Abstract
The heteroclinic bifurcation and chaos of a Duffing oscillator with forcing excitation under both delayed displacement feedback and delayed velocity feedback are studied by Melnikov method. The Melnikov function is analytically established to detect the necessary conditions for generating chaos. Through the analysis of the analytical necessary conditions, we find that the influences of the delayed displacement feedback and delayed velocity feedback are separable. Then the influences of the displacement and velocity feedback parameters on heteroclinic bifurcation and threshold value of chaotic motion are investigated individually. In order to verify the correctness of the analytical conditions, the Duffing oscillator is also investigated by numerical iterative method. The bifurcation curves and the largest Lyapunov exponents are provided and compared. From the analysis of the numerical simulation results, it could be found that two types of period-doubling bifurcations occur in the Duffing oscillator, so that there are two paths leading to the chaos in this oscillator. The typical dynamical responses, including time histories, phase portraits, and Poincare maps, are all carried out to verify the conclusions. The results reveal some new phenomena, which is useful to design or control this kind of system.
Highlights
It is well known that many kinds of nonlinearities exist in engineering systems, such as parametric excitation, nonsmoothness, time delay, discontinuity, and large deformation [1,2,3,4,5,6]
Amer et al [16] investigated the Duffing oscillator with parametric excitation under timedelay feedback based on the multiple scales perturbation method and analyzed the influences of the system parameters
Through all the above analysis, it could be found that tendencies of the analytical solutions for the influences of all the delayed feedback parameters are consistent with the numerical iterative simulations
Summary
It is well known that many kinds of nonlinearities exist in engineering systems, such as parametric excitation, nonsmoothness, time delay, discontinuity, and large deformation [1,2,3,4,5,6]. The complicated Duffing nonlinear system with time delay is possible to generate more complex bifurcation and chaotic dynamic phenomena [11,12,13,14]. Yang and Sun et al [27, 28] investigated the necessary condition for the generating chaos of a double-well Duffing oscillator with bounded noise excitations and time-delay feedback by Melnikov theory. The firstorder exact analytical solution of the necessary condition for generating chaos in sense of Smale horseshoes in a Duffing oscillator with both delayed displacement feedback and delayed velocity feedback is obtained based on Melnikov theory.
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