Turing instability and attractor bifurcation for the general Brusselator model

In this paper, we analyze the dynamic bifurcation of the general Brusselator model when the order of reaction is $ p \in (1,\infty) $. We verify that the Turing instability occurs above the critical control number and obtain a rigorous formula for the bifurcated stable patterns. We define a constant $ s_N $ that gives a criterion for the continuous transition. We obtain continuous transitions for $ s_N>0 $, but jump transitions for $ s_N<0 $. By using this criterion, we prove mathematically that higher-molecular reactions are rarely observed. We also provide some numerical results that illustrate the main results.