ABSTRACT This paper studies the global dynamics of piecewise smooth differential equations defined in the two-dimensional torus and sphere in the case when the switching manifold breaks the manifold into two connected components. Over the switching manifold, we consider the Filippov's convention for discontinuous differential equations. The study of piecewise smooth dynamical systems over torus and sphere is common for maps and up to where we know this is the first characterization for piecewise smooth flows arising from solutions of differential equations. We provide conditions under generic families of piecewise smooth equations to get periodic and dense trajectories. Considering these generic families of piecewise differential equations, we prove that a non-deterministic chaotic behaviour appears. Global bifurcations are also classified.

中文翻译：

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二维圆环面和球面分段光滑微分方程的混沌行为

摘要 本文研究了在切换流形将流形分解为两个连通分量的情况下，定义在二维环面和球面中的分段光滑微分方程的全局动力学。在切换流形上，我们考虑不连续微分方程的 Filippov 约定。圆环面和球面上分段光滑动力系统的研究对于地图很常见，直到我们知道这是对由微分方程解产生的分段光滑流的第一个表征。我们在分段平滑方程的通用族下提供条件，以获得周期性和密集的轨迹。考虑到这些分段微分方程的通用族，我们证明出现了非确定性混沌行为。全局分叉也被分类。