Stability analysis of the breathing circle billiard

Stability is a fundamental problem in time dependent billiards. In this work, we prove that the breathing circle billiard has invariant tori near infinity preventing the unboundedness of energy when the motion of boundary is regular enough. The proof also implies the boundedness of the energy of all solutions for a new class of Fermi-Ulam model with one of the walls replaced by a potential which is growing to infinity as the position coordinate approaches to the origin. When the motion of boundary is piecewise smooth, the dynamics near infinity is either elliptic or hyperbolic depending on an explicit parameter, which is similar to the results in [11] for the piecewise smooth Fermi-Ulam model. Moreover, we show the existence of unbounded orbits when this parameter is within some intervals. The numerical simulations are supported by our mathematical analysis.