Homoclinic orbit bifurcations in a pattern formation system

An extensive numerical exploration of delicate global dynamics of the pulse self-replication and the stability of singular homoclinic stationary solutions and its bifurcations in the one-dimensional Gray-Scott model are carried out. A careful analysis of the scenario of the global bifurcation diagram suggests that the dynamics of self-replicating system is related to a hierarchy structure of folding bifurcation branches in parameter regions. The numerics suggests the Bogdanov-Takens points together with a presence of critical points emanating from the particular codimension-two homoclinic orbit play a central role for global bifurcation of periodic obits and the homoclinic solutions and the complex chaotic dynamics. Numerical simulation also reveals the existence of the modulating two-pulse or multi-pulse, which accompanies the procedure of pulse self-replicating in reaction-diffusion systems