Hopf bifurcation in anisotropic reaction diffusion systems posed in large rectangles

Abstract

The oscillatory instability (Hopf bifurcation) for anisotropic reaction diffusion equations posed in large (but finite) rectangles is investigated. The work pursued in this dissertation extends previous studies for infinitely extended 2D systems to include finite-size effects. For the case considered, the solution of the reaction diffusion system is represented in terms of slowly modulated complex amplitudes of four wave-trains propagating in four oblique directions. While for the infinitely extended system the modulating amplitudes are independent dynamical variables, the finite size of the domain leads to relations between them induced by wave reflections at the boundaries. This leads to a single amplitude equation for a doubly periodic function that captures all four envelopes in different regions of its fundamental domain. The amplitude equation is derived by matching an asymptotic bulk solution to an asymptotic boundary layer solution. While for the corresponding infinitely extended system no further parameters generically remain in the amplitude (envelope) equations above the onset value of the control parameter, the finite-size amplitude equation retains a dependence on a rescaled version of this parameter. Numerical simulations show that the dynamics of the bounded system shows different behavior at onset in comparison to the unbounded system, and the complexity of the solutions significantly increases when the rescaled control parameter is increased. As an application of the technique developed, an anisotropic Activator-Inhibitor model with higher order diffusion is studied, and parameter values of the amplitude equations are calculated for several parameter sets of the model equations.