Chaos detection of Duffing system with fractional-order derivative by Melnikov method
Chaos detection of Duffing system with fractional-order derivative by Melnikov method
复制标题
分数阶导数Duffing系统的Melnikov法混沌检测
DOI：
10.1063/1.5124367
复制
发表时间：
2019-12-01
期刊：
影响因子：
2.9
通讯作者：
Yang, Shaopu
中科院分区：
文献类型：
--
作者：
Niu, Jiangchuan;Liu, Ruyu;Yang, Shaopu
利用Melnikov方法研究了具有分数阶导数的Duffing系统在外部简谐激励下的混沌检测问题。为了将Melnikov方法应用于含分数阶导数的Duffing系统的混沌检测，利用谐波平衡法将其转化为一阶近似等效的整数阶系统，使其具有与原系统相同的稳态幅频响应方程。通过数值解比较了分数阶导数Duffing系统与其一阶近似等效整数阶系统的幅频响应，两者吻合较好。然后，利用Melnikov函数得到了具有分数阶导数的Duffing系统的解析混沌判据。详细分析了具有分数阶导数和整数阶导数的Duffing系统的分岔和混沌现象，并通过分岔分析和相图验证了Melnikov函数得出的混沌判据。分析结果表明，Melnikov方法通过将分数阶导数的Duffing系统等效为整数阶系统，可以有效地检测出其混沌。由AIP Publishing授权出版。
The chaos detection of the Duffing system with the fractional-order derivative subjected to external harmonic excitation is investigated by the Melnikov method. In order to apply the Melnikov method to detect the chaos of the Duffing system with the fractional-order derivative, it is transformed into the first-order approximate equivalent integer-order system via the harmonic balance method, which has the same steady-state amplitude-frequency response equation with the original system. Also, the amplitude-frequency response of the Duffing system with the fractional-order derivative and its first-order approximate equivalent integer-order system are compared by the numerical solutions, and they are in good agreement. Then, the analytical chaos criterion of the Duffing system with the fractional-order derivative is obtained by the Melnikov function. The bifurcation and chaos of the Duffing system with the fractional-order derivative and an integer-order derivative are analyzed in detail, and the chaos criterion obtained by the Melnikov function is verified by using bifurcation analysis and phase portraits. The analysis results show that the Melnikov method is effective to detect the chaos in the Duffing system with the fractional-order derivative by transforming it into an equivalent integer-order system. Published under license by AIP Publishing.