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.
The dynamical properties of fractional-order Duffing–van der Pol oscillator are studied, and the amplitude–frequency response equation of primary resonance is obtained by the harmonic balance method. The stability condition for steady-state solution is obtained based on Lyapunov theory. The comparison of the approximate analytical results with the numerical results is fulfilled, and the approximations obtained are in good agreement with the numerical solutions. The bifurcations of primary resonance for system parameters are analyzed. The results show that the harmonic balance method is effective and convenient for solving this problem, and it provides a reference for the dynamical analysis of similar nonlinear systems.
The superharmonic resonance of fractional-order Mathieu–Duffing oscillator subjected to external harmonic excitation is investigated. Based on the Krylov–Bogolubov–Mitropolsky (KBM) asymptotic method, the approximate analytical solution for the third superharmonic resonance under parametric-forced joint resonance is obtained, where the unified expressions of the fractional-order term with fractional order from 0 to 2 are gained. The amplitude–frequency equation for steady-state solution and corresponding stability condition are also presented. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional-order term, excitation amplitudes, and nonlinear stiffness coefficient on the superharmonic resonance response of the system are analyzed in detail. The results show that the KBM method is effective to analyze dynamic response in a fractional-order Mathieu–Duffing system.
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