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.

中文翻译：

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参数激励对具有分数次非线性的双分数阶阻尼系统的影响。

在复杂的双分数阶阻尼系统中，研究了线性/非线性共振现象和超临界/亚临界干草叉分叉机制，该系统能承受高频参数激励并包含分数功率非线性。通过使用解析方法和迭代公式，可以得出初级频率处的线性响应幅度以及二次谐波和三次谐波频率处的超谐波响应幅度的近似理论表达式。介绍了一种基于Caputo导数的数值逼近方案，用于系统仿真，显示了足够的精度。由于参数激励，当指数不是整数时，可以明确给出稳定平衡点的解析近似表达式，从而可以用解析的方式研究音叉分叉，非线性共振，并且比外部激励情况具有更多的可操作性。发现分数阶导数可能带来新的多分支和新的多共振现象，这是以前尚未报道的。随着系统不同控制参数的变化，等效的慢速变化系统可以从双稳态转换为单稳态，最后转换为双稳态。与通过双频外部激励来激励系统的情况不同，参数激励系统的最佳响应幅度相对于指数值不是单调的。