Numerical bifurcation and stability analysis of solitary pulses in an excitable reaction—diffusion medium

Abstract

We present a systematic, computer-assisted study of the bifurcations and instabilities of solitary pulses in an excitable medium capable of displaying both stable pulse propagation and spatiotemporally chaotic dynamics over intervals of parameter space. The reaction—diffusion model used is of the activator-inhibitor type; only the activator diffuses in this medium. The control parameters are the ratio of time scales of the activator and inhibitor dynamics and the excitation threshold. This study focuses on travelling pulses, their domain of existence and the bifurcations that render them unstable. These pulses are approximated as: (a) homoclinic orbits in a travelling wave ODE frame; and (b) as solutions of the full partial differential equation (PDE) with periodic boundary conditions in large domains. A variety of bifurcations in the travelling wave ODE frame are observed (including heteroclinic loops, so-called T-points [A.R. Champneys and Y.A. Kuznetsov, Numerical detection and continuation of codimension-2 homoclinic bifurcations, Int. J. Bif. Chaos 4 (1994) 785; H. Kokobu, Homoclinic and heteroclinic bifurcations of vectorfields, Japan J. Appl. Math. 5 (1988) 455]). Instabilities in the full PDE frame include both Hopf bifurcations to modulated travelling waves (involving the discrete pulse spectrum) as well as transitions involving the continuous spectrum (such as the so-called ‘backfiring’ transition [M. Bär, M. Hildebrand, M. Eiswirth, M. Falcke, H. Engel and M. Neufeld, Chemical turbulence and standing waves in a surface reaction model: The influence of global coupling and wave instabilities, Chaos 4 (1994) 499]). The stability of modulated pulses is computed through numerical Floquet analysis and a cascade of period doubling bifurcations is observed, as well as certain global bifurcations. These results, corroborated by observations from direct numerical integration, provide a ‘skeleton’ around which many features of the overall complex spatiotemporal dynamics of the PDE are organized.