The Poincaré map is widely used to study the qualitative behavior of dynamical systems. For instance, it can be used to describe the existence of periodic solutions. The Poincaré map for dynamical systems with impulse effects (SIEs) was introduced in the last decade and mainly employed to study the existence of limit cycles (periodic gaits) for the locomotion of bipedal robots. We investigate sufficient conditions for the existence and uniqueness of Poincaré maps for dynamical SIEs evolving on a differentiable manifold. We apply the results to show the existence and uniqueness of Poincaré maps for systems with multiple domains.

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关于具有脉冲效应系统的庞加莱映射的存在性和唯一性

庞加莱映射被广泛用于研究动力系统的定性行为。例如，它可以用来描述周期解的存在。具有脉冲效应（SIE）的动力系统的庞加莱映射是在过去十年中引入的，主要用于研究双足机器人运动的极限循环（周期性步态）的存在。我们研究了在可微流形上演化的动态 SIE 的 Poincaré 映射存在和唯一性的充分条件。我们应用结果来显示具有多个域的系统的庞加莱映射的存在性和唯一性。