Chaotic Motions in Forced Mixed Rayleigh–Liénard Oscillator with External and Parametric Periodic-Excitations

Miwadinou, C. H.; Monwanou, A. V.; Koukpémèdji, A. A.; Kpomahou, Y. J. F.; Orou, J. B. Chabi

doi:10.1142/s0218127418300057

Cited by 13 publications

(14 citation statements)

References 16 publications

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“…). Substituting equation ( 17) into (18) and evaluating the integral by using the standard integral tables [63], we obtain the Melnikov function:…”

Section: Fixed Points and Melnikov Criterion For Homoclinicmentioning

confidence: 99%

“…In recent years, 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.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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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.

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“…). Substituting equation ( 17) into (18) and evaluating the integral by using the standard integral tables [63], we obtain the Melnikov function:…”

Section: Fixed Points and Melnikov Criterion For Homoclinicmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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“…If M(τ 0 ) � 0 and dM/dτ 0 ≠ 0 for some τ 0 and some sets of parameters, then horseshoes exist, and chaos occurs [21][22][23][24][25]. Using this Melnikov criterion for the appearance of the intersection between the perturbed and unperturbed separatrixes, it is found that chaos appears when the following condition is satisfied:…”

Section: Horseshoe Chaosmentioning

confidence: 99%

“…e object of chaos theory is the study of nonlinear phenomena governed by simple and deterministic laws whose behavior under certain conditions becomes unpredictable. Since its discovery in the 20th century, chaos has been one of the most interesting for dynamic systems in areas such as physics, mathematics, chemistry, biochemistry, economics and finance, epidemiology, and engineering [19][20][21][22][23][24][25]. Depending on the field of study, it is sometimes useful or undesirable to the point where many researchers are interested in its prediction and/or control.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Dynamics of the Quadratic-Damping Helmholtz Oscillator

et al. 2020

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In this paper, the Helmholtz equation with quadratic damping themes is used for modeling the dynamics of a simple prey-predator system also called a simple Lotka–Volterra system. From the Helmholtz equation with quadratic damping themes obtained after modeling, the equilibrium points have been found, and their stability has been analyzed. Subsequently, the harmonic oscillations have been studied by the harmonic balance method, and the phenomena of resonance and hysteresis are observed. The primary and secondary resonances have been researched by the multiple-scale method, and the conditions of stability of the amplitudes of oscillations are determined. Chaos is detected analytically by the Melnikov method and numerically using the basin of attraction, the bifurcation diagram, the Lyapunov exponent, the phase portrait, and the Poincaré section. The effects of all the parameters of the system are analyzed in detail, and special emphasis is placed on the new parameters. Through this analysis, the complex phenomena such as hysteresis, bistability, amplitude jump, resonances, and chaos have been obtained. The control of the parameters and the necessary conditions to control the aforementioned phenomena have been found.

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“…For instance, nonequilibrium phenomena such as oscillations, bistability, complex oscillations, and quasichaotic behavior of the reaction are revealed by these studies. One of the main challenges has been to predict and to control these phenomena in nonlinear chemical oscillations for potential applications (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15]). e study of these oscillations has been made with a periodically external excitation or a parametric excitation [11,12].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Dynamics in a Chemical Reaction under an Amplitude-Modulated Excitation: Hysteresis, Vibrational Resonance, Multistability, and Chaos

et al. 2020

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This paper deals with the effects of an amplitude-modulated (AM) excitation on the nonlinear dynamics of reactions between four molecules. The computation of the ﬁxed points of the autonomous nonlinear chemical system has been made in detail using the Cardan’s method. Hopf bifurcation has been also successfully checked. Routes to chaos have been investigated through bifurcations structures, Lyapunov exponent, phase portraits, and Poincaré section. The effects of the control force on chaotic motions have been strongly analyzed, and the control efficiency is found in the cases g=0 (unmodulated case) and g≠0 with Ω=ω and Ω/w≠p/q; p and q are simple positive integers. Vibrational resonance (VR), hysteresis, and coexistence of several attractors have been studied in detail based on the relationship between the frequencies of the AM force. Results of analytical investigations are validated and complemented by numerical simulations.

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Chaotic Motions in Forced Mixed Rayleigh–Liénard Oscillator with External and Parametric Periodic-Excitations

Cited by 13 publications

References 16 publications

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

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

Nonlinear Dynamics of the Quadratic-Damping Helmholtz Oscillator

Nonlinear Dynamics in a Chemical Reaction under an Amplitude-Modulated Excitation: Hysteresis, Vibrational Resonance, Multistability, and Chaos

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