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.

Amplitude modulation leads to the disappearance of relaxation oscillations in the Duffing system

“…Moreover, amplitude modulation is also applied in studying chaotic phenomena and fractal structures in modulation [32,33]. By simulating the amplitude modulation processes in these systems, researchers can reveal key features of system dynamics, such as stability, chaotic behavior, and frequency response characteristics [34][35][36][37]. Therefore, amplitude modulation continues to play a significant role in the research of nonlinear dynamics.…”

Complex Rayleigh–van-der-Pol–Duffing Oscillators: Dynamics, Phase, Antiphase Synchronization, and Image Encryption

“…Consequently, the dynamics of these models are mathematically depicted using nonlinear ODEs [19][20][21][22][23]. Kpomahou et al [24] introduced the real Rayleigh-van-der-Pol-Duffing oscillator (RVDO) as: 2 )…”

Effects of periodic parametric damping and amplitude-modulated signal on vibrational resonance and torus-doubling bifurcations occurrence in an asymmetric mixed Rayleigh-Liénard oscillator

“…Nonlinear parametrically excited systems with the influence of a periodic external excitation constitute an important another class of dynamical systems which exhibit a rich and complicated dynamical behaviors. The dynamic study of this class of nonlinear dynamical systems has been intensively investigated [18][19][20][21][22][23][24]. However, the interaction between the periodic parametric damping and self excited vibrations of nonlinear systems for chaotic dynamics has received less research attention.…”