On self-excited auto-parametric systems

We consider a Rayleigh type of self-excited auto-parametric system. We study the semi-trivial solution and its domain of instability where non-trivial solutions are initiated. We are interested in the existence and stability of the non-trivial solutions and we analyze the behaviour of the solutions by examining it for various values of some parameters. We divide the discussion on the non-trivial solution in exact resonance and near resonance cases. In the analysis we use both normal forms (or averaging) and numerical bifurcation path-following techniques. The system displays a rich pattern of different bifurcations, a robust heteroclinic cycle and instability behaviour.