Towards a general theory of nonlinear flow-distributed oscillations

Abstract

We outline a general theory for the analysis of flow-distributed standing and travelling wave patterns in one-dimensional, open plug-flows of oscillatory chemical media. We treat both the amplitude and phase dynamics of small and large-amplitude waves, considering both travelling and stationary waves on an equal footing and emphasizing features that are generic to a variety of kinetic models. We begin with a linear stability analysis for constant and periodic boundary forcing, drawing attention to the implications for systems far from a Hopf bifurcation. Among other results, we show that for systems far from a Hopf bifurcation, the first absolutely unstable mode may be a stationary wave mode. We then introduce a non-linear formalism for studying both travelling and stationary waves and show that the wave forms and their amplitudes depend on a single reduced transport parameter. Our formalism sheds light on cases where neither the linearized analyis nor the kinematic theory of phase waves give an adequate description, and it can be applied to study some of the more complex types of bifurcations (Canards, period-doublings, etc.) in open flow systems.