Analytical study on the chaos threshold of a Duffing oscillator with a fractional-order derivative term by the Melnikov method

被引：0

作者：

Qin H. ^{[1
]}

Wen S. ^{[1
]}

Shen Y. ^{[2
]}

Xing H. ^{[2
]}

Wang J. ^{[2
]}

机构：

[1] State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Department of Traffic and Transportation, Shijiazhuang Tiedao University, Shijiazhuang

[2] Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang

来源：

Zhendong yu Chongji/Journal of Vibration and Shock | 2021年 / 40卷 / 06期

关键词：

Duffing oscillator; Fractional-order derivative; Heteroclinic orbit; Melnikov method;

D O I：

10.13465/j.cnki.jvs.2021.06.004

中图分类号：

学科分类号：

摘要：

The bifurcations and chaos of a Duffing oscillator with a fractional-order derivative term was studied. The equivalent stiffness and equivalent damping were used to deal with the fractional-order derivative, where the derivative term was made equivalent to a term in the form of trigonometric function and exponential function. Then, the Melnikov method was used to analyze the necessary conditions for the bifurcation and chaos generation of the fractional-order Duffing oscillator. The approximate analytical solution of the fractional-order Duffing oscillator was obtained. Finally, the comparison between the analytical solution and the numerical solution was investigated, and the accuracy of the analytical result was proved. The influences of fractional order and coefficient of the fractional-order derivative on the necessary condition of chaos were studied by simulation. Additionally, it is found that there is a bistability characteristic in the fractional-order Duffing oscillator. Starting from the two steady-state solutions, the system can reach the chaos state through the period-doubling bifurcation with the change of external excitation parameter f, which was then confirmed by analyzing its dynamic response. © 2021, Editorial Office of Journal of Vibration and Shock. All right reserved.

引用

页码：33 / 40and134

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