Heteroclinic dynamics in the nonlocal parametrically driven nonlinear Schrِdinger equation

Faraday waves are described, under appropriate conditions, by a damped nonlocal parametrically driven nonlinear Schrِdinger 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.