Eur. Phys. J. Spec. Top. (2022) 231: 2183-2197
https://doi.org/10.1140/epjs/s11734-021-00369-6

Regular Article

Chaos threshold analysis of Duffing oscillator with fractional-order delayed feedback control

¹ State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University, 050043, Shijiazhuang, China
² School of Mechanical Engineering, Shijiazhuang Tiedao University, 050043, Shijiazhuang, China

^a wsf39811@163.com

Received: 19 May 2021
Accepted: 7 December 2021
Published online: 20 January 2022

Abstract

In this paper, the bifurcation and chaotic threshold of Duffing oscillator with fractional-order delayed feedback control is studied. The fractional-order delayed feedback control is equivalent to the approximate integer-order control. It is found that the fractional-order delayed feedback control has the function of displacement feedback and velocity feedback. Then, the analytically necessary condition for generating chaos in Duffing oscillator with fractional-order delayed feedback control is obtained by Melnikov method. The accuracy of the analytically necessary condition by Melnikov method is verified by numerical simulation and the largest Lyapunov exponents of the system. From the analysis of the numerical simulation results, it is found that there are two paths leading to the chaos after period-doubling bifurcations due to different initial values in Duffing oscillator with fractional-order delayed feedback. Finally, the influence of the parameters of the fractional-order delayed feedback control on bifurcation and chaos is analyzed. The increase of the fractional-order delayed feedback gain will resist the generation of chaos. Both time delay and the fractional-order affect the threshold of chaos in the form of trigonometric functions. The better control performance in the system can be obtained by choosing the reasonable fractional order and time delay. Those results present some new system characteristics which provide theoretic guidance to design and control of this kind system.