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The dynamic characteristics of the 3:1 super-harmonic resonance response of the Duffing oscillator with the fractional derivative are studied. Firstly, the approximate solution of the amplitude-frequency response of the system is obtained by using the periodic characteristic of the response. Secondly, a set of critical parameters for the qualitative change of amplitude-frequency response of the system is derived according to the singularity theory and the two types of the responses are obtained. Finally, the components of the 1X and 3X frequencies of the system’s time history are extracted by the spectrum analysis, and then the correctness of the theoretical analysis is verified by comparing them with the approximate solution. It is found that the amplitude-frequency responses of the system can be changed essentially by changing the order and coefficient of the fractional derivative. The method used in this paper can be used to design a fractional order controller for adjusting the amplitude-frequency response of the fractional dynamical system.

The modulation of response caused by the fractional derivative in the Duffing system under super-harmonic resonance