Effect of parametric excitation on a bifractional-order damped system with a fractional-power nonlinearity.

Investigation on linear/nonlinear resonance phenomena and supercritical/subcritical pitchfork bifurcation mechanism is reported in a complex bifractional-order damped system which endures a high-frequency parametric excitation and contains fractional-power nonlinearity. The approximate theoretical expression of the linear response amplitude at the primary frequency and the superharmonic response amplitude at the second and third harmonic frequencies are obtained by utilizing an analytical method and an iterative formula. A numerical approximation scheme based on the Caputo derivative for the simulation of the system is introduced, showing sufficient precision. Due to the parametric excitation, analytical approximation expressions of the stable equilibrium points are given explicitly when the exponent is not an integer so that the pitchfork bifurcation, nonlinear resonance can be studied in an analytical way, exhibiting much more operability than the external excitation case. It is found that the fractional-order derivative may bring new multibifurcation and new multiresonance phenomena, which have not yet been reported before. With the variation of different control parameters of the system, the equivalent slow-varying system can be converted from bistability to monostability and finally to bistability. Unlike the cases of the system excited by bifrequency external excitation, the optimum response amplitude of the parametric excited system is not monotonous with respect to the values of the exponent. For a range of parameters of the system, it is also found that the superharmonic resonance at the second and third harmonic frequencies is affected deeply by the parametric excitation.