Two-link billiard trajectories: Extremal properties and stability

Two-link periodic trajectories of a plane convex billiard, when a point mas s moves along a segment which is orthogonal to the boundary of the billiard at its end points, are considered. It is established that, if the caustic of the boundary lies within the billiard, then, in a typical situation, the re is an even number of two-link trajectories and half of them are hyperbol ic (and, consequently, unstable) and the other half are of elliptic type. A n example is given of a billiard for which the caustic intersects the bound ary and all of the hue-link trajectories are hyperbolic. The analysis of th e stability is based on an analysis of the extremum of a function of the le ngth of a segment of a convex billiard which is orthogonal to the boundary at one of its ends. (C) 2001 Elsevier Science Ltd. All rights reserved.