Numerical methods for locating stable periodic orbits embedded in a largely chaotic system

Abstract

Monte Carlo methods are combined with a Newton method to construct an efficient numerical procedure for locating stable periodic orbits embedded in a largely chaotic system. We find that the Newton method effectively enlarges the basin of attraction of the stable orbit by orders of magnitude relative to the stable region surrounding the orbit. Three variants of the Newton method are tested. We conclude that an all-points finite difference version is the optimal choice. Use of a Monte Carlo search with importance sampling and combined with the Newton method proves to be the most efficient search procedure. Application to the two and three dimensional quartic oscillator leads to previously unknown stable orbits. © 1994 American Institute of Physics.