How to escape a sawtooth: transition to chaos in a simple scattering model

Abstract

The author discusses a two-parameter family of truncated sawteeth serving as the asymptotics of a simple scattering model which shows a transition from a regular to a chaotic dynamics. The question as to how the set of trapped trajectories ('the invariant set') evolves with varying family parameters ('the transition problem') proves to be the key to the understanding of this transition. This invariant set follows a hyperbolic cascade of bifurcations of boundary type. At the critical point, where chaos sets in, the invariant set can be represented as the limit set of a generalized cellular automaton (GCA). Alternatively it can be generated from a single seed by (and identified with) the closure of a transformation semi-group of GCAs. This representation can be used to calculate the rotation numbers of cycles belonging to the invariant set. Statistical measures like the topological entropy indicate the occurrence of a phase transition.