In this paper, by introducing the definition of the solution, we prove the existence and uniqueness of the solution for nonlinear on–off switched systems with switching at variable times (NVTSS). Based on this result, the continuous dependence and differentiability of the solution of NVTSS (4) with respect to the initial state are presented. Besides, the switching phenomenon (the integral curve of NVTSS (4) may hit some surface finite or infinite times causing a rhythmical beating) and periodic solution of NVTSS (4) are also discussed.

中文翻译：

---

可变时间切换的非线性开关系统解的存在性和唯一性

在本文中，通过介绍解决方案的定义，我们证明了具有可变时间切换（NVTSS）的非线性开关系统的解的存在性和唯一性。基于此结果，提出了NVTSS（4）解相对于初始状态的连续依赖性和可微性。此外，还讨论了切换现象（NVTSS（4）的积分曲线可能会在表面上有限或无限次数地引起有节奏的跳动）并讨论了NVTSS（4）的周期解。