Dynamical analysis of fractional-order nonlinear oscillator by incremental harmonic balance method

Abstract

In this paper, the incremental harmonic balance (IHB) method is extended to analyze the dynamical properties of fractional-order nonlinear oscillator, and the general forms of the periodic solutions for the oscillator are founded based on IHB method, which is useful to obtain the solutions with higher precision. The fractional-order Duffing oscillator is selected at first to test the feasibility of IHB method for this kind of system. The comparisons between the numerical results with the approximate analytical solutions by IHB method and averaging method verify the correctness and higher precision of IHB method. Then the nonlinear dynamics of fractional-order Mathieu–Duffing oscillator is researched by IHB method, where the coupling effects of the two kinds of typical nonlinearities are especially analyzed. Moreover, the influences of the system parameters in the fractional-order derivative on the amplitude–frequency curves are also researched by IHB method. At last, the detailed results are summarized and the conclusions are made, which presents some useful information to analyze and/or control the dynamical response of this kind of system.