Orientation-dependent pinning and homoclinic snaking on a planar lattice
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...
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| Format: | Article |
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Society for Industrial and Applied Mathematics
2015
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| Online Access: | https://eprints.nottingham.ac.uk/28656/ |
| _version_ | 1848793619010420736 |
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| author | Dean, Andrew David Matthews, Paul C. Cox, Stephen M. King, John. R. |
| author_facet | Dean, Andrew David Matthews, Paul C. Cox, Stephen M. King, John. R. |
| author_sort | Dean, Andrew David |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:03:10Z |
| format | Article |
| id | nottingham-28656 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:03:10Z |
| publishDate | 2015 |
| publisher | Society for Industrial and Applied Mathematics |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-286562020-05-04T17:04:22Z https://eprints.nottingham.ac.uk/28656/ Orientation-dependent pinning and homoclinic snaking on a planar lattice Dean, Andrew David Matthews, Paul C. Cox, Stephen M. King, John. R. 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. Society for Industrial and Applied Mathematics 2015-03-19 Article PeerReviewed Dean, Andrew David, Matthews, Paul C., Cox, Stephen M. and King, John. R. (2015) Orientation-dependent pinning and homoclinic snaking on a planar lattice. SIAM Journal on Applied Dynamical Systems, 14 (1). pp. 481-521. ISSN 1536-0040 Homoclinic Snaking Direction-Dependent Pinning Exponential Asymptotics Square Lattice http://dx.doi.org/10.1137/140966897 doi:10.1137/140966897 doi:10.1137/140966897 |
| spellingShingle | Homoclinic Snaking Direction-Dependent Pinning Exponential Asymptotics Square Lattice Dean, Andrew David Matthews, Paul C. Cox, Stephen M. King, John. R. Orientation-dependent pinning and homoclinic snaking on a planar lattice |
| title | Orientation-dependent pinning and homoclinic snaking on a planar lattice |
| title_full | Orientation-dependent pinning and homoclinic snaking on a planar lattice |
| title_fullStr | Orientation-dependent pinning and homoclinic snaking on a planar lattice |
| title_full_unstemmed | Orientation-dependent pinning and homoclinic snaking on a planar lattice |
| title_short | Orientation-dependent pinning and homoclinic snaking on a planar lattice |
| title_sort | orientation-dependent pinning and homoclinic snaking on a planar lattice |
| topic | Homoclinic Snaking Direction-Dependent Pinning Exponential Asymptotics Square Lattice |
| url | https://eprints.nottingham.ac.uk/28656/ https://eprints.nottingham.ac.uk/28656/ https://eprints.nottingham.ac.uk/28656/ |