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Chaos in Traveling Waves of Lattice Systems of Unbounded Media

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Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 119))

Abstract

We describe coupled map lattices (CMLs) of unbounded media corresponding to some well-known evolution partial differential equations (including reaction-diffusion equations and the Kuramoto-Sivashinsky, Swift-Hohenberg and Ginzburg-Landau equations). Following Kaneko we view CMLs also as phenomenological models of the medium and present the dynamical systems approach to studying the global behavior of solutions of CMLs. In particular, we establish spatio-temporal chaos associated with the set of traveling wave solutions of CMLs as well as describe the dynamics of the evolution operator on this set. Several examples are given to illustrate the appearance of Smale horseshoes and the presence of the dynamics of Morse-Smale type.

This research was partially supported by the National Science Foundation grant DMS9704564 and by the NATO Collaborative Research grant 970161.

D.R. Orendovici died in a tragic accident in the mountains near Marseille-Lumini on July 6 1998. He was a graduate student at Penn State University and was invited to give a talk at the International Conference“From Cristal to Chaos.”

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Authors and Affiliations

  1. Department of Mathematics, Eberly College of Science, Pennsylvania State University, University Park, PA, 16802, USA

    D. R. Orendovici & Ya B. Pesin

Authors
  1. D. R. Orendovici
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  2. Ya B. Pesin
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Editors and Affiliations

  1. Applied Math 217-50, California Institute of Technology, Pasadena, CA, 91125, USA

    Eusebius Doedel

  2. Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur (LIMSI), B.P. 133, 91403, Orsay, France

    Laurette S. Tuckerman

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Orendovici, D.R., Pesin, Y.B. (2000). Chaos in Traveling Waves of Lattice Systems of Unbounded Media. In: Doedel, E., Tuckerman, L.S. (eds) Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems. The IMA Volumes in Mathematics and its Applications, vol 119. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1208-9_15

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  • DOI: https://doi.org/10.1007/978-1-4612-1208-9_15

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7044-7

  • Online ISBN: 978-1-4612-1208-9

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