秩1奇异吸引子与异宿缠结
Rank-one strange attractors versus Heteroclinic tangles
摘要：
本文给出了平面上一类双参数摄动微分方程族中奇异吸引子（可观测混沌）出现的一种机制。这两个参数在循环中连续鞍点的不变流形中相互独立，作用方式不同。当这两个参数都为零时，该流动表现出与两个平衡点相关联的吸引异宿环。第一个参数使循环中连续鞍点的二维不变流形拉开，第二个参数使循环中连续鞍点的二维不变流形横截相交。这些相对位置可以使用Melnikov方法来确定。在已有理论的基础上，我们证明了双参数族中许多复杂的动力学对象的存在性，从支持SRB（Sinai-Ruelle-Bowen）测度的“大”奇异吸引子到超稳定汇和Hénon-type吸引子.我们画出一个合理的分岔图与考虑中的问题，我们表明，异宿缠结的发生是一个\emph{prevalent}现象。
英文摘要：
We present a mechanism for the emergence of strange attractors (observable chaos) in a two-parameter periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the invariant manifolds of consecutive saddles in the cycle. When both parameters are zero, the flow exhibits an attracting heteroclinic cycle associated to two equilibria. The first parameter makes the two-dimensional invariant manifolds of consecutive saddles in the cycle to pull apart; the second forces transverse intersection. These relative positions may be determined using the Melnikov method. Extending the previous theory on the field, we prove the existence of many complicated dynamical objects in the two-parameter family, ranging from "large" strange attractors supporting SRB (Sinai-Ruelle-Bowen) measures to superstable sinks and Hénon-type attractors. We draw a plausible bifurcation diagram associated to the problem under consideration and we show that the occurrence of heteroclinic tangles is a \emph{prevalent} phenomenon.
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