Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues

Abstract

In this paper, we study the dynamic phase transition for one dimensional Brusselator model. By the linear stability analysis, we define two critical numbers lambda(0) and lambda(1) for the control parameter lambda in the equation. Motivated by [9], we assume that lambda(0) < lambda(1) and the linearized operator at the trivial solution has multiple critical eigenvalues beta(+)(N) and beta(+)(N+1) . Then, we show that as lambda passes through lambda(0), the trivial solution bifurcates to an S-1-attractor A(N). We verify that A(N) consists of eight steady state solutions and orbits connecting them. We compute the leading coefficients of each steady state solution via the center manifold analysis. We also give numerical results to explain the main theorem.