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

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Journal of Vibration and Shock ›› 2021, Vol. 40 ›› Issue (6) : 33-40.

QIN Hao1，WEN Shaofang1，SHEN Yongjun2，XING Haijun2，WANG Jun2

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Abstract

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.

Key words

Melnikov method / fractional-order derivative / Duffing oscillator / heteroclinic orbit

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QIN Hao1，WEN Shaofang1，SHEN Yongjun2，XING Haijun2，WANG Jun2. Analytical study on the chaos threshold of a Duffing oscillator with a fractional-order derivative term by the Melnikov method[J]. Journal of Vibration and Shock, 2021, 40(6): 33-40

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