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Topological Approach to Rigorous Numerics of Chaotic Dynamical Systems with Strong Expansion of Error Bounds

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Abstract

A new algorithm for obtaining rigorous results concerning the existence of chaotic invariant sets of dynamical systems generated by non-autonomous, time-periodic differential equations is presented. Unlike all other algorithms the presented algorithm does not require the numerical integration of the solutions and as a consequence it is insensitive to the rapid error growth in the case of long integration. The result is based on a new theoretical approach to the computation of the homology of the Poincaré map. A concrete numerical example concerning a time-periodic differential equation in the complex plane is provided.

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Authors and Affiliations

  1. Institute of Computer Science, Jagiellonian University, ul. St. Łojasiewicza 6, 30-348, Kraków, Poland

    Marian Mrozek

  2. Chair of Computational Mathematics, WSB-NLU, ul. Zielona 27, 33-300, Nowy Sa̧cz, Poland

    Marian Mrozek

  3. Institute of Mathematics, Jagiellonian University, ul. St. Łojasiewicza 6, 30-348, Kraków, Poland

    Roman Srzednicki

Authors
  1. Marian Mrozek
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  2. Roman Srzednicki
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Correspondence to Marian Mrozek.

Additional information

Communicated by Peter Kloeden.

Both authors supported by MNiSzW grant N201 037 31/3151.

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Mrozek, M., Srzednicki, R. Topological Approach to Rigorous Numerics of Chaotic Dynamical Systems with Strong Expansion of Error Bounds. Found Comput Math 10, 191–220 (2010). https://doi.org/10.1007/s10208-009-9053-5

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  • DOI: https://doi.org/10.1007/s10208-009-9053-5

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