In this note, we consider a class of homogeneous second order systems which contains two relevant sub-classes: the first one in a controller form and the second one in an observer form. For both of them, we provide necessary and sufficient conditions on the gain parameters to guarantee oscillatory or non-oscillatory behavior of the solutions. This is in the same spirit of tuning of second order linear systems, where the gains can be designed to obtain an overdamped, critically damped or underdamped behavior. The analysis of all the cases is performed in a unified framework based on a search for invariant sets which are analogous to the invariant eigenspaces of linear systems.

在这篇文章中，我们考虑了一类齐次二阶系统，它包含两个相关的子类：第一个以控制器形式出现，第二个以观察者形式出现。对于它们两者，我们提供了增益参数的必要和充分条件，以保证解决方案的振荡或非振荡行为。这与调整二阶线性系统的精神相同，其中可以设计增益以获得过阻尼、临界阻尼或欠阻尼行为。所有案例的分析都是在一个统一的框架中进行的，该框架基于对类似于线性系统的不变特征空间的不变集的搜索。