A Two - Parameter Study of the Extent ofChaos in

The billiard system of Benettin and Strelcyn 2] is generalized to a two-parameter family of diierent shapes. Its boundaries are composed of circular segments. The family includes the integrable limit of a circular boundary, convex boundaries of various shapes with mixed dynamics, stadiums, and a variety of non-convex boundaries, partially with ergodic behaviour. The extent of chaos has been measured in two ways: (i) in terms of phase space volume occupied by the main chaotic band, and (ii) in terms of the Lyapunov exponent of that same region. The results are represented as a kind of phase diagram of chaos. We observe complex regularities, related to the bifurcation scheme of the most prominent resonances. A detailed stability analysis of these resonances up to period six explains most of these features. The phenomenon of breathing chaos 13], observed earlier in a one-parameter study of the gravitational wedge billiard, is part of the picture, giving support to the conjecture that this is a fairly common global scenario.