Nonlinear dynamics of system oscillations modeled by a forced Van der Pol generalized oscillator

TL;DR

This paper investigates the nonlinear oscillations of a forced Van der Pol generalized oscillator, analyzing amplitude responses, resonance phenomena, hysteresis, jump phenomena, and bifurcations using analytical and numerical methods.

Contribution

It introduces a detailed analysis of a generalized Van der Pol oscillator with forcing, including amplitude responses and bifurcation behavior, using harmonic balance, multiple scales, and numerical simulations.

Findings

Identification of amplitude responses for harmonic, superharmonic, and subharmonic oscillations.

Observation of hysteresis and jump phenomena in the system.

Bifurcation sequences characterized for different oscillatory states.

Abstract

This paper considers the oscillations modeled by a forced Van der Pol generalized oscillator. These oscillations are described by a nonlinear differential equation of the form $ \ddot{x}+x-\varepsilon\left(1-ax^2-b\dot{x}^2\right)\dot{x}=E\sin{{\Omega}t}.$ The amplitudes of the forced harmonic, primary resonance superharmonic and subharmonic oscillatory states are obtained using the harmonic balance technique and the multiple time scales methods. We obtain also the hysteresis and jump phenomena in the system oscillations. Bifurcation sequences displayed by the model for each type of oscillatory states are performed numerically through the fourth-order Runge- Kutta scheme.