Chaotic Dynamics of a Mixed Rayleigh–Liénard Oscillator Driven by Parametric Periodic Damping and External Excitations

Abstract

In this paper, chaotic dynamics of a mixed Rayleigh–Liénard oscillator driven by parametric periodic damping and external excitations is investigated analytically and numerically. The equilibrium points and their stability evolutions are analytically analyzed, and the transitions of dynamical behaviors are explored in detail. Furthermore, from the Melnikov method, the analytical criterion for the appearance of the homoclinic chaos is derived. Analytical prediction is tested against numerical simulations based on the basin of attraction of initial conditions. As a result, it is found that for ω = ν , the chaotic region decreases and disappears when the amplitude of the parametric periodic damping excitation increases. Moreover, increasing of F 1 and F 0 provokes an erosion of the basin of attraction and a modification of the geometrical shape of the chaotic attractors. For ω ≠ ν and η = 0.8 , the fractality of the basin of attraction increases as the amplitude of the external periodic excitation and constant term increase. Bifurcation structures of our system are performed through the fourth-order Runge–Kutta ode 45 algorithm. It is found that the system displays a remarkable route to chaos. It is also found that the system exhibits monostable and bistable oscillations as well as the phenomenon of coexistence of attractors.

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

“…is an advanced version of the one used to study chaotic dynamics, bursting oscillations and mixed-mode oscillations [25,26]. From this equation, the asymmetric potential function can be written as follows:…”

“…Then, the effect of time-varying periodic damping of a nonlinear system under the influence of external excitation on the dynamical response becomes an interesting research topic. In this perspective, bursting and mixed-mode oscillations, coexisting attractors and homoclinic bifurcation have been investigated in a mixed Rayleigh-Liénard oscillator under periodic parametric damping and external excitations [25,26]. Recently, regular, chaotic behaviors and coexisting attractors have been investigated in a new nonlinear dissipative chemical oscillator subjected to periodic parametric damping and external excitations [27,28].…”

Two fractional order Langevin equation with new chaotic dynamics

“…Example 5 (see [24]). We examine the Mixed Rayleigh Lienard Oscillator Driven by Parametric Periodic Pimping and External Excitation given by: D α D β − α 1 + η cos vt y(t) =ω 2 0 (F 0 + F 1 cos ωt) − β 0 (D β y(t)) 2 − β 1 (D β y(t)) 3 + ω 2 0 y(t) − γy(t) 3 .…”

Extreme bursting events via pulse-shaped explosion in mixed Rayleigh-Lienard nonlinear oscillator

“…The class of mixed Rayleigh-Liénard system is widespread in the areas of science, and engineering. The dynamics of this hybrid system have been intensively studied in the literature, and exciting results such as period-doubling bifurcation leading to chaotic dynamics, strange attractors, symmetry breaking, and so on have been reported using various excitation models [81,82,83,84]. Also, the quadratic nonlinear damping term αx ẋ appeared in many real-world models and was found to play a crucial role mostly in microelectromechanical, and nanoelectromechanical oscillators [85,86], fluid mechanics [87], and have implications in mass and force sensing applications [88,89], mechanical noise squeezing in laser cooling technology [90].…”