Chaos detection of Duffing system with fractional-order derivative by Melnikov method.

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.