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Nonlinear dynamics of a periodically driven Duffing resonator coupled to a van der Pol oscillator. (English) Zbl 1250.34032
Summary: We explore the dynamics of a periodically driven Duffing resonator coupled elastically to a van der Pol oscillator in the case of $1:1$ internal resonance in the cases of weak and strong coupling. Whilst strong coupling leads to dominating synchronization, the weak coupling case leads to a multitude of complex behaviours. A two-time scales method is used to obtain the frequency-amplitude modulation. The internal resonance leads to an antiresonance response of the Duffing resonator and a stagnant response (a small shoulder in the curve) of the van der Pol oscillator. The stability of the dynamic motions is also analyzed. The coupled system shows a hysteretic response pattern and symmetry-breaking facets. Chaotic behaviour of the coupled system is also observed and the dependence of the system dynamics on the parameters are also studied using bifurcation analysis.
##### MSC:
 34C15 Nonlinear oscillations, coupled oscillators (ODE) 34D06 Synchronization 34C28 Complex behavior, chaotic systems (ODE) 70K28 Parametric resonances (nonlinear dynamics) 70K40 Forced nonlinear motions (general mechanics)