In order to solve the general multidimensional perturbed oscillatory system \(y'' + \varOmega y = f(y, y')\) with \(K\in {\mathbb {R}}^{d\times d}\), the order conditions for the ERKN (extended Runge–Kutta–Nyström) methods and some effective ERKN methods were presented in the literature. These methods integrate exactly the multidimensional unperturbed oscillator \(y'' + \varOmega y = 0\). In this paper, we analyze the stability of ERKN methods for general oscillatory second-order initial value problems whose right-hand-side functions depend on both y and \(y'\). Based on the linear test model \(y''(t)+\omega ^2y(t)+\mu y'(t)=0\) with \(\mu <2\omega \), further discussion and analysis on the linear stability of ERKN methods for general oscillatory problems are presented. A new conception of \(\alpha \)-stability region is proposed to investigate how well the numerical methods respect the damping rate of the general oscillatory systems. It gains more insight to the numerical methods when applied to the systems involving \(y'\). Numerical experiments are carried out to show the significance of the theory.

为了求解一般多维扰动振荡系统\(y'' + \varOmega y = f(y, y')\)与\(K\in {\mathbb {R}}^{d\times d}\ )，文献中介绍了 ERKN（扩展的 Runge-Kutta-Nyström）方法和一些有效的 ERKN 方法的阶条件。这些方法精确地整合了多维无扰动振荡器\(y'' + \varOmega y = 0\)。在本文中，我们分析了 ERKN 方法对于一般振荡二阶初值问题的稳定性，其右手边函数取决于y和\(y'\)。基于线性测试模型\(y''(t)+\omega ^2y(t)+\mu y'(t)=0\)与\(\mu <2\omega \)，进一步讨论和分析一般振荡问题的 ERKN 方法的线性稳定性。提出了\(\alpha \)稳定区的新概念来研究数值方法对一般振荡系统阻尼率的尊重程度。当应用于涉及\(y'\)的系统时，它对数值方法有更深入的了解。进行了数值实验以表明该理论的重要性。