Multi-parameter Hopf bifurcations of rimming flows

Abstract

In order to investigate the emergence of periodic oscillations of rimming flows, we study analytically the stability of steady states for the model of Benilov, Kopteva, O'Brien (2005) [7], which describes the dynamics of a thin fluid film coating the inner wall of a rotating cylinder and includes effects of surface tension, gravity, and hydrostatic pressure. We apply multi-parameter perturbation methods to eigenvalues of Fréchet derivatives and prove the transition of a pair of conjugate eigenvalues from the stable to the unstable complex half-plane under appropriate variations of parameters. In order to prove rigorously the corresponding branching of periodic in time solutions from critical equilibria, we extend the multi-parameter Hopf-bifurcation theory to the case of infinite-dimensional dynamical systems.