On the stability of the two-link trajectory of a parabolic Birkhoff billiard (Q2821426)

The author considers a dynamical system representing the planar motion of a material point in a domain bounded by two parabolas. It is assumed that the point moves along straight lines inside the domain, and the collision with the boundary is perfectly elastic. The dynamical system under consideration admits a periodic trajectory corresponding to the motion along a segment whose endpoints belong to different boundary parabolas. The stability of this periodic trajectory is studied by analyzing properties the flow of the dynamical system. For this purpose, expansions of the flow are derived in a neighborhood of the periodic trajectory up to third-order terms. Stability conditions with respect to the linear approximation are studied in detail. A crucial point in nonlinear stability analysis is related to the existence of a fourth-order resonance. Sufficient stability conditions are obtained for the model with and without resonances. These results are illustrated with plots of the stability domain in the space of dimensionless parameters of the parabolas.