Homoclinic bifurcation from heteroclinic cycles with periodic orbits and tracefiring of pulses.

摘要

Heteroclinic networks can form a skeleton for the nearby dynamics in terms of other heteroclinic, homoclinic or periodic solutions. In many cases such solutions occur as spatial profiles of fronts, pulses or wave-trains of certain spatially one-dimensional partial differential equations in a comoving frame. The present thesis was motivated by the numerical discovery of a self-organized periodic replication process of travelling pulses, termed 'tracefiring', in the three-component Oregonator model of the light-sensitive Belousov-Zhabotinskij reaction.; In the first part of this thesis we consider ordinary differential equations in three or higher dimensions and analyze homoclinic orbits bifurcating from certain heteroclinic cycles between an equilibrium and a periodic orbit. Such heteroclinic cycles differ significantly from heteroclinic cycles between equilibria, in particular the periodicity induces a lack of hyperbolicity. We establish existence and uniqueness of countably infinite families of curves of 1-homoclinic orbits accumulating at certain codimension-1 or -2 heteroclinic cycles of this type. The main result shows the bifurcation of finitely many curves of 1-homoclinic orbits from such codimension-2 heteroclinic cycles depending on global topological properties of the heteroclinic sets. In addition, a leading order expansion of the associated curves in parameter space is derived.; These heteroclinic cycles occur in spatial dynamics of spatially one-dimensional reaction diffusion systems. Codimension-1 corresponds to fronts connecting stable states and codimension-2 certain stable and unstable ones. The second part of this thesis provides an analysis of the structure of relevant essential spectra, and boundary as well as absolute spectra in the sense of Sandstede and Scheel (Physica D 145, 233--277, 2000). Our analysis includes vanishing diffusion rates as well as the case of asymptotically periodic fronts and parts of their absolute spectra.; In the third part, the theoretical results are used to partially explain the aforementioned tracefiring, which also occurs in other models. Codimension-1 heteroclinic cycles can be viewed as a general framework for the constituents of tracefiring. For the Oregonator model, a codimension-2 heteroclinic cycle and the spectral theory can explain the onset of tracefiring. This is corroborated by numerical computation of relevant spectra and solutions.