Fractional-order harmonic resonance in a multi-frequency excited fractional Duffing oscillator with distributed time delay

摘要

The general investigation on the vibration mechanism at arbitrary fraction-order harmonic resonance of the fractional Mathieu-Duffing system driven by multiple periodic excita-tions with distributed delay is reported theoretically and numerically. In terms of numer-ical scheme, a precise numerical algorithm based on the definition of Caputo derivative is applied. The unified analytical results of the steady-state response amplitude at each resonant frequency are obtained by utilizing the method of direct partition of motions, the self-consistent technique and the Krylov-Bogolubov-Mitropolsky asymptotic method. It relies on two kinds of parametric-forced pattern. It is found that for some parametric setting, both the amplitude of the external high-frequency excitation and the frequency of the external low-frequency excitation can give rise to saddle-node bifurcation of the steady-state amplitude in the analytical results. It is equivalent to amplitude jump phe-nomenon in the numerical observation. While comparing to the integer-order system, it is more likely to happen in the fractional-order situation. With the variant of the frac-tional order of the derivative, the steady-state amplitude solutions can be converted from monostability to bistability, then to monostability, and finally to the bistability again. This new multi-saddle-node bifurcation has not yet been reported before. The novel transcrit-ical bifurcation induced by the strength of the distributed delay is discussed and verified in detail for the first time.(c) 2021 Elsevier B.V. All rights reserved.