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Heteroclinic dynamics in the nonlocal parametrically driven nonlinear Schrödinger equation. (English) Zbl 0988.35155
Summary: Faraday waves are described, under appropriate conditions, by a damped nonlocal parametrically driven nonlinear Schrödinger equation. As the strength of the applied forcing increases this equation undergoes a sequence of transitions to chaotic dynamics. The origin of these transitions is explained using a careful study of a two-mode Galerkin truncation and linked to the presence of heteroclinic connections between the trivial state and spatially periodic standing waves. These connections are associated with cascades of gluing and symmetry-switching bifurcations; such bifurcations are located in the partial differential equations as well.

##### MSC:
 35Q55 NLS-like (nonlinear Schrödinger) equations 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 37J25 Stability problems (finite-dimensional Hamiltonian etc. systems)
AUTO
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