In smooth systems, the form of the heteroclinic Melnikov chaotic threshold is similar to that of the homoclinic Melnikov chaotic threshold. However, this conclusion may not be valid in nonsmooth systems with jump discontinuities. In this paper, based on a newly constructed nonsmooth pendulum, a kind of impulsive differential system is introduced, whose unperturbed part possesses a nonsmooth heteroclinic solution with multiple jump discontinuities. Using the recursive method and the perturbation principle, the effects of the nonsmooth factors on the behaviors of the nonsmooth dynamical system are converted to the integral items which can be easily calculated. Furthermore, the extended Melnikov function is employed to obtain the nonsmooth heteroclinic Melnikov chaotic threshold, which implies that the existence of the nonsmooth heteroclinic orbits may be due to the breaking of the nonsmooth heteroclinic loops under the perturbation of damping, external forcing and nonsmooth factors. It is worth pointing out that the form of the nonsmooth heteroclinic Melnikov function is different from the one of the nonsmooth homoclinic Melnikov function, which is quite different from the classical Melnikov theory.

中文翻译：

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具有跳跃不连续性的非光滑系统中的异宿混沌阈值

在光滑系统中，异宿 Melnikov 混沌阈值的形式与同宿 Melnikov 混沌阈值的形式相似。然而，这个结论在具有跳跃不连续性的非光滑系统中可能无效。本文基于一种新构造的非光滑摆，介绍了一种脉冲微​​分系统，其未扰动部分具有具有多个跳跃间断的非光滑异宿解。利用递推法和摄动原理，将非光滑因素对非光滑动力系统行为的影响转化为易于计算的积分项。此外，使用扩展的 Melnikov 函数来获得非光滑异宿 Melnikov 混沌阈值，这意味着非光滑异宿轨道的存在可能是由于非光滑异宿环在阻尼、外力和非光滑因素的扰动下断裂。值得指出的是，非光滑异宿 Melnikov 函数的形式不同于非光滑同宿 Melnikov 函数的一种，这与经典的 Melnikov 理论有很大的不同。