Resonance analysis of fractional-order Mathieu oscillator

The resonant behavior of fractional-order Mathieu oscillator subjected to external harmonic excitation is investigated. Based on the harmonic balance (HB) method, the first-order approximate analytical solutions for primary resonance and parametric-forced joint resonance are obtained, and the higher-order approximate steady-state solution for parametric-forced joint resonance is also obtained, where the unified forms of the fractional-order term with fractional order between 0 and 2 are achieved. The correctness of the approximate analytical results is verified by numerical results. The effects of the fractional order and parametric excitation frequency on the resonance response of the system are analyzed in detail. The results show that the HB method is effective to analyze dynamic response in a fractional-order Mathieu system.