Quasi-Periodic Bifurcations of Higher-Dimensional Tori
Quasi-Periodic Bifurcations of Higher-Dimensional Tori
复制标题
DOI：
10.1142/s0218127416300160
复制
发表时间：
2016-07
期刊：
影响因子：
--
通讯作者：
M. Komuro;K. Kamiyama;T. Endo;K. Aihara
中科院分区：
文献类型：
--
作者：
M. Komuro;K. Kamiyama;T. Endo;K. Aihara
本文对映射中拟周期d维环面的局部分叉进行了分类。MTD)和流动中(简写为D≥1)。这些分叉可以方便地分为正常分叉和共振分叉。MTD的正常分叉可以分为四类：鞍结型、倍周期、二重覆盖和Neimark-Sacker分叉。此外，FTD的正规分叉可以分为三类：鞍结型分叉、二重覆盖分叉和Neimark-Sacker分叉。这些分支是由主导Lyapunov丛的类型决定的。共振分叉被称为准周期解的锁相。这些分支被分为两类：(d−1)维环面的鞍结圈分支和异宿环分支。前者是可逆的，而后者是不可逆转的。此外，我们还提出了一种分析高维环面的方法，该方法在截面上使用一维环面。St1)和零维环面(简写为ST0)。ST1的分叉可以分为五类：鞍结型分叉、倍周期分叉、分量倍化分叉、二重覆盖分叉和Neimark-Sacker分叉。ST0的分叉可以分为四类：鞍结型分叉、倍周期分叉、分量倍化分叉和Neimark-Sacker分叉。此外，我们还阐明了ST1/ST0分叉与MTD/FTD分叉之间的关系。我们给出了所有这些分支的例子。
We classify the local bifurcations of quasi-periodic d-dimensional tori in maps (abbr. MTd) and in flows (abbr. FTd) for d ≥ 1. It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MTd can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark–Sacker bifurcations. Furthermore, normal bifurcations of FTd can be classified into three classes: saddle-node, double covering, and Neimark–Sacker bifurcations. These bifurcations are determined by the type of the dominant Lyapunov bundle. Resonance bifurcations are well known as phase locking of quasi-periodic solutions. These bifurcations are classified into two classes for both MTd and FTd: namely, saddle-node cycle and heteroclinic cycle bifurcations of the (d − 1)-dimensional tori. The former is reversible, while the latter is irreversible. In addition, we propose a method for analyzing higher-dimensional tori, which uses one-dimensional tori in sections (abbr. ST1) and zero-dimensional tori in sections (abbr. ST0). The bifurcations of ST1 can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and Neimark–Sacker bifurcations. The bifurcations of ST0 can be classified into four classes: saddle-node, period doubling, component doubling, and Neimark–Sacker bifurcations. Furthermore, we clarify the relationship between the bifurcations of ST1/ST0 and the bifurcations of MTd/FTd. We present examples of all of these bifurcations.