Orbits in Highly Perturbed Dynamical Systems. 111. Nonperiodic Orbits
Abstract
The intersections of orbits by a surface of section either form closed invariant curves or are scattered in some zones of instability." We follow the evolution of invariant curves in the potential V = 21 (A x2+By2) - exy2 as the perturbation increases while the total energy is constant. The zones of instability increase with increasing e and some of them combine; then we see "heteroclinic" doubly asymptotic orbits and new periodic orbits belonging to "irregular" families. There are also open invariant curves of very complicated form. The points of intersection of most orbits in a zone of instability seem ergodically distributed. How- ever there are many exceptional regions around stable invariant points; it is doubtful whether ergodicity is ever attained, even locally. We have also calculated the rotation number versus the distance of each invariant curve from the central invariant point along a straight line. In the zones of instability, only the rotation numbers of regular periodic orbits are defined unambiguously. The abrupt change of the rotation number near unstable invariant points indicates that in each zone of instability we have interaction of more than one resonance.
- Publication:
-
The Astronomical Journal
- Pub Date:
- March 1971
- DOI:
- 10.1086/111098
- Bibcode:
- 1971AJ.....76..147C