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.
Highlights
Many dynamical systems in various scientific fields are represented in terms of nonlinear ordinary differential equations which appear in a vast range of applications [1,2,3,4,5,6,7]
The nonlinear dynamics of resonant RLC series circuit has been described using a general class of mixed Rayleigh–Lienard oscillators [16]. is new class of oscillators contains the class of oscillators given by equation (1)
The nonlinear dynamics of this class of oscillators has been intensively studied, and interesting results such as perioddoubling leading to chaotic motion, strange attractors, reverse period-doubling bifurcation, symmetry breaking, antimonotonicity, existence of horseshoe chaos, and so on have been obtained [17,18,19,20,21,22,23,24,25]. In most of these studies, the Melnikov perturbation method [18, 24, 26, 27] has been widely used to detect chaotic dynamics and to analyze nearhomoclinic motion with deterministic or random perturbation. is method is today considered as a powerful analytical tool to provide an approximate criterion for the occurrence of hetero/homoclinic chaos in a wide class of dynamical systems. In these nonlinear dynamics works, the considered nonlinear oscillators are usually subjected to linear and nonlinear damping with constant coefficients. is damping combination is necessary since experiments showed a strong nonlinear dependence of damping with the vibration amplitude of nonlinear vibrating systems
Summary
Yelome Judicael Fernando Kpomahou ,1 Laurent Amoussou Hinvi, Joseph Adebiyi Adechinan, and Clement Hodevewan Miwadinou 4. Chaotic dynamics of a mixed Rayleigh–Lienard oscillator driven by parametric periodic damping and external excitations is investigated analytically and numerically. 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. It is found that for ω ], the chaotic region decreases and disappears when the amplitude of the parametric periodic damping excitation increases. Increasing of F1 and F0 provokes an erosion of the basin of attraction and a modification of the geometrical shape of the chaotic attractors. For ω ≠ ] and η 0.8, the fractality of the basin of attraction increases as the amplitude of the external periodic excitation and constant term increase. It is found that the system exhibits monostable and bistable oscillations as well as the phenomenon of coexistence of attractors
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