2007 Fiscal Year Final Research Report Summary
Exploration for Complex Dynamics of Particle Patterns in Dissipative Systems
| Project/Area Number |
16204008
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Hokkaido University |
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Principal Investigator |
NISHIURA Yasumasa Hokkaido University, Research Institute for Electronic Science, Professor (00131277)
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| Co-Investigator(Kenkyū-buntansha) |
YANAGITA Tatsuo Hokkaido University, Research Institute for Electronic Science, Assi. (80242262)
IIMA Makoto Hokkaido University, Research Institute for Electronic Science, Assi. (90312412)
EI Shinichiro Kyushu University, Faculty of Mathematics, Professor (30201362)
UEDA Keiichiro Kyoto University, Research Institute for Mathematical Science, Assi. (00378960)
TERAMOTO Takashi Chitose Institute of Science and Technology, Faculty of Photonics Science, Lect. (40382543)
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| Project Period (FY) |
2004 – 2007
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| Keywords | reaction diilffusion systems / pulse / spot / scattering / global bifurcation / heteroclinic connection |
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| Research Abstract |
Particle patterns mean any spatially localized structures sustained by the balance between inflow and outflow of energy/material which arise in the form of chemical blob, discharge pattern, morphological spot, and binary convection cell. These are modeled by typically three-component reaction diffusion systems or a couple of complex GL equations With concentration field.
Strong interaction such as collision among particle patterns is a big challenge, since dissipative systems do not have many conservative quantities. Unlike weak-interaction through tails of those objects, there are so far no Systematic methods to handle them partly because of large deformation of patterns during the collision process. We present a new approach to clarify a backbone structure behind the complicated transient collision process A key ingredient lies in a hidden network of unstable, solutions called scattors which play a crucial role to understand the input-output relation for collision process (namely the relation of two dynamics before and after collision).More precisely, the associated network of scattors via heteroclinic connections forms a backbone for the whole collisional dynamics. The viewpoint of scattor network seems quite useful for many classes of model systems including reaction-diffusion systems, CGLE, and binary fluid convection.
Another interesting achievement is that the above viewpoint from a network of unstable patterns also works quite well in order to understand the propagating manner of waves in heterogeneous media. There appear defects induced by heterogeneities and propagation of particle patterns is equivalent to the collision problem between particle patterns and defects. In both cases, reduction PDE to finite dimensional system is possible, which allows us to make most of the numerical results rigorous.
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