谐波平衡法的理论基础
A theoretical basis for the Harmonic Balance Method
摘要：
谐波平衡法提供了一种寻找截断傅里叶级数作为常微分方程周期解的近似的启发式方法。另一种获得这种近似值的自然方法是应用数值方法。在本文中，我们恢复开创性的结果斯托克斯和浦部，提供了一个理论基础，证明在这些截断系列附近，无论是他们已经获得的方式，有实际的周期解的方程。我们将把我们的注意力限制在一维非自治常微分方程上，并将所得结果应用于来自平面自治系统的几个具体例子。
英文摘要：
The Harmonic Balance method provides a heuristic approach for finding truncated Fourier series as an approximation to the periodic solutions of ordinary differential equations. Another natural way for obtaining these type of approximations consists in applying numerical methods. In this paper we recover the pioneering results of Stokes and Urabe that provide a theoretical basis for proving that near these truncated series, whatever is the way they have been obtained, there are actual periodic solutions of the equation. We will restrict our attention to one-dimensional non-autonomous ordinary differential equations and we apply the results obtained to a couple of concrete examples coming from planar autonomous systems.
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