Bifurcations of strongly non-linear self-excited oscillations

Abstract

Perturbation of a single-degree-of-freedom conservative oscillator leads to the emergence and vanishing of periodic solutions and to various types of self-excited oscillations. Using techniques from dynamical systems theory, in particular a certain Poincaré map, we establish the presence of Hopf bifurcations, various types of homoclinic bifurcations and saddle-node bifurcations of the associated Poincaré map. The corresponding bifurcation sets in parameter space are computed explicitly by perturbation methods. The theory is applied to the generalized van der Pol and the generalized Rayleigh oscillator, and to the case of a non-linear spring attached to a conveyor belt.