Exploration of billiards with Keplerian potential

We study a class of elliptic billiards with a Keplerian potential inside, considering two cases: a reflective one, where the particle reflects elastically on the boundary, and a refractive one, where the particle can cross the billiard's boundary entering a region with a harmonic potential. In the latter case the dynamics is given by concatenations of inner and outer arcs, connected by a refraction law. In recent papers (e.g. Barutello et al 2023 Nonlinearity 36 4209; De Blasi and Terracini 2022 Nonlinear Anal. 218 112766; De Blasi and Terracini 2022 Discrete Contin. Dyn. Syst. 43 1269–318; Takeuchi and Zhao 2024 Adv. Math. 436 109411) these billiards have been extensively studied in order to identify which conditions give rise to either regular or chaotic dynamics. In this paper we complete the study by analysing the non focused reflective case, thus complementing the results obtained in Takeuchi and Zhao (2024 Adv. Math. 436 109411) in the focused one. We then analyse the focused and non focused refractive case, where no results on integrability are known except for the centred circular case, by providing an extensive numerical analysis. We present also a theoretical result regarding the linear stability of homothetic equilibrium orbits in the reflective case for general ellipses, highlighting the possible presence of bifurcations already in the integrable framework.