State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University, Shijiazhuang 050043, China.
School of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China.
Chaos. 2019 Dec;29(12):123106. doi: 10.1063/1.5124367.
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.
基于 Melnikov 方法研究了在外部谐波激励下具有分数阶导数的 Duffing 系统的混沌检测。为了将 Melnikov 方法应用于检测具有分数阶导数的 Duffing 系统的混沌，通过谐波平衡法将其转化为一阶近似等效整数阶系统，该系统具有与原始系统相同的稳态幅频响应方程。此外，通过数值解比较了分数阶导数的 Duffing 系统及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，通过 Melnikov 函数得到了分数阶导数的 Duffing 系统的解析混沌准则。详细分析了分数阶导数和整数阶导数的 Duffing 系统的分岔和混沌，通过分岔分析和相图验证了 Melnikov 函数得到的混沌准则。分析结果表明，通过将分数阶导数的 Duffing 系统转化为等效的整数阶系统，Melnikov 方法可以有效地检测其混沌。
