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 of the reactor as a bifurcation parameter. We define a parameter space of for which the Turing instability may happen. Then, we show that it really occurs below the critical number 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 if is negative (resp. positive). We prove that when lies near the Bogdanov–Takens point . When the critical eigenvalue is double, we have a supercritical bifurcation that produces an ‐attractor . We prove that 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.