Ordered and chaotic behavior of two coupled van der Pol oscillators

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Abstract
A physically intuitive, highly symmetric coupling of two van der Pol oscillators 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 points appear and evidence is presented pointing to the existence of infinite periodic trajectories. A complete characterization of the dynamics is done on three specific cases, as a function of the coupling parameters. It is found that several attractors coexist in phase space, either having the symmetry of the model equations or appearing in pairs that 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, and 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 oscillators lock onto an ordered behavior, the period and amplitude surfaces are computed as a function of the (two) coupling parameters and compared with those periods and amplitudes for the corresponding unperturbed oscillators.

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