Chaotic dynamics of a Rayleigh-Duffing oscillator with periodically external and parametric excitations*

Chaotic motions of a Rayleigh-Duffing oscillator with periodically external and parametric excitations are investigated rigorously. Chaos arising from intersections of homoclinic orbits is analyzed with the Melnikov method. The critical curves separating the chaotic and non-chaotic regions are obtained. The chaotic feature on the system parameters is discussed. Chaotic dynamics are also compared for the systems with a periodically external excitation or a parametric excitation. Some new dynamical phenomena including”controllable frequency” are presented. Numerical simulations verify the analytical results. Introduction The Rayleigh-Duffing oscillator models are widely used in physics, engineering, electronics, and many other disciplines. The nonlinear dynamics for this class of oscillators has been investigated in recent years. Using numerical and analytical approaches, nonlinear dynamics of a non-ideal Duffing-Rayleigh oscillator is studied by Felix et al [1]. Kanai and Yabuno [2] investigated creation-annihilation process of limit cycles in the Rayleigh-Duffing oscillator with negative linear damping and negative linear stiffness by the perturbation method. Qriouet and Mira [3] studied bifurcation structures related to families of fractional harmonics solutions generated by the Duffing-Rayleigh equation with a non-symmetrical periodic external force. With path integration based on the Gauss-Legendre integration scheme, Xie et al [4, 5] studied a Duffing-Rayleigh oscillator subject to harmonic and stochastic excitations. By using the asymptotic perturbation method, Siewe et al [6] studied the principal parametric resonance of a Rayleigh-Duffing oscillator with time-delayed feedback position and linear velocity terms. Mihara and Kawakami [7] studied synchronization and chaos of coupled Duffing-Rayleigh oscillators. Ma et al [8] investigated the synchronization of self-sustained Rayleigh-Duffing oscillator by the synchronization criteria based on Lyapunov direct method and the stability theory of linear time-varied systems. With the Melnikov method and Numerical methods, effects of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh-Duffing oscillator were investigated by Siewe et al [9]. Using the Melnikov method, Siewe et al [10] investigated the chaotic behavior of the Rayleigh-Duffing oscillator under a harmonic external excitation. Using the Melnikov method, Zhang and Luo [11] studied chaos of Rayleigh-Duffing like system. They also investigated the synchronization of two fractional Rayleigh Duffing-like systems with active control technology. With the Melnikov method, the effect of nonlinear dissipation on the basin boundaries of a driven two-well modified Rayleigh-Duffing oscillator was studied by Miwadinou et al [12]. With stochastic averaging method, Zhang et al [13] investigated the response of a Duffing-Rayleigh system with a fractional derivative under Gaussian white noise excitation. The effects of different system parameters and noise intensity on the response of the system are also discussed there. By using the composite cell coordinate system method, Yue et al [14] studied the global bifurcations including the crisis and metamorphosis of the Rayleigh-Duffing oscillator. In this paper, chaotic motions of the Rayleigh-Duffing oscillator with periodically external and parametric excitations are studied analytically with the Melnikov method. The critical curves 6th International Conference on Mechatronics, Materials, Biotechnology and Environment (ICMMBE 2016) © 2016. The authors Published by Atlantis Press 286 separating the chaotic and non-chaotic regions are plotted. The chaotic feature on the system parameters is discussed in detail and some new dynamical phenomena are presented. The phase portraits and Poincaré sections are numerically computed, which verify the analytical results. Formulation of the problem Consider the Rayleigh-Duffing oscillator with periodically external and parametric excitations 2 3 (1 ) (1 )cos x x x x x f x t μ α β ω − − − + = +    (1) where μ , α and β are nonlinear damping, linear and nonlinear restoring parameters, f and ω are the amplitude and frequency of the excitation, respectively. Assuming the damping μ and excitation amplitude f are small, settingμ εμ = , f f ε = , where ε is a small parameter, then Eq.(1) can be written as 3 2 (1 ) (1 )cos x y y x x x x f x t α β εμ ε ω =   = − + − + +  