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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 and coupled oscillators for ordinary differential equations
34D06 Synchronization of solutions to ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
70K28 Parametric resonances for nonlinear problems in mechanics
70K40 Forced motions for nonlinear problems in mechanics
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