&nbsp;一类分数阶分段Duffing振子的混沌研究

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振动与冲击 ›› 2022, Vol. 41 ›› Issue (13) : 8-16.

论文

一类分数阶分段Duffing振子的混沌研究

王军1，申永军1，张建超1，王晓娜2

作者信息 +

Chaos of a class of piecewise Duffing oscillator with fractional-order derivative term

WANG Jun1, SHEN Yongjun1, ZHANG Jianchao1, WANG Xiaona2

Author information +

文章历史 +

摘要

研究了谐波激励下含有分数阶微分项的分段Duffing振子的混沌运动，分数阶微分项采用Caputo定义进行计算，并利用等效刚度和等效阻尼的概念对其进行处理。运用Melnikov方法，建立了Smale马蹄意义下混沌运动的必要条件，得到了系统发生混沌运动的临界条件，并进行了解析解和数值解的比较，结果证明了解析必要条件的正确性，最后通过数值模拟，研究了系统线性刚度系数、阻尼系数、分数阶阶次、分数阶系数以及分段Duffing刚度系数对系统混沌运动的影响。
关键词：Melnikov方法；分数阶微分；分段Duffing 振子；混沌

Abstract

This paper presents an investigation of the chaos in a piecewise Duffing oscillator with fractional-order derivative under harmonic excitation. The Caputo definition is used to calculate the fractional-order derivative, and the concepts of equivalent stiffness and equivalent damping are used to process the fractional-order derivative. Based on the Melnikov method, the analytically necessary condition for the chaos in the sense of Smale horseshoes is established and then the chaotic critical condition curve is obtained. The comparison between the analytical solution and the numerical solution is investigated, and the results verify the correctness of the analytically necessary condition. Finally, the influence of linear stiffness coefficient, damping coefficient, fractional-order order, fractional-order coefficient and piecewise Duffing stiffness coefficient on the necessary conditions for chaos are studied by numerical simulation.
Key words：Melnikov method; fractional-order derivative; piecewise Duffing oscillator; chaos

导出引用

王军1，申永军1，张建超1，王晓娜2. 一类分数阶分段Duffing振子的混沌研究[J]. 振动与冲击, 2022, 41(13): 8-16

WANG Jun1, SHEN Yongjun1, ZHANG Jianchao1, WANG Xiaona2. Chaos of a class of piecewise Duffing oscillator with fractional-order derivative term[J]. Journal of Vibration and Shock, 2022, 41(13): 8-16

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