NUMERICAL-METHODS FOR LOCATING STABLE PERIODIC-ORBITS EMBEDDED IN A LARGELY CHAOTIC SYSTEM

Monte Carlo methods are combined with a Newton method to construct an efficient numerical procedure for locating stable periodic orbits embe dded in a largely chaotic system. We find that the Newton method effec tively enlarges the basin of attraction of the stable orbit by orders of magnitude relative to the stable region surrounding the orbit. Thre e variants of the Newton method are tested. We conclude that an all-po ints finite difference version is the optimal choice. Use of a Monte C arlo search with importance sampling and combined with the Newton meth od proves to be the most efficient search procedure. Application to th e two and three dimensional quartic oscillator leads to previously unk nown stable orbits.