A Minimal Model for Spatio-Temporal Patterns in Thin Film Flow

We consider the development of spatio-temporal oscillations in the Kuramoto-Sivashinsky amplitude model of thin film flow. These develop from Hopf bifurcations off of steady state solutions and are observed to undergo symmetry breaking and period doubling bifurcations. Oscillatory branches in the parameter regime studied apparently terminate in Silnikov type homo-clinic connections. A minimal, three mode nonlinear Galerkin discretization, capable of capturing this bifurcation behavior is constructed. A simple shooting algorithm which exploits this sharp reduction in dimensionality is used to accurately locate the homoclinic connections.