Chaos and Regularity in an Anisotropic Soft Squircle Billiard

Abstract: A hard-wall billiard is a mathematical model describing the confinement of a free particle that collides specularly and instantaneously with boundaries and discontinuities. Soft billiards are a generalization that includes a smooth boundary whose dynamics are governed by Hamiltonian equations and overcome overly simplistic representations. We study the dynamical features of an anisotropic soft-wall squircle billiard. This curve is a geometric figure that seamlessly blends the angularity of a square with the smooth curves of a circle. We characterize the billiard's emerging trajectories, exhibiting the onset of chaos and its alternation with regularity in the parameter space. We characterize the transition to chaos and the stabilization of the dynamics by revealing the nonlinearity of the parameters (squarness, ellipticity, and hardness) via the computation of Poincar\'e surfaces of section and the Lyapunov exponent across the parameter space. We expect our work to introduce a valuable tool to increase understanding of the onset of chaos in soft billiards.