Homoclinic bifurcation of heteroclinic cycles with periodic orbits and self-replication of pulses

  • Jens Rademacher (University of Minnesota)
G3 10 (Lecture hall)


New results on homoclinic bifurcation from certain generic codimension-1 and -2 heteroclinic cycles between an equilibrium and a periodic orbit are presented. We show implications for spectral stability of associated travelling waves in spatially onedimensional reaction-diffusion systems, and new results concerning the absolute spectrum of periodic wave trains. These are used to partially explain the phenomenon of 'tracefiring', which is the bifurcation of a stable excitation pulse to a self-replicating pulse-chain. Here, secondary pulses periodically grow out of the wake of the rear pulse and travel in the same direction as the chain. The aforemention codimension-1 case serves as an organizing center for some of the spatial patterns involved. We present this phenomenon in the Oregonator model of the light-sensitive BZ reaction. For this case, the primary pulse's instability is explained through its interaction with a small amplitude periodic wave train. This relies on the above codimension-2 and spectral structure results, as well as numerical computations of spectra for the pulse and periodic wave train.