AI 中文总结
研究分段光滑向量场，通过扩展相关定理，证明若T - 周期向量场有横截同宿点，其时间 - T映射与伯努利移位共轭，进而证明该向量场混沌，此为相关理论在分段光滑向量场领域的扩展。
AI 中文摘要
本文给出了无滑动运动的分段光滑向量场的伯克霍夫 - 斯梅尔同宿定理和λ - 引理的扩展。证明了若向量场在自变量上是T - 周期的且有一个横截同宿点，那么它的时间 - T映射与一个伯努利移位共轭，从而该向量场是混沌的。这里考虑的伯努利移位可以有任意有限个符号或可数无穷多个。
英文摘要
In this paper we provide extensions of the Birkhoff-Smale Homoclinic Theorem and the $λ$-Lemma for piecewise smooth vector fields free of sliding motion. We prove that if the vector field is $T$-periodic in its independent variable and has a transverse homoclinic point, then its time-$T$-map is conjugated with a Bernoulli Shift and thus the vector field is chaotic. The Bernoulli Shifts considered here may have any finite number of symbols, or countably infinite many.
