Numerical Exploration of Bifurcations and Chaos in Coupled Oscillators

Abstract

A system of coupled oscillators has recently been of great interests since it serves as a paradigm for a variety of problems of competing periods and associated phenomena occurring in nature. A common approach is to study a system of coupled oscillators through models of circle and torus mappings and attempt to emulate rich phenomena of quasiperiodicity, mode-locking, and the transition to chaos often observed in these systems. We attempt to classify types of chaos, which are often observed in three coupled oscillations, and explain some mechanisms for their appearance. Through an interactive use of computational and visual tools especially developed for studying torus maps, we have explored interesting cross-sections of the bifurcation set in the parameter space of the rotation parameter and the nonlinearity. We also present some preliminary numerical results on the structure of the parameter space of the torus maps with an exchange symmetry.