1999 Fiscal Year Final Research Report Summary
Infinite Dimensional Transient Dynamics in Dissipative Systems
| Project/Area Number |
09440071
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 |
NISHLURA Yasumasa Hokkaido Univ., Res. Inst. For Electronic Science, Prof., 電子科学研究所, 教授 (00131277)
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| Co-Investigator(Kenkyū-buntansha) |
YANAGITA Tatsuo Hokkaido Univ., Res. Inst. For Electronic Science, Inst., 電子科学研究所, 助手 (80242262)
KOBAYASHI Ryo Hokkaido Univ., Res. Inst. For Electronic Science, Assoc. Prof., 電子科学研究所, 助教授 (60153657)
TSUDA Ichiro Hokkaido Univ., Graduate School of Math., Prof., 理学研究科, 教授 (10207384)
UEYAMA Daishin Hiroshima Univ., Department of Math. And Life Sci., Inst., 大学院・理学研究科, 助手 (20304389)
SAKAMOTO Kunimochi Hiroshima Univ., Department of Math., Assoc. Prof., 大学院・理学研究科, 助教授 (40243547)
NII Syun-saku Saitama Univ., Department of Math., Inst. (50282421)
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| Project Period (FY) |
1997 – 1999
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| Keywords | self-replication pattern / pattern formation / reaction diffusion system / pulse interaction / bifurcation / chaos / rugged landscape / transient dynamics |
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| Research Abstract |
(a) A mechanism of self-replicating pattern(SRP) was clarified from a global bifurcational view point. The main difficulty lies in the fact that SRP is a transient dynamics and contains a large deformation of the pattern (I.e., splitting), hence the conventional approaches do not work to understand its mechanism from mathematical view point. It turned out that the hierarchy structure of saddle-node points drives SRP.
(b) A new geometrical understanding for the spatio-temporal chaos arising in the Gray-Scott model was proposed. This is based on the interrelationship of global bifurcating branches of ordered patterns with respect to the parameters contained in the above model, especially their locations of saddle-node points. At the onset point there exists a generalized heteroclinic cycle on the whole line and spatio-temporal chaos emerges by unfolding this cycle.
(c) Pulse interaction approach was employed to prove the existence of invariant manifold for the initiation of self-replicatio
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n. Pulse interaction approach was developped to describe weak interaction among pulses, nevertheless it was shown that it is also powerful near bifurcation points such as saddle-node point. Combining this with natural assumptions, we are able to describe the whole process of self-replication.
(d) AUTO software is a well-known tool to search for the bifurcating branches for ODE, however one can't use it directly to PDE systems. This not only contains technical difficulties but also requires us to make AUTO accessible to spatial information, for instance, profiles of patterns and eigenfunctions. Moreover we explored the method to obtain, say only stable branches emanating from specific branch, otherwise AUTO outputs huge number of spaghetti like branches from which one can't extract useful information.
(e) Existence of rugged landscape for the fourth order conserved equation with nonlocal term is proved, which arises in block-copolymer dynamics. Spectral comparison theorem between the fourth and the second order equations was also shown.
(f) Localized chaotic pulse of some reaction-diffusion system on a lattice was studied numerically. It is well-known that a variety of complex patterns are observed for the dissipative systems on a discrete lattice, however it is new that localized pulse behaves chaotically. Several chaotic pulses form a molecullar state, I.e., time-periodic multi-pulse. Such chaotic pulses are born from usual standing or traveling pulses as parameters are changed appropriately. Less
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