Approaches

  • Sergey P. Kuznetsov

Abstract

Physical and technical devices usually are not specially suited to admitting simple mathematical proofs. So, it is vital to employ numerical tools for verification of the hyperbolicity in systems, which potentially may possess uniformly hyperbolic chaotic attractors. Substantiation of hyperbolicity is essential to accounting relevant conclusions of the mathematical theory, like availability of description in terms of Markov partitions with a finite alphabet, or structural stability of the chaotic altraclors, which may be of natural practical significance. Section 7.1 is devoted to approaches based on verification of a fundamental property of the hyperbolic invariant sets that is transversality of mutual location of stable and unstable manifolds for all orbits belonging to the invariant set. In Sect. 7.2 we turn to a technique of visualization of natural invariant measures along filaments of attractors, which shows presence or absence of singularities in the distributions. For comparison, besides models with uniformly hyperbolic attractors, some systems not relating to this class are considered in terms of the same approaches. Final part of the chapter is devoted to the cone criterion based on a rigorous mathematical result and appropriate for validation of hyperbolicity in compulations. Here it is reexamined in some detail, with explanation of technical hints, and with concrete examples of its application.

Keywords

Invariant Measure Unstable Manifold Chaotic Attractor Stable Manifold Cone Criterion 
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.

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Copyright information

© Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Sergey P. Kuznetsov
    • 1
  1. 1.Kotel’nikov’s Institute of Radio-Engineering and Electronics of RASSaratov BranchSaratovRussian Federation

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