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Effect of vertical high-frequency parametric excitation on self-excited motion in a delayed van der Pol oscillator. (English) Zbl 1142.34332
Summary: We investigate the interaction effect of fast vertical parametric excitation and time delay on self-oscillation in a van der Pol oscillator. We use the method of direct partition of motion to derive the main autonomous equation governing the slow dynamic and then we apply the averaging technique on this slow dynamic to derive a slow flow. In particular we analyze the slow flow to analytically approximate regions where self-excited vibrations can be eliminated. Numerical integration is performed and compared to the analytical results showing a good agreement for small time delay. It was shown that vertical parametric excitation, in the presence of delay, can suppress self-excited vibrations. These vibrations, however, persist for all values of the excitation frequency in the case of a fast vertical parametric excitation without delay [R. Bourkha, M. Belhaq, Chaos Solitons Fractals 34, No. 2, 621–627 (2007; Zbl 1167.70012)].

MSC:
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
70K99 Nonlinear dynamics in mechanics
Software:
dde23
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References:
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