Explosion of limit cycles and chaotic waves in a simple nonlinear chemical system.

Abstract

We consider a simple model of an autocatalytic chemical reaction where a limit cycle rapidly increases to infinite period and amplitude, and disappears under variation of a parameter. We show that this bifurcation can be understood from seeing the system as a singular perturbation problem, and we find the bifurcation point by an asymptotic analysis. Scaling laws for period and amplitude are derived. The unphysical bifurcation to infinity disappears under generic modifications of the model, and for a simple example we show is replaced by a canard explosion, that is, a narrow parameter interval with an explosive growth of the amplitude. The bifurcation to infinity introduces a strong sensitivity that may result in chaotic dynamics if diffusion is added. We show that this behavior persists even if the kinetics is modified to preclude the bifurcation to infinity.