Dynamical analysis of Duffing oscillator with fractional-order feedback with time delay

Abstract

With increasingly strict requirements for control speed and system performance, the unavoidable time delay becomes a serious problem. Fractional-order feedback is constantly adopted in control engineering due to its advantages, such as robustness, strong de-noising ability and better control performance. In this paper, the dynamical characteristics of an autonomous Duffing oscillator under fractional-order feedback coupling with time delay are investigated. At first, the first-order approximate analytical solution is obtained by the averaging method. The equivalent stiffness and equivalent damping coefficients are defined by the feedback coefficient, fractional order and time delay. It is found that the fractional-order feedback coupling with time delay has the functions of both delayed velocity feedback and delayed displacement feedback simultaneously. Then, the comparison between the analytical solution and the numerical one verifies the correctness and satisfactory precision of the approximately analytical solution under three parameter conditions respectively. The effects of the feedback coefficient, fractional order and nonlinear stiffness coefficient on the complex dynamical behaviors are analyzed, including the locations of bifurcation points, the stabilities of the periodic solutions, the existence ranges of the periodic solutions, the stability of zero solution and the stability switch times. It is found that the increase of fractional order could make the delay-amplitude curves of periodic solutions shift rightwards, but the stabilities of the periodic solutions and the stability switch times of zero solution cannot be changed. The decrease of the feedback coefficient makes the amplitudes and ranges of the periodic solutions become larger, and induces the stability switch times of zero solution to decrease, but the stabilities of the periodic solutions keep unchanged. The sign of the nonlinear stiffness coefficient determines the stabilities and the bending directions of delay-amplitude curves of periodic solutions, but the bifurcation points, the stability of zero solution and the stability switch times are not changed. It could be concluded that the primary system parameters have important influences on the dynamical behavior of Duffing oscillator, and the results are very helpful to design, analyze or control this kind of system. The analysis procedure and conclusions could provide a reference for the study on the similar fractional-order dynamic systems with time delays.