Hypermeander of spirals: local bifurcations and statistical properties

In both experimental studies and numerical simulations of waves in excitable media, rigidly rotating spiral waves are observed to undergo transitions to complicated spatial dynamics with long-term Brownian-like motion of the spiral tip. This phenomenon is known as hypermeander. In this paper, we review a number of recent results on dynamics with noncompact group symmetries and make the case that hypermeander may occur at a codimension two bifurcation from a rigidly rotating spiral wave. Our predictions are based on center bundle reduction (Sandstede, Scheel and Wulff), and on central limit theorems and invariance principles for group extensions of hyperbolic dynamical systems. These predictions are confirmed by numerical simulations of the center bundle equations.