Turing Instability and Dynamic Bifurcation for the One-Dimensional Gray–Scott Model

Abstract

We study the dynamic bifurcation of the one-dimensional Gray–Scott model by taking the diffusion coefficient $\lambda$ of the reactor as a bifurcation parameter. We define a parameter space $\Sigma$ of $(k,F)$ for which the Turing instability may happen. Then, we show that it really occurs below the critical number ${\lambda }_0$ and obtain rigorous formula for the bifurcated stable patterns. When the critical eigenvalue is simple, the bifurcation leads to a continuous (resp. jump) transition for $\lambda <{\lambda }_0$ if $A_m(k,F)$ is negative (resp. positive). We prove that $A_m(k,F)>0$ when $(k,F)$ lies near the Bogdanov–Takens point $(\frac{1}{16},\frac{1}{16})$. When the critical eigenvalue is double, we have a supercritical bifurcation that produces an $S^1$-attractor ${\Omega }_m$. We prove that ${\Omega }_m$ consists of four asymptotically stable static solutions, four saddle static solutions, and orbits connecting them. We also provide numerical results that illustrate the main theorems.