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

Research Project

Project/Area Number 18540200
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKitami Institute of Technology

Principal Investigator

SANNAMI Atsuro  Kitami Institute of Technology, Faculty of Engineering, Professor (30154157)

Co-Investigator(Kenkyū-buntansha) KOUNO Masaharu  Kitami Institute of Technology, Faculty of Engineering, Professor (40170203)
YAMADA Hiroshi  Kitami Institute of Technology, Faculty of Engineering, Professor (50210472)
Project Period (FY) 2006 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,430,000 (Direct Cost: ¥3,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2006: ¥2,000,000 (Direct Cost: ¥2,000,000)
Keywordsdynamical systems / non-linear / symbolic dynamics / Henon map / KAM theory / homoclinic bifurcation / chaos / attractor
Research Abstract

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.

Report

(3 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • Research Products

    (2 results)

All 2006

All Journal Article (2 results)

  • [Journal Article] 楕円曲線上の不安定主G-束のある特徴付け2006

    • Author(s)
      山田 浩嗣
    • Journal Title

      数理解析研究所講究録 1501

      Pages: 79-95

    • Related Report
      2006 Annual Research Report
  • [Journal Article] DS-diagramのいくつかの例2006

    • Author(s)
      河野 正晴
    • Journal Title

      HAKONE SEMINAR 22

      Pages: 49-75

    • Related Report
      2006 Annual Research Report

URL: 

Published: 2006-04-01   Modified: 2016-04-21  

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