Homoclinic Snakes Bounded by a Saddle-Center Periodic Orbit

Champneys, A. R., Knobloch, Edgar, Ma, Yi-Ping and Wagenknecht, T. (2012) Homoclinic Snakes Bounded by a Saddle-Center Periodic Orbit. SIAM Journal on Applied Dynamical Systems, 11 (4). pp. 1583-1613. ISSN 1536-0040

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Official URL: http://dx.doi.org/10.1137/110855429


We describe a new variant of the so-called homoclinic snaking mechanism for the generation of infinitely many distinct localized patterns in spatially reversible partial differential equations on the real line. In standard snaking a branch of localized states undergoes infinitely many folds as the pattern grows in length by adding cells at either side. In the cases studied here the localized states have a defect or hump in the middle corresponding to an additional orbit homoclinic to the underlying spatially periodic orbit, and the folds accumulate on a parameter value where the periodic orbit undergoes a saddle-center transition. By analyzing an appropriate normal form in a spatial dynamics approach, it is shown that convergence of the folds is algebraic rather than exponential. Specifically the parameter value of the $n$th fold scales like $n^{-4}$. The transition from this saddle-center mediated snaking to regular snaking is described by a codimension-two bifurcation that is also analyzed. The results are compared with numerical computations on two distinct complex Ginzburg--Landau models, one of which is variational and so represents a conservative system in space, while the other is nonvariational. Good agreement with the theory is found in both cases, and the connection between the theory and the recently identified defect-mediated snaking is established.

Item Type: Article
Uncontrolled Keywords: homoclinic snaking, localized patterns, global bifurcations
Subjects: F300 Physics
G100 Mathematics
Department: Faculties > Engineering and Environment > Mathematics, Physics and Electrical Engineering
Depositing User: Dr Yi-Ping Ma
Date Deposited: 23 Nov 2016 15:30
Last Modified: 12 Oct 2019 22:28
URI: http://nrl.northumbria.ac.uk/id/eprint/28409

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