Harmonic Balance is one of the most popular methods for computing periodic solutions of nonlinear dynamical systems. In this work, we address two of its major shortcomings: First, we investigate to what extent the computational burden of stability analysis can be reduced by consistent use of Chebyshev polynomials. Second, we address the problem of a rigorous error bound, which, to the authors' knowledge, has been ignored in all engineering applications so far. Here, we rely on Urabe's error bound and, again, use Chebyshev polynomials for the computationally involved operations. We use the error estimate to automatically adjust the harmonic truncation order during numerical continuation, and confront the algorithm with a state-of-the-art adaptive Harmonic Balance implementation. Further, we rigorously prove, for the first time, the existence of some isolated periodic solutions of the forced-damped Duffing oscillator with softening characteristic. We find that the effort for obtaining a rigorous error bound, in its present form, may be too high to be useful for many engineering problems. Based on the results obtained for a sequence of numerical examples, we conclude that Chebyshev-based stability analysis indeed permits a substantial speedup. Like Harmonic Balance itself, however, this method becomes inefficient when an extremely high truncation order is needed as, e.g., in the presence of (sharply regularized) discontinuities.

翻译：谐波平衡法是计算非线性动力系统周期解最常用的方法之一。本文针对该方法的两个主要缺陷进行研究：首先，探讨通过一致采用切比雪夫多项式能在多大程度上降低稳定性分析的计算负担；其次，解决严格误差界的问题——据作者所知，该问题迄今在所有工程应用中均被忽视。本研究基于浦部误差界，再次运用切比雪夫多项式执行计算密集型操作。我们利用误差估计在数值延拓过程中自动调整谐波截断阶数，并将该算法与当前最先进的自适应谐波平衡实现进行对比。此外，首次严格证明了具有软化特性的受迫阻尼杜芬振荡器中某些孤立周期解的存在性。研究发现，以当前形式获取严格误差界所需计算量可能过高，难以适用于众多工程问题。基于一系列数值实验的结果，我们得出结论：基于切比雪夫多项式的稳定性分析确实能实现显著加速。然而，与谐波平衡法本身类似，该方法在处理需要极高阶截断的场景（例如存在尖锐正则化的间断点时）时效率会显著下降。