Abstract
Abstract In this paper, the harmonic balance method and its variants are used to analyze the response of Mathieu–Duffing oscillator with Caputo derivative. First, the exact and approximate expressions of the Caputo derivatives of trigonometric function and composite function are derived. Next, using the approximate expression of the Caputo derivative of the composite function, the resonance of Duffing oscillator with Caputo derivative is analyzed by the harmonic balance method. Finally, Mathieu–Duffing oscillator with Caputo derivative is approximated by three kinds of methods, i.e., the harmonic balance method, the residue harmonic balance method and the improved harmonic balance method. The corresponding numerical simulations are given to illustrate the performance of these methods as well. The results show that the residue harmonic balance method is more precise than the harmonic balance method and the improved harmonic balance method in analyzing the dynamic response of Mathieu–Duffing oscillator with Caputo derivative.
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