Critical parameters of an oval billiard with an elliptical component

We investigate the critical parameters governing the transition from integrability to chaos in a family of billiards combining elliptical and oval deformations. Unlike standard oval billiards, where a single critical parameter determines the destruction of the last invariant spanning curve, the introduction of an integrable elliptic component adds a second deformation axis. We derive an analytical expression for the critical parameter ɛ c ɛ for this combined system and validate it numerically using Slater’s theorem. Our results show that increasing the elliptical component lowers the critical threshold for the onset of global chaos. Moreover, we identify a previously unexplored regime: when the two deformation components are in phase, the elliptic contribution progressively suppresses chaos, leading to the reemergence of invariant curves and periodic orbits. A first-order analytical approximation captures this behavior and is supported by numerical simulations. Overall, our findings demonstrate how the interplay between distinct boundary deformations enriches phase-space structure and provides enhanced control over chaotic dynamics in billiard systems.