Asymptotic and numerical study of Brusselator chaos

We investigate the Brusselator reaction–diffusion equations with periodic boundary conditions. We consider the range of values of the parameters used by Kuramoto in his study of chaotic concentration waves. We determine numerically the bifurcation diagram of the long-time travelling and standing wave solutions using a highly accurate Fourier pseudo-spectral method. For moderate values of the bifurcation parameter, we have found a sequence of instabilities leading either to periodic and quasiperiodic standing waves, or to chaotic regimes. However, for large values of the control parameter, we have found only uniform time-periodic solutions or time-periodic travelling wave solutions. Our numerical study motivates a new asymptotic analysis of the Brusselator equations for large values of the control parameter and small diffusion coefficients. This analysis explains the numerical predictions. The chaotic regime is limited to moderate values of the control parameter and periodic solutions are the only solutions for large values of the control parameter. We identify the stabilizing mechanism as the relaxation oscillations which appear when the control parameter is large. Our asymptotic result on the stability of periodic solutions is then generalized to a class of two-variable reaction-diffusion equations.