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Isolas Versus Snaking of Localized Rolls

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Abstract

We analyze the bifurcation diagrams of spatially localized stationary patterns that exhibit a long spatially periodic interior plateau (referred to as localized rolls). In a wide variety of contexts, these bifurcation diagrams consist of isolas or of intertwined s-shaped curves that are commonly referred to as snaking branches. These diagrams have been rigorously analyzed by connecting the existence curves of localized rolls with the bifurcation structure of fronts that connect the rolls to the trivial state. Previous work assumed that the stable and unstable manifolds of rolls were orientable. Here, we extend these results to the nonorientable case and also discuss topological barriers that prevent snaking, thus allowing only isolas to occur. The results are applied to the Swift–Hohenberg system for which we show that nonorientable roll patterns cannot snake.

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Acknowledgements

The authors would like to thank the Institute for Computational and Experimental Research in Mathematics (ICERM) in Providence, RI, where the majority of this work was carried out as part of the 2016 Summer@ICERM program. This program also provided support for Surabhi Desai and Aric Wheeler; we acknowledge support from the NSF via the Grant DMS-1148284 that supported Melissa Stadt. Beck was partially supported by the NSF through the Grant DMS-1411460, and Sandstede was partially supported by the NSF through DMS-1408742.

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Authors and Affiliations

  1. Department of Mathematics, Brown University, Providence, RI, USA

    Tarik Aougab

  2. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Margaret Beck

  3. Department of Mathematics, University of Arizona, Tucson, AZ, USA

    Paul Carter

  4. Mathematical Institute, University of St Andrews, St Andrews, Scotland

    Surabhi Desai

  5. Division of Applied Mathematics, Brown University, Providence, RI, USA

    Björn Sandstede

  6. Department of Mathematics, University of Washington, Seattle, WA, USA

    Melissa Stadt

  7. Department of Mathematics, University of North Carolina, Chapel Hill, NC, USA

    Aric Wheeler

Authors
  1. Tarik Aougab
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  2. Margaret Beck
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  3. Paul Carter
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  4. Surabhi Desai
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  5. Björn Sandstede
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  6. Melissa Stadt
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  7. Aric Wheeler
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Corresponding author

Correspondence to Björn Sandstede.

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In honor and memory of George Sell

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Aougab, T., Beck, M., Carter, P. et al. Isolas Versus Snaking of Localized Rolls. J Dyn Diff Equat 31, 1199–1222 (2019). https://doi.org/10.1007/s10884-017-9624-0

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  • DOI: https://doi.org/10.1007/s10884-017-9624-0

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