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To snake or not to snake in the planar Swift-Hohenberg equation. (English) Zbl 1200.37014
In this paper, based on their previous study of snakes-and-ladders bifurcation structures of one-dimensional Swift-Hohenberg equation, the authors extended their investigation to the two-dimensional Swift-Hohenberg equation posed on a cylinder and the plane. On cylinders, they found localized roll, square, and stripe patches that exhibited snaking and nonsnaking behavior on the same bifurcation branch. A subtle finding was that asymmetric structures could be stable. On the plane, they investigated the bifurcation structure of fully localized roll structures known as “worms”. The profiles of planar patterns and the shape of their bifurcation branches were explained using simple ideas from the theory of dynamical systems. For one-dimensional structures, the behavior of snaking or nonsnaking can be predicted using only their profiles. However, for planar patterns, the authors showed that it was in general not possible to predict whether a given pattern snakes or not by inspecting their profiles.

37C29 Homoclinic and heteroclinic orbits for dynamical systems
35B32 Bifurcations in context of PDEs
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
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