High‐Codimension Bifurcations in Gierer‐Meinhardt System

This paper investigates the ODE system corresponding to the Gierer‐Meinhardt system, examining its kinetic behavior and possible bifurcation phenomenon. The Gierer‐Meinhardt system describes the spatio‐temporal evolution of activators and substrates in a reaction‐diffusion system and has a wide range of applications. Through the analysis of the system's equilibria and stability, various bifurcation phenomena are observed, such as Hopf bifurcation, saddle‐node bifurcation, and Bogdanov–Takens bifurcation. Notably, Bogdanov–Takens bifurcations of codimension‐three are observed, which encompass both cusp‐type and degenerate focus‐type cases. The numerical simulation results demonstrate the complex dynamical behavior of bifurcation phenomena, including the coexistence of two stable equilibria and stable limit cycles, as well as the emergence of stable and unstable homoclinic loop. This study offers a new perspective for understanding the complex dynamical behavior in reaction‐diffusion systems, with implications for the fields of biology, mathematics, and applied sciences.