Bifurcation and Stability Analyses for a Coupled Brusselator Model

This paper addresses the dynamic behaviour of a chemical oscillator arising from the series coupling of two Brusselators. Of particular interest is the study of the associated Hopf bifurcation and double-Hopf bifurcations. The motion of the oscillator may either be periodic (bifurcating from a Hopf-type critical point), or quasi-periodic (bifurcating from a compound critical point). Furthermore, bifurcation analysis reveals that the limit cycles associated with the "rst Brusselator are always stable, while that generated by the second Brusselator may be unstable if the parameter values are chosen far from the stability boundary. It is interesting to note that in the vicinity of the double-Hopf compound critical point, there exist periodic as well as quasi-periodic solutions. The quasi-periodic motion is stable for a small parameter region. A robust Gauss}Seidel like implicit "nite-di!erence method (GS1) has been developed and used for the solution of the resulting initial-value problem (IVP). In addition to being of comparable accuracy (judging by the similarity of the pro"les generated) with the fourth order Runge}Kutta method (RK4), the GS1 method will be seen to have better numerical stability property than RK4. Unlike the RK4, which fails when large time steps are used to integrate the IVP, extensive numerical simulations with appropriate initial data suggest that the GS1 method is unconditionally convergent. Moreover, it is more economical computationally. ( 2001 Academic Press