Bifurcations in the Family of Billiards Associated with the Curvature Flow

Abstract We describe some dynamical properties of one parameter families of billiards on convex curves (ovals) which are deformed by the curvature (curve-shortening) flow. We obtain the bifurcations of the period two orbits and some special non-Birkhoff orbits, the normal periodic orbits. We prove the destruction of non-convex caustics of the ellipse by deforming it through the curvature flow. This work has some results from the doctoral thesis Damasceno (Família de aplicações bilhares geradas pelo fluxo de curvature. Tese de Doutorado, UNICAMP, 2011) and Salazar (Região de Instabilidade em Bilhares. tese de Doutorado, UFMG, 2019).