To what extent can symbolic dynamics represent the structure of non-linear dynamics?

符号动力学在多大程度上可以代表非线性动力学的结构？

• 批准号: 18540200
• 负责人: SANNAMI Atsuro
• 金额: $ 2.2万
• 依托单位: Kitami Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2006
• 资助国家: 日本
• 起止时间: 2006 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540200/
• 关键词: dynamical systems; non-linear; symbolic dynamics; Henon map; KAM theory; homoclinic bifurcation; chaos; attractor

The purpose of the research is to find an appropriate method to represent the dynamics of non-linear systems by using symbolic dynamics.For diffeomophisms of dimension two and more, it is quite difficult to construct an itinerary representation, mainly because for diffeomorphisms there exists no special point such as critical point of one-dimensional maps. For the Henon map, which is the simplest non-linear diffeomorphism, there is an idea called “pruning front" which is that one may regard the non-wandering set of non-horseshoe Henon map as a subshift of two-symbols full-shift. In 1990, Davis-MacKay-Sannami gave a mechanism and its promising evidence for the dynamics on the non-wandering set being represented by “missing blocks expression", which is a kind of pruning front. Recently, Arai gave a rigorous proof for that including many other parameter values cases. But, those all examples are the case of complex full-horseshoe, and missing blocks expression (and so pruning front representation) for the cases of having tangencies and sinks have not been known. In this research, I investigated the case of having sinks, and I succeeded to find a missing blocks expression for such case. For the moment, my method gives missing blocks expression for almost any finite orbit. For example, a certain missing blocks gives only one periodic orbit of period 3, which can not occur for the Henon map. There must be more restrictions for missing blocks expression of the real Henon map.By pursuing the direction obtained by this research, we may make some progress in the important problem of giving a symbolic representation to dynamics of non-linear systems.

该研究的目的是找到一种适当的方法来通过使用符号动力学来表示非线性系统的动力学。对于二级尺寸的差异性，很难构建行程表示非常困难，这主要是因为对于差异性而言，没有像一维映射的关键点那样的特殊点。对于最简单的非线性差异形态的亨逊地图，有一个称为“修剪阵线”的想法，即可能关注非伴奏的非杂音非杂音的亨逊地图，作为两个符号的次移动。 1990年，戴维斯·玛基·桑纳米（Davis-Mackay-Sannami）提供了一种机制及其有望证明非随机套装动态的证据，以“丢失的块表达”为代表，这是一种修剪的阵线。最近，Arai对其中包括许多其他参数值情况给出了严格的证明。但是，所有这些示例都是复杂的全能曲，而对于具有切线和下沉的情况的情况，尚未知道块的表达（以及较修剪的前面表示）。在这项研究中，我调查了有水槽的案例，并成功地为这种情况找到了丢失的块表达。目前，我的方法给出了几乎所有有限轨道的块表达式。例如，某些丢失的块仅给出一个周期性3的周期轨道，而亨逊地图无法发生。通过追求这项研究所获得的方向，我们可能会在重要的问题中取得一些进展，即在为非线性系统的动态提供符号表示方面，我们可能会取得一些进展，因此必须有更多限制。

项目成果

DS-diagramのいくつかの例

DS 图的一些示例

• 发表时间: 2006
• 期刊: HAKONE SEMINAR 22
• 作者: Y.Tamura;H.Tanaka;相原義弘;Y. Aihara;Atsushi Atsuji;Atsushi Atsuji;厚地淳;Atsushi Atsuji;Atsushi Atsuji;Y. Aihara;厚地淳;山田 浩嗣;河野 正晴
• 通讯作者: 河野 正晴

楕円曲線上の不安定主G-束のある特徴付け

椭圆曲线上不稳定主 G 丛的表征

• 发表时间: 2006
• 期刊: 数理解析研究所講究録 1501
• 作者: Y.Tamura;H.Tanaka;相原義弘;Y. Aihara;Atsushi Atsuji;Atsushi Atsuji;厚地淳;Atsushi Atsuji;Atsushi Atsuji;Y. Aihara;厚地淳;山田 浩嗣
• 通讯作者: 山田 浩嗣