A usual way to find positive invariant sets of ordinary differential equations is to restrict the search to predefined finitely generated shapes, such as linear templates, or ellipsoids as in classical quadratic Lyapunov function based approaches. One then looks for generators or parameters for which the corresponding shape has the property that the flow of the ODE goes inwards on its border. But for non-linear systems, where the structure of invariant sets may be very complicated, such simple predefined shapes are generally not well suited. The present work proposes a more general approach based on a topological property, namely Ważewski's property. Even for complicated non-linear dynamics, it is possible to successfully restrict the search for isolating blocks of simple shapes, that are bound to contain non-empty invariant sets. This approach generalizes the Lyapunov-like approaches, by allowing for inwards and outwards flow on the boundary of these shapes, with extra topological conditions. We developed and implemented an algorithm based on Ważewski's property, SOS optimization and some extra combinatorial and algebraic properties, that shows very nice results on a number of classical polynomial dynamical systems.

查找常微分方程的正不变集的常用方法是将搜索限制为预定义的有限生成的形状，例如线性模板或椭球，这是基于经典二次Lyapunov函数的方法。然后，寻找具有相应形状具有ODE流在其边界处向内流动的属性的生成器或参数。但是对于非线性系统，其中不变集合的结构可能非常复杂，因此这种简单的预定义形状通常不太适合。本工作提出了一种基于拓扑属性（即Ważewski的属性）的更通用的方法。即使对于复杂的非线性动力学，也可以成功地限制对隔离简单形状的块的搜索，这些形状必须包含非空不变集。这种方法通过允许在具有额外拓扑条件的情况下在这些形状的边界上向内和向外流动来概括类Lyapunov方法。我们基于Ważewski的性质，SOS优化以及一些额外的组合和代数性质，开发并实现了一种算法，该算法在许多经典的多项式动力学系统上均显示出非常好的结果。