Global bifurcations at the onset of pulse self-replication.
In this work, we carried out an extensive numerical exploration of the delicate global dynamics of pulse self-replication and analyzed the stability of singular homoclinic stationary solutions and their bifurcations in the one-dimensional Gray-Scott model. This stability analysis has several implications for understanding the recently discovered phenomena of self-replicating pulses. The solutions of the ordinary differential equation are organized around a codimension-2 global bifurcation from which two or N branches of homoclinic orbits originate, corresponding to solitary pulse solutions in the partial differential equation. A careful analysis of the bifurcation scenarios in the global bifurcation diagram suggests that the dynamics of the self-replicating system are related to a hierarchy structure of folding bifurcation branches in parameter space. The numerical simulation suggests that the Bogdanov-Takens points, together with the presence of critical points emanating from a particular codimension-2 homoclinic orbit, play a central role for the global bifurcation of periodic orbits, the homoclinic solutions, and the complex chaotic dynamics. The numerical simulation also reveals the existence of a modulating two-pulse or multipulse, which accompanies the procedure of pulse self-replication in reaction-diffusion systems.
Yue
Baozeng
B
Department of Mechanics, School of Science, Beijing Institute of Technology, Beijing 100081, China. bzyue@sohu.com