Widely separated frequencies in coupled oscillators with energy-preserving quadratic nonlinearity

In this paper we present an analysis of a system of coupled oscillators suggested by atmospheric dynamics. We make two assumptions for our system. The first assumption is that the frequencies of the characteristic oscillations are widely separated and the second is that the nonlinear part of the vector field preserves the distance to the origin. Using the first assumption, we prove that the reduced normal form of our system has an invariant manifold which exists for all values of the parameters. This invariant manifold cannot be perturbed away by including higher order terms in the normal form. Using the second assumption, we view the normal form as an energy-preserving three-dimensional system which is linearly perturbed. Restricting ourselves to a small perturbation, the flow of the energy-preserving system is used to study the flow in general. We present a complete study of the flow of the energy-preserving system and its bifurcations. Using these results, we return to the dissipative system and provide the condition for having a Hopf bifurcation of one of the two equilibria of the perturbed system. We also numerically follow the periodic solution created via the Hopf bifurcation and find a sequence of period-doubling and fold bifurcations, also a torus (or Neimark–Sacker) bifurcation.