Vibrations of a nonlinear self-excited system driven by parametric and/or external excitations are studied in the paper. The model is composed of a self-excitation term represented by a nonlinear van der Pol function, periodically varied stiffness which represents parametric excitation and external harmonic force. Interactions between self and parametric or/and external force lead to complex behaviours observed by quasi-periodic or chaotic motions but under specific conditions, near resonance zones the response is harmonic. The transition from quasi-periodicity to periodic oscillations is caused by so called the frequency locking phenomenon which in fact corresponds to the second kind Hopf bifurcation (Neimark-Sacker bifurcation). The periodic resonances can be determined analytically, quasi-periodic oscillations however are investigated mainly numerically. The goal of this paper is to determine quasi-periodic dynamics and Hopf bifurcations analytically by using the multiple time scale method (MSM) in two steps: (1) to determine periodic solutions of the fast flow by the first order MSM and (2) to determine periodic solutions of the slow–flow by the second order MSM. The analytical solutions obtained in both scales allow determining bifurcation points of the system.

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