Bogdanov-Takens Bifurcation of Codimension 3 in the Gierer-Meinhardt Model

Bifurcation of the local Gierer-Meinhardt model is analyzed in this paper. It is found that the degenerate Bogdanov-Takens bifurcation of codimension 3 exists in the model, except for the saddle-node bifurcation and the Hopf bifurcation. That was not reported in the literature about this model. The existence of equilibria, their stability, the bifurcation and the induced complicated and interesting dynamics are explored in detail, by using stability analysis, normal form method and bifurcation theory. Numerical results are also presented to validate the theoretical results.

Automated summary

This paper analyzes the bifurcation of the local Gierer-Meinhardt model and discovers the existence of a degenerate Bogdanov-Takens bifurcation of codimension 3 in the model, which has not been reported in the literature. The paper explores the existence, stability, bifurcation, and the resulting complex and interesting dynamics of equilibria using stability analysis, normal form method, and bifurcation theory. Numerical results are also provided to validate the theoretical findings.