It is a significant and challenging task to detect both the coexistence of singular cycles, mainly homoclinic and heteroclinic cycles, and chaos induced by the coexistence in nonsmooth systems. By analyzing the dynamical behaviors on manifolds, this paper proposes some criteria to accurately locate the coexistence of homoclinic cycles and of heteroclinic cycles in a class of three-dimensional (3D) piecewise affine systems (PASs), respectively. It further establishes the existence conditions of chaos arising from such coexistence, and presents a mathematical proof by analyzing the constructed Poincaré map. Finally, the simulations for two numerical examples are provided to validate the established results.

中文翻译：

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一类具有共存奇异循环的 3D 三区分段仿射系统产生的混沌

检测奇异循环（主要是同宿循环和异宿循环）的共存以及非光滑系统共存引起的混沌是一项重要且具有挑战性的任务。通过分析流形上的动力学行为，本文提出了一些标准，以分别准确定位一类三维（3D）分段仿射系统（PAS）中同宿循环和异宿循环的共存。它进一步确立了这种共存所产生的混沌的存在条件，并通过分析所构建的庞加莱图给出了数学证明。最后，提供了两个数值例子的模拟来验证所建立的结果。