Bounds on the Hausdorff dimension of a renormalisation map arising from an excitable reaction-diffusion system on a fractal lattice

A renormalisation approach to investigate travelling wave solutions of an excitable reaction-diffusion system on a deterministic fractal structure has recently been derived. The dynamics of a particular class of solutions which are governed by a two-dimensional subspace of these renormalisation recursion relationships are discussed in this paper. The bifurcations of this mapping are discussed with reference to the discontinuities which arise at the singularities. The map is chaotic for a bounded region in parameter space and bounds on the Hausdorff dimension of the associated invariant hyperbolic set are calculated.