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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:
34C15Nonlinear oscillations, coupled oscillators (ODE)
34D06Synchronization
34C28Complex behavior, chaotic systems (ODE)
70K28Parametric resonances (nonlinear dynamics)
70K40Forced nonlinear motions (general mechanics)