Frequency responses of multi-degree-of-freedom mechanical systems with weak forcing and damping can be studied as perturbations from their conservative limit. Specifically, recent results show how bifurcations near resonances can be predicted analytically from conservative families of periodic orbits (nonlinear normal modes). However, the stability of forced-damped motions is generally determined a posteriori via numerical simulations. In this paper, we present analytic results on the stability of periodic orbits that perturb from conservative nonlinear normal modes. In contrast with prior approaches to the same problem, our method can tackle strongly nonlinear oscillations, high-order resonances and arbitrary types of non-conservative forces affecting the system, as we show with specific examples.
在作者最近的论文中（S.~元永和K.~ Yagasaki，Obstructions to integrability of nearly integrable动力系统near regular level sets，已提交出版，和K.~ Yagasaki，Nonintegrability of nearly integrable动力系统near resonant periodic orbits，submitted for publication），提供了两种不同的技术，使我们能够证明具有单自由度哈密顿系统时间周期扰动形式的受迫非线性振子的实解析或复亚纯不可积性。这里采用Bogoyavlenskij意义下的不可积性概念，并且要求首次积分和交换向量场对小参数的实解析或复亚纯依赖性.在本文中，我们回顾了这些理论并继续证明它们的有用性。特别地，我们考虑了周期受迫阻尼摆，并证明了它在上述意义下的不可积性。
Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits
在作者最近的一篇论文中（K。Yagasaki，Nonintegrability of the限制三体问题，submitted for publication），开发了一种技术，用于确定几乎可积系统是否不是亚纯Bogoyavlenskij可积的，使得首次积分和交换向量场也亚纯依赖于小参数。在这里，我们将继续证明某些类别的动力系统的技术。特别是，我们考虑时间周期扰动的单自由度哈密顿系统和讨论的关系的技术与次谐波Melnikov方法，这使我们能够检测周期轨道的存在性和它们的稳定性。我们说明了理论的周期性强迫Duffing振荡器和两个额外的例子：二阶耦合振荡器和二维系统受到恒定的扭矩的双曲型。
