The study of some nonlinear dynamical systems modelled by a more general Rayleigh-Van Der Pol equation

Mircea LupuFlorin Isaia



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. In this paper, we study the mathematical model for nonlinear dynamical systems with distributed parameters given by a generalized Rayleigh-Van der Pol equation. In the autonomous case and in the non-autonomous case, conditions of stability, bifurcations, self-oscillations are studied using criteria of Liapunov, Bendixon, Hopf [11], [12]. Asymptotic and numerical methods are often used [5]. The equation has the form x¨ + ω 2x = α − βx2 − γx˙ 2  x˙ + f (t) , where resonance and limit cycles can be remarked [1]. Note that for β = 0, α 6= 0, γ 6= 0 we have the Rayleigh equation [1], while for γ = 0, α 6= 0, β 6= 0 we have the Van der Pol equation [2],[3]. Besides the theoretical study, the applications to techniques are very important: dynamical systems in the mechanics of vibrations, oscillations in electromagnetism and transistorized circuits [6], the aerodynamics of the flutter with two degrees of freedom [8], are modelled by this hybrid equation proposed by authors.

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Isaia, Florin, Lupu, Mircea