Nonlinear dynamics in the study of a hybrid system of Rayleigh-Van der Pol type

Abstract

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, as well as in the non-autonomous case, conditions for stability, bifurcations, and self-oscillations are studied using some criteria of Lyapunov, Bendixon, Hopf. Asymptotic and numerical methods are often used. The equation has the form
\[
\ddot{x}+\omega^2x=\left(\alpha-\beta x^2-\gamma\dot{x}^2\right)\dot{x}+f\left(t\right),
\]
where resonance and limit cycles can be noticed. Note that for $\beta=0$, $\alpha\neq0$, $\gamma\neq0$ we have the Rayleigh equation, while for $\gamma=0$, $\alpha\neq0$, $\beta\neq0$ we have the Van der Pol equation. Besides the theoretical study, the applications to techniques are very important: dynamical systems in the mechanics of vibrations, oscillations in electromagnetism and transistorized circuits, aerodynamics of the flutter with two degrees of freedom, are modelled by this hybrid equation that we propose.