ORDERED AND CHAOTIC BEHAVIOR OF 2 COUPLED VANDERPOL OSCILLATORS

A physically intuitive, highly symmetric coupling of two van der Pol o scillators is considered here. The structure of the equilibrium points and the discrete symmetries of the model equations are discussed. For some combinations of the parameters, infinitely many equilibrium poin ts appear and evidence is presented pointing to the existence of infin ite periodic trajectories. A complete characterization of the dynamics is done on three specific cases, as a function of the coupling parame ters. It is found that several attractors coexist in phase space, eith er having the symmetry of the model equations or appearing in pairs th at restore such symmetry. The possibility that the asymptotic dynamics is different in the coexisting symmetric and asymmetric attractors is investigated, along with their creation or destruction, splitting, an d merging, when a control parameter is varied. The presence of several attractors allows the points in phase space to change from one basin to another when a control parameter is changed. The route to chaos is through period doubling when only one attractor is explored. When osci llators lock onto an ordered behavior, the period and amplitude surfac es are computed as a function of the (two) coupling parameters and com pared with those periods and amplitudes for the corresponding unpertur bed oscillators.