The stability of two-link trajectories of a Birkhoff billiard

The inertial motion of a particle in a plane region bounded by two analytical curves is studied. Within the region the particle moves in a straight line and the collisions with the boundary curves are considered to be absolutely elastic. It is assumed that the boundary curves allow the existence of a two-link periodic trajectory. The nonlinear problem of stability of this trajectory is analysed. An algorithm for constructing the area-preserving mapping corresponding to this problem in the form of series is explained. General conditions are obtained for the stability and instability of a two-link trajectory, expressed in terms of the coefficients of the series specifying the boundary curves. Some specific examples are considered.