The coexistence of random periodic solutions with various periods (i.e. subharmonics) is proved to random differential equations on a circle with random impulses of all integer orders. One of the theorems is also extended to random differential inclusions on a circle with multivalued deterministic impulses. These results can be roughly characterized as a further application of the randomized Sharkovsky type theorems to random impulsive differential equations and inclusions on a circle.

中文翻译：

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圆上随机脉冲微分方程和包含的随机次谐波解的共存

证明了具有各种周期（即次谐波）的随机周期解的共存性，证明了具有所有整数阶随机脉冲的圆上的随机微分方程。其中一个定理还扩展到具有多值确定性脉冲的圆上的随机微分包含。这些结果可以粗略地描述为随机 Sharkovsky 类型定理对随机脉冲微分方程和圆上包含物的进一步应用。