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Discrete-Time Nonautonomous Dynamical Systems

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Stability and Bifurcation Theory for Non-Autonomous Differential Equations

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2065))

Abstract

These notes present and discuss various aspects of the recent theory for time-dependent difference equations giving rise to nonautonomous dynamical systems on general metric spaces:First, basic concepts of autonomous difference equations and discrete-time (semi-) dynamical systems are reviewed for later contrast in the nonautonomous case. Then time-dependent difference equations or discrete-time nonautonomous dynamical systems are formulated as processes and as skew products. Their attractors including invariants sets, entire solutions, as well as the concepts of pullback attraction and pullback absorbing sets are introduced for both formulations. In particular, the limitations of pullback attractors for processes is highlighted. Beyond that Lyapunov functions for pullback attractors are discussed.Two bifurcation concepts for nonautonomous difference equations will be introduced, namely attractor and solution bifurcations.Finally, random difference equations and discrete-time random dynamical systems are investigated using random attractors and invariant measures.

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Acknowledgements

This work was partially supported by the DFG grant KL 1203/7-1, the Ministerio de Ciencia e Innovación project MTM2011-22411, the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under the Ayuda 2009/FQM314 and the Proyecto de Excelencia P07-FQM-02468 (Peter E. Kloeden). Martin Rasmussen was supported by an EPSRC Career Acceleration Fellowship.

The authors thank Dr. Thomas Lorenz for carefully reading parts of the article.

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Authors and Affiliations

  1. Institut für Mathematik, Goethe-Universität, 11 19 32, 60054, Frankfurt a.M., Germany

    P. E. Kloeden

  2. Institut für Mathematik, Universität Klagenfurt, Universitätsstraße 65–67, 9020, Klagenfurt, Austria

    C. Pötzsche

  3. Department of Mathematics, Imperial College, London, SW7 2AZ, UK

    M. Rasmussen

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  1. P. E. Kloeden
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  2. C. Pötzsche
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  3. M. Rasmussen
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Correspondence to P. E. Kloeden .

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Kloeden, P.E., Pötzsche, C., Rasmussen, M. (2013). Discrete-Time Nonautonomous Dynamical Systems. In: Stability and Bifurcation Theory for Non-Autonomous Differential Equations. Lecture Notes in Mathematics(), vol 2065. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32906-7_2

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