Global dynamics of two coupled parametrically excited van der Pol oscillators (Q1761620)

The authors investigate the behavior of two nonlinearly coupled and parametrically excited van der Pol oscillators with near one-to-one internal resonance. The additional assumptions are such that, after introducing a double time scale, they obtain for the slow flow a 4-dimensional cubic first order system whose linearization at the origin has a double zero eigenvalue of nilpotent type and a pair of purely imaginary eigenvalues. Further transformations lead to a perturbed Hamiltonian system, and for the corresponding unperturbed system there exists a planar subspace with two centers and one saddle point with two adjacent homoclinic orbits. Moreover, in a convenient invariant complementary 2-subspace, a Bogdanov-Takens bifurcation situation with center C arises. In the perturbed 4-dimensional system, this center C becomes the starting point of a Shilnikov-type homoclinic orbit. This is shown by constructing a higher-dimensional Melnikov function. The resulting chaotic behavior in the whole 4-space is made obvious by numerical simulations and corresponding two- and three-dimensional phase portraits.