Bifurcations in a General Delay Sel’kov–Schnakenberg Reaction–Diffusion System
Abstract
The dynamics of a delay Sel’kov–Schnakenberg reaction–diffusion system are explored. The existence and the occurrence conditions of the Turing and the Hopf bifurcations of the system are found by taking the diffusion coefficient and the time delay as the bifurcation parameters. Based on that, the existence of codimension-2 bifurcations including Turing–Turing, Hopf–Hopf and Turing–Hopf bifurcations are given. Using the center manifold theory and the normal form method, the universal unfolding of the Turing–Hopf bifurcation at the positive constant steady-state is demonstrated. According to the universal unfolding, a Turing–Hopf bifurcation diagram is shown under a set of specific parameters. Furthermore, in different parameter regions, we find the existence of the spatially inhomogeneous steady-state, the spatially homogeneous and inhomogeneous periodic solutions. Discretization of time and space visualizes these spatio-temporal solutions. In particular, near the critical point of Hopf–Hopf bifurcation, the spatially homogeneous periodic and inhomogeneous quasi-periodic solutions are found numerically.
This work is supported by the National Natural Science Foundation of China [Grant Numbers 11301263, 11701306 and 11701275]; the China Postdoctoral Science Foundation [Grant Number 2018M630547]; the Talent Introduction Project of Shanghai Institute of Technology [Grant Number YJ2022-26]; the Natural Science Fund Development Project by Nanjing Tech University; the China Scholarship Council.
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