Codimension-Two Bifurcation Analysis on a Discrete Gierer–Meinhardt System

The stability and the two-parameter bifurcation of a two-dimensional discrete Gierer–Meinhardt system are investigated in this paper. The analysis is carried out both theoretically and numerically. It is found that the model can exhibit codimension-two bifurcations ([Formula: see text], [Formula: see text], and [Formula: see text] strong resonances) for certain critical values at the positive fixed point. The normal forms are obtained by using a series of affine transformations and bifurcation theory. Numerical simulations including bifurcation diagrams, phase portraits and basins of attraction are conducted to validate the theoretical predictions, which can also display some interesting and complex dynamical behaviors.