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Abstract: This paper investigates the chaotic behavior of an extended Duffing Van der pol oscillator in a ϕ 6 potential under additive harmonic and bounded noise excitations for a specific parameter choice. From Melnikov theorem, we obtain the conditions for the existence of homoclinic or heteroclinic bifurcation in the case of the ϕ 6 potential is bounded, which are complemented by the numerical simulations from which we illustrate the bifurcation surfaces and the fractality of the basins of attraction. The results show that the threshold amplitude of bounded noise for onset of chaos will move upwards as the noise intensity increases, which is further validated by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the less possible chaotic domain in parameter space. The effect of bounded noise on Poincare maps is also investigated. [Copyright &y& Elsevier]

Abstract: The chaotic behavior of Duffing oscillator possessing both homoclinic and heteroclinic orbits and subjected to harmonic and bounded noise excitations is investigated. By means of the random Melnikov technique together with associated mean-square criterion, necessary conditions for onset of chaos resulting from homoclinic or heteroclinic bifurcation are derived semi-analytically. The results reveal that for larger noise intensity the threshold amplitude of bounded noise for onset of chaos will move upward as the noise intensity increases, which is further verified by the top Lyapunov exponents of the system. Thus the larger the noise intensity results in the less possible chaotic domain in parameter space. The effects of bounded noise on Poincare maps of the system responses are also discussed, together with the numerical simulation of the top Lyapunov exponents. [Copyright &y& Elsevier]

Abstract: Time delays are often sources of complex behavior in dynamic systems. Yet its complexity needs to be further explored, particularly when multiple time delays are present. As a purpose to gain insight into such complexity under multiple time delays, we investigate the mechanism for the action of multiple time delays on a particular non-autonomous system in this paper. The original mathematical model under consideration is a Duffing oscillator with harmonic excitation. A delayed system is obtained by adding delayed feedbacks to the original system. Two time delays are involved in such system, one of which in the displacement feedback and the other in the velocity feedback. The time delays are taken as adjustable parameters to study their effects on the dynamics of the system. Firstly, the stability of the trivial equilibrium of the linearized system is discussed and the condition under which the equilibrium loses its stability is obtained. This leads to a critical stability boundary where Hopf bifurcation or double Hopf bifurcation may occur. Then, the chaotic behavior of such system is investigated in detail. Particular emphasis is laid on the effect of delay difference between two time delays on the chaotic properties. A Melnikov’s analysis is employed to obtain the necessary condition for onset of chaos resulting from homoclinic bifurcation. And numerical analyses via the bifurcation diagram and the top Lyapunov exponent are carried out to show the actual time delay effect. Both the results obtained by the two analyses show that the delay difference between two time delays plays a very important role in inducing or suppressing chaos, so that it can be taken as a simple but efficient “switch” to control the motion of a system: either from order to chaos or from chaos to order. [Copyright &y& Elsevier]

Abstract: The chaotic behavior of a double-well Duffing oscillator with both delayed displacement and velocity feedbacks under a harmonic excitation is investigated. By means of the Melnikov technique, necessary condition for onset of chaos resulting from homoclinic bifurcation is derived analytically. The analytical results reveal that for negative feedback the presence of time delay lowers the threshold and enlarges the possible chaotic domain in parameter space; while for positive feedback the presence of time delay enhances the threshold and reduces the possible chaotic domain in parameter space, which are further verified numerically through Poincare maps of the original system. Furthermore, the effect of the control gain parameters on the chaotic motion of the original system is studied in detail. [Copyright &y& Elsevier]

Abstract: The chaotic behaviors of a particle in a triple well ϕ 6 potential possessing both homoclinic and heteroclinic orbits under harmonic and Gaussian white noise excitations are discussed in detail. Following Melnikov theory, conditions for the existence of transverse intersection on the surface of homoclinic or heteroclinic orbits for triple potential well case are derived, which are complemented by the numerical simulations from which we show the bifurcation surfaces and the fractality of the basins of attraction. The results reveal that the threshold amplitude of harmonic excitation for onset of chaos will move downwards as the noise intensity increases, which is further verified by the top Lyapunov exponents of the original system. Thus the larger the noise intensity results in the more possible chaotic domain in parameter space. The effect of noise on Poincare maps is also investigated. [Copyright &y& Elsevier]