To what extent can symbolic dynamics represent the structure of non-linear dynamics?
| Project/Area Number |
18540200
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Kitami Institute of Technology |
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Principal Investigator |
SANNAMI Atsuro Kitami Institute of Technology, Faculty of Engineering, Professor (30154157)
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| 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)
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| Project Period (FY) |
2006 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| 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)
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| Keywords | dynamical systems / non-linear / symbolic dynamics / Henon map / KAM theory / homoclinic bifurcation / chaos / attractor |
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| 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.
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Report
(3 results)
Research Products
(2 results)
