Dual billiards, twist maps, and impact oscillators

Abstract

In this paper techniques of twist map theory are applied to the annulus maps arising from dual billiards on a strictly convex closed curve G in the plane. It is shown that there do not exist invariant circles near G when there is a point on G where the radius of curvature vanishes or is discontinuous. In addition, when the radius of curvature is not $C^1$ there are examples with orbits that converge to a point of G. If the derivative of the radius of curvature is bounded, such orbits cannot exist. The final section of the paper concerns an impact oscillator whose dynamics are the same as a dual billiards map. The appendix is a remark on the connection of the inverse problems for invariant circles in billiards and dual billiards.