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Isolas of multi-pulse solutions to lattice dynamical systems

Part of: General theory in ordinary differential equations Smooth dynamical systems: general theory Qualitative theory

Published online by Cambridge University Press:  20 July 2020

Jason J. Bramburger
Jason J. Bramburger*
Affiliation:
Department of Mathematics Statistics, University of Victoria, Victoria, BC, V8P 5C2, Canada and Division of Applied Mathematics, Brown University, Providence, RI02906, USA ([email protected])
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Abstract

This work investigates the existence and bifurcation structure of multi-pulse steady-state solutions to bistable lattice dynamical systems. Such solutions are characterized by multiple compact disconnected regions where the solution resembles one of the bistable states and resembles another trivial bistable state outside of these compact sets. It is shown that the bifurcation curves of these multi-pulse solutions lie along closed and bounded curves (isolas), even when single-pulse solutions lie along unbounded curves. These results are applied to a discrete Nagumo differential equation and we show that the hypotheses of this work can be confirmed analytically near the anti-continuum limit. Results are demonstrated with a number of numerical investigations.

Keywords

BifurcationLattice Dynamical SystemLocalized StructureSnakingIsola

MSC classification

Primary: 34C23: Bifurcation
Secondary: 34A33: Lattice differential equations 37C05: Smooth mappings and diffeomorphisms
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 3 , June 2021 , pp. 916 - 952
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Copyright
Copyright © The Author(s) 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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