Resonance and bifurcation are prominent and significant features observed in various nonlinear systems, often leading to catastrophic failure in practical engineering. This paper investigates, under an analytical and numerical perspective, the dynamical characteristics of a fractional Rayleigh oscillator with distributed time delay. Firstly, through the application of the multiple scales method, we derive approximated analytical solutions and amplitude–frequency equations for the regions near both primary and secondary resonances. The stability conditions of steady-state motions and the existence region of the subharmonic response are also obtained. Furthermore, to validate the accuracy of the approximated solutions, the results are compared with numerical solutions derived from the Caputo scheme, revealing a high concordance between them. Then, a comprehensive study on response curves is conducted for the system under different nonlinear damping, fractional parameters and delay strength. Finally, we identify and discuss the presence of the forked bifurcation within the system.

中文翻译：

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分布式时滞分数瑞利振荡器的谐波谐振与分岔

共振和分岔是在各种非线性系统中观察到的突出且重要的特征，通常会导致实际工程中的灾难性故障。本文从分析和数值的角度研究了具有分布式时滞的分数瑞利振荡器的动态特性。首先，通过应用多尺度方法，我们推导了初级和次级共振附近区域的近似解析解和幅频方程。还得到了稳态运动的稳定性条件和次谐波响应的存在区域。此外，为了验证近似解的准确性，将结果与从 Caputo 方案得出的数值解进行了比较，揭示了它们之间的高度一致性。然后，对系统在不同非线性阻尼、分数参数和延迟强度下的响应曲线进行了综合研究。最后，我们识别并讨论系统内分叉的存在。