Abstract:

Phase models arise when oscillators are coupled weakly and have proven to be a valuable tool for the analysis of small and large systems of coupled oscillators. Pivotal to this theory is the notion of the phase response curve (PRC) which is related to the eigenfunctions of a certain linear operator. Recently, with Dan Wilson, we have developed an extension of weak coupling theory that is able to incorporate higher order effects and involves so called isostable response functions. These, too are related to the same linear operator as the PRC. Using the higher order reductions, we are able to study various bifurcations that are not evident at the lowest order phase reduced models. We apply this to the normal form for the Hopf bifurcation and for a neural model. Furthermore, they allow one to better analyze the effects of noise on oscillators. Finally, using isostables extended to a reaction diffusion equation, we are able to give a three line derivation of the Kuramoto-Shivashinsky equations.

427 Thackeray Hall