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 0951-7715/9/6/002/img1 in the plane. It is shown that there do not exist invariant circles near 0951-7715/9/6/002/img1 when there is a point on 0951-7715/9/6/002/img1 where the radius of curvature vanishes or is discontinuous. In addition, when the radius of curvature is not 0951-7715/9/6/002/img4 there are examples with orbits that converge to a point of 0951-7715/9/6/002/img1. If the derivative of the radius of curvature is bounded, such orbits cannot exist. There is also a remark on the connection of the inverse problems for invariant circles in billiards and dual billiards. The final section of the paper concerns an impact oscillator whose dynamics are shown to be the same as a dual billiards map.
- Publication:
-
Nonlinearity
- Pub Date:
- November 1996
- DOI:
- 10.1088/0951-7715/9/6/002
- Bibcode:
- 1996Nonli...9.1411B