Influence of Amplitude-Modulated Force and Nonlinear Dissipation on Chaotic Motions in a Parametrically Excited Hybrid Rayleigh–Van der Pol–Duffing Oscillator

Abstract

The generation and evolution of chaotic motions in a hybrid Rayleigh–Van der Pol–Duffing oscillator driven by parametric and amplitude-modulated excitation forces are investigated analytically and numerically. By using the Melnikov method, the conditions for the appearance of horseshoe chaos in our system are derived in the case where the modulation frequency [Formula: see text] and the forcing frequency [Formula: see text] are the same [Formula: see text]. The obtained results show that the chaotic region decreases and increases in certain ranges of frequency. The numerical simulations based on the basin of attraction of initial conditions validate the obtained analytical predictions. It is also found that in the case where [Formula: see text] is irrational, the increase of amplitude-modulated force accentuates the fractality of the basin of attraction. The global dynamical changes of our model are numerically examined. It is found that our model displays a rich variety of bifurcation phenomena and remarkable routes to chaos. In addition, the presence of the hybrid Rayleigh–Van der Pol damping force reduces the chaotic domain in the absence of amplitude-modulated force. But when the amplitude-modulated force acts on the system, the chaotic oscillations decrease and disappear. Further, the geometric shape of the chaotic attractors considerably decreases in the presence of the amplitude-modulated excitation force. On the other hand, the system presents transient chaos, torus-chaos and torus of different topologies when [Formula: see text] is irrational.