Chaos and strange attractors in coupled oscillators with energy-preserving nonlinearity

We present an analysis for a particular singularly perturbed conservative system. This system comes from the normal form of two coupled oscillator systems with widely-separated frequencies and energy-preserving nonlinearity. The analysis is done in this paper for a degenerate case of such a system, while the generic one has been treated in the literature. To understand the relation with the strong resonance case, we have computed the normal form of the 2: 1 resonance, and found that the latter is contained in our system. We present a theorem that gives the existence of a nontrivial equilibrium for a general singularly perturbed conservative system. We detect that the nontrivial equilibrium undergoes two Hopf bifurcations. Furthermore, the periodic solutions created through these Hopf bifurcations undergo a sequence of period doubling bifurcations. This leads to the presence of chaotic dynamics through Shil'nikov bifurcation of a homoclinic orbit. Also, we measure the size of the chaotic attractor which is created in our system.