A. H. Osbaldestin

The response of an excitable neuron to trains of electrical spikes is relevant to the understanding of the neural code. In this paper, we study a neurobiologically motivated relaxation oscillator, with appropriately identified fast and slow coordinates, that admits an explicit mathematical analysis. An application of geometric singular perturbation theory shows the existence of an attracting invariant manifold, which is used to construct the Fenichel normal form for the system. This facilitates the calculation of the response of the system to pulsatile stimulation and allows the construction of a so-called extended isochronal map. The isochronal map is shown to have a single discontinuity and be of a type that can admit three types of response: mode-locked, quasiperiodic, and chaotic. The bifurcation structure of the system is seen to be extremely rich and supports period-adding bifurcations separated by windows of both chaos and periodicity. A bifurcation analysis of the isochronal map is presented in conjunction with a description of the various routes to chaos in this system.

We construct a renormalization fixed point corresponding to the strong coupling limit of the golden mean Harper equation. We give an analytic expression for this fixed point, establish its existence and uniqueness, and verify properties previously seen only in numerical calculations. The spectrum of the linearization of the renormalization operator at this fixed point is also explicitly determined. This strong coupling fixed point also helps describe the onset of a strange nonchaotic attractor in quasiperiodically forced systems.

We consider the s-state Potts model on the diamond hierarchical lattice for large s. We show that the generalized dimensions Dq of the density of zeros supported by the associated Julia set are given by Dq = 1 - (q - 1)|s|-2/3/4 log 2+O(s-1). The information dimension D1 equals 1 to all orders.