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

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.