Dynamical response of Mathieu–Duffing oscillator with fractional-order delayed feedback

Highlights: • The analytical solution for Mathieu–Duffing oscillator with fractional-order delayed feedback is obtained. • The fractional-order delayed feedback has both the functions of delayed velocity feedback and delayed displacement feedback. • The special effects of time delay on nonzero periodic solutions are analyzed in detail. • The effects of the fractional-order parameters on system response are characterized. - Abstract: In this paper, the dynamical response of Mathieu–Duffing oscillator under fractional-order delayed feedback is investigated. At first, the approximate analytical solution and the amplitude-frequency equation are obtained based on the averaging method. The equivalent stiffness coefficient and equivalent damping coefficient are defined by the feedback coefficient, fractional order and time delay et al. The effects of feedback coefficient, fractional order and time delay on these two equivalent parameters are analyzed. It is found that the fractional-order delayed feedback has not only the function of delayed velocity feedback, but also the function of delayed displacement feedback. Then, the comparison of the amplitude-frequency curves obtained by the analytical and numerical solutions verifies the correctness and satisfactory precision of the approximate analytical solution. The effects of the parameters in the fractional-order delayed feedback on the complex dynamical behaviors of Mathieu–Duffing oscillator are studied. It could be found that fractional-order delayed feedback has important influences on the dynamical behavior of Mathieu–Duffing oscillator, and the results are very helpful to design, analyze or control in vibration engineering.