Global analysis of periodic orbit bifurcations in coupled Morse oscillator systems: time-reversal symmetry, permutational representations and codimension-2 collisions.

Abstract

In this paper we study periodic orbit bifurcation sequences in a system of two coupled Morse oscillators. Time-reversal symmetry is exploited to determine periodic orbits by iteration of symmetry lines. The permutational representation of Tsuchiya and Jaffe is employed to analyze periodic orbit configurations on the symmetry lines. Local pruning rules are formulated, and a global analysis of possible bifurcation sequences of symmetric periodic orbits is made. Analysis of periodic orbit bifurcations on symmetry lines determines bifurcation sequences, together with periodic orbit periodicities and stabilities. The correlation between certain bifurcations is explained. The passage from an integrable limit to nointegrability is marked by the appearance of tangent bifurcations; our global analysis reveals the origin of these ubiquitous tangencies. For period-1 orbits, tangencies appear by a simple disconnection mechanism. For higher period orbits, a different mechanism involving 2-parameter collisions of bifurcations is found. (c) 1999 American Institute of Physics.