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 happens in the model, except that teh saddle-node bifurcation and the Hopf bifurcation. That was not reported in the existing results about this model. The existence of equilibria, their stability, the bifurcation and the induced complicated and interesting dynamics are explored in detail, by using the stability analysis, the normal form method and bifurcation theory. Numerical results are also presented to validate theoretical results.