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Characteristics of Poincaré Recurrences

  • Vadim S. Anishchenko
  • Tatyana E. Vadivasova
  • Galina I. Strelkova
Chapter
Part of the Springer Series in Synergetics book series (SSSYN)

Abstract

The analysis of Poincaré recurrences is one of the fundamental problems in the theory of dynamical systems. Poincaré recurrence means that practically any phase trajectory starting from some point of the system phase space passes arbitrarily close to the initial state an infinite number of times. H. Poincaré called these phase trajectories stable according to Poisson. Since Poincaré’s day, the analysis of the dynamics of Poisson stable systems has been an active topic of research in both mathematics and physics. The fundamental importance of this problem is evidenced by the fact that the very idea that a system should return over time to a neighborhood of its initial state is used much more widely than in mathematical theory alone. Thus, in a certain sense, it has become one of the philosophical concepts of modern science.

Keywords

Lyapunov Exponent Return Time Phase Trajectory Local Approach Topological Entropy 
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.

Reference

  1. 1.
    Afraimovich, V., Ugalde, E., Urias, J.: Fractal Dimension for Poincaré Recurrences. Elsevier, Amsterdam/London (2006)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Vadim S. Anishchenko
    • 1
  • Tatyana E. Vadivasova
    • 1
  • Galina I. Strelkova
    • 1
  1. 1.Department of Physics Instiute of Nonlinear DynamicsSaratov State UniversitySaratovRussia

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