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Symmetry and Pattern Formation in Coupled Cell Networks

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Pattern Formation in Continuous and Coupled Systems

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 115))

Abstract

We describe some basic concepts and techniques from symmetric bifurcation theory in the context of coupled systems of cells (‘oscillator networks’). These include criteria for the existence of symmetry-breaking branches of steady and periodic states. We emphasize the role of symmetry as a general framework for such analyses. As well as overt symmetries of the network we discuss internal symmetries of the cells, ‘hidden’ symmetries related to Neumann boundary conditions, and spatio-temporal symmetries of periodic states. The methods are applied to a model central pattern generator for legged animal locomotion.

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Authors and Affiliations

  1. Mathematics Department, University of Houston, Houston, TX, 77204-3476, USA

    Martin Golubitsky

  2. Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

    Ian Stewart

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  1. Martin Golubitsky
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  2. Ian Stewart
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Editors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204-4792, USA

    Martin Golubitsky

  2. Department of Chemical Engineering, University of Houston, Houston, TX, 77204-4792, USA

    Dan Luss

  3. Theoretical and Applied Mechanics, Cornell University, Ithaca, NY, 14853, USA

    Steven H. Strogatz

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Golubitsky, M., Stewart, I. (1999). Symmetry and Pattern Formation in Coupled Cell Networks. In: Golubitsky, M., Luss, D., Strogatz, S.H. (eds) Pattern Formation in Continuous and Coupled Systems. The IMA Volumes in Mathematics and its Applications, vol 115. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1558-5_6

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  • DOI: https://doi.org/10.1007/978-1-4612-1558-5_6

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7192-5

  • Online ISBN: 978-1-4612-1558-5

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