This study introduces a novel homotopy-based semi-analytical approach designed to tackle challenges posed by strong nonlinear frequency mixing effects and mixed periodicity and quasi-periodicity intervals, which remain unsolved in the literature. This method, termed the iterative residue-regulating homotopy method (IRRHM), starting from a simple initial guess, seeks to achieve an asymptotic convergence toward a highly accurate approximate solution involving hundreds or even more influential frequency components by iteratively constructing homotopies between the original equations and a series of residue functions. A mode-switching module automatically discerns the solution’s quasi-periodic or periodic nature. The IRRHM no longer relies on empirically chosen linear operators used in the homotopy analysis method and achieves markedly higher convergence rates than the latter. Meanwhile, different from the iterative solver utilized in existing harmonic-balance-based methods, each iteration of the IRRHM generates a series of approximations by constructing a homotopy, facilitating more effective monitoring of the computational status and providing the essential basis for improving the convergence performance of the iterative strategy. The mixed periodicity and quasi-periodicity regions of a forced van der Pol-Duffing oscillator and the frequency comb response generated in a two-degree-of-freedom modal-coupling system are solved, showcasing the merits of IRRHM. Rarely reported long-periodic (∼180 T) responses are revealed in the former. Using this approach, the application scope of frequency domain methods can be significantly broadened to encompass problems characterized by significant spectral complexity and unknown properties regarding periodicity or quasi-periodicity.

中文翻译：

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用于强非线性混频系统的自区分和求解混合周期性和准周期性

本研究引入了一种新的基于同伦的半解析方法，旨在解决强大的非线性频率混合效应以及混合周期性和准周期性区间所带来的挑战，这些挑战在文献中仍未得到解决。这种方法被称为迭代余基调节同伦法 （IRRHM），从一个简单的初始猜测开始，旨在通过迭代构造原始方程和一系列余基函数之间的同伦，实现向涉及数百个甚至更多有影响力的频率分量的高精度近似解的渐近收敛。模式切换模块可自动识别解决方案的准周期性或周期性。IRRHM 不再依赖于同伦分析方法中使用的经验选择的线性算子，并且实现了明显高于后者的收敛速率。同时，与现有基于谐波平衡的方法中使用的迭代求解器不同，IRRHM 的每次迭代都通过构建同伦生成一系列近似，有助于更有效地监测计算状态，为提高迭代策略的收敛性能提供必要的基础。求解了强制范德波尔-达芬振荡器的混合周期性和准周期性区域以及在二自由度模态耦合系统中产生的频率梳响应，展示了 IRRHM 的优点。在前者中揭示了很少报道的长周期 （∼180 T） 反应。使用这种方法，频域方法的应用范围可以显著扩大，以包括以显著的频谱复杂性和关于周期性或准周期性未知的性质为特征的问题。