Skip to main content

Research Repository

Advanced Search

Orientation-Dependent Pinning and Homoclinic Snaking on a Planar Lattice

Dean, Andrew David; Matthews, Paul C.; Cox, Stephen M.; King, John. R.

Authors

Andrew David Dean

Paul C. Matthews

Stephen M. Cox stephen.cox@nottingham.ac.uk

JOHN KING john.king@nottingham.ac.uk
Professor of Theoretical Mechanics



Abstract

We study homoclinic snaking of one-dimensional, localized states on two-dimensional, bistable lattices via the method of exponential asymptotics. Within a narrow region of parameter space, fronts connecting the two stable states are pinned to the underlying lattice. Localized solutions are formed by matching two such stationary fronts back-to-back; depending on the orientation relative to the lattice, the solution branch may “snake” back and forth within the pinning region via successive saddle-node bifurcations. Standard continuum approximations in the weakly nonlinear limit (equivalently, the limit of small mesh size) do not exhibit this behavior, due to the resultant leading-order reaction-diffusion equation lacking a periodic spatial structure. By including exponentially small effects hidden beyond all algebraic orders in the asymptotic expansion, we find that exponentially small but exponentially growing terms are switched on via error function smoothing near Stokes lines. Eliminating these otherwise unbounded beyond-all-orders terms selects the origin (modulo the mesh size) of the front, and matching two fronts together yields a set of equations describing the snaking bifurcation diagram. This is possible only within an exponentially small region of parameter space—the pinning region. Moreover, by considering fronts orientated at an arbitrary angle ψ to the x-axis, we show that the width of the pinning region is nonzero only if tan ψ is rational or infinite. The asymptotic results are compared with numerical calculations, with good agreement.

Journal Article Type Article
Publication Date Mar 19, 2015
Journal SIAM Journal on Applied Dynamical Systems
Print ISSN 1536-0040
Electronic ISSN 1536-0040
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 14
Issue 1
Pages 481-521
APA6 Citation Dean, A. D., Matthews, P. C., Cox, S. M., & King, J. R. (2015). Orientation-Dependent Pinning and Homoclinic Snaking on a Planar Lattice. SIAM Journal on Applied Dynamical Systems, 14(1), 481-521. https://doi.org/10.1137/140966897
DOI https://doi.org/10.1137/140966897
Keywords Homoclinic Snaking, Direction-Dependent Pinning, Exponential Asymptotics, Square Lattice
Publisher URL https://epubs.siam.org/doi/10.1137/140966897
Related Public URLs
https://www.maths.notti...ersonal/smc/pubdate.htm
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf
Additional Information (c) 2015 Society for Industrial and Applied Mathematics

Files

96689.pdf (811 Kb)
PDF

Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





You might also like



Downloadable Citations

;