From quasiperiodicity to chaos in the Belousov–Zhabotinskii reaction. II. Modeling and theory

Abstract

We discuss the quasiperiodic behavior experimentally observed in the Belousov–Zhabotinskii reaction as the result of the interaction of two elementary instabilities, namely the Hopf bifurcation at the origin of the oscillating nature of this reaction and the hysteresis bifurcation which accounts for the phenomenom of bistability. We use a normal form approach to understand the evolution of the dynamics when the BZ system is moved away from the local situation where both these instabilities are competing. We first discuss the transition to chaos which comes with the breaking up of the underlying torus into a fractal object. Then we emphasize that nonlocally such a two-frequency dynamics manifests through alternating periodic–chaotic sequences which look very much like the sequences observed in bench experiments. We propose a seven-variable Oregonator type model which not only accounts for these sequences but also for those which involve only one fundamental frequency as observed in the 1980 Texas experiment. We refer to the dynamical system theory to definitively establish the existence of deterministic chaos in both of these sequences. We conclude with the very promising perspective of using a normal form approach to reduce the evolution equations to their simplest form by selecting the relevant instabilities which control the dynamics of the BZ reaction in the regions of parameter space explored so far.