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.

通过使用符号动力学找到一种适当的方法来找到非线性系统动力学的目的。对于二级尺寸的HISMS，构建行程表示特殊点（例如一维映射的临界点）是非常分散的。修剪前部“这就是一个人可能将非腰部的非杂音熊图集视为对两符号的全班缩影。1990年，戴维斯 - 玛基 - 萨纳米（Davis-Mackay-Sannami在悬挂式集合中，“缺失的块表达式”是一种修剪的阵线。静止的情况是，我的方法几乎为任何有限的轨道提供了丢失的块表达式通过追求这项研究获得的方向，在给出符号的重要问题中取得了一些进展，这代表了非线性系统的动态。

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