Project/Area Number  09440071 
Research Category 
GrantinAid for Scientific Research (B).

Section  一般 
Research Field 
General mathematics (including Probability theory/Statistical mathematics)

Research Institution  HOKKAIDO UNIVERSITY 
Principal Investigator 
NISHLURA Yasumasa Hokkaido Univ., Res. Inst. For Electronic Science, Prof., 電子科学研究所, 教授 (00131277)

CoInvestigator(Kenkyūbuntansha) 
NII Syunsaku Saitama Univ., Department of Math., Inst., 理学部, 助手 (50282421)
YANAGITA Tatsuo Hokkaido Univ., Res. Inst. For Electronic Science, Inst., 電子科学研究所, 助手 (80242262)
SAKAMOTO Kunimochi Hiroshima Univ., Department of Math., Assoc. Prof., 大学院・理学研究科, 助教授 (40243547)
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)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥13,200,000 (Direct Cost : ¥13,200,000)
Fiscal Year 1999 : ¥2,600,000 (Direct Cost : ¥2,600,000)
Fiscal Year 1998 : ¥2,100,000 (Direct Cost : ¥2,100,000)
Fiscal Year 1997 : ¥8,500,000 (Direct Cost : ¥8,500,000)

Keywords  selfreplication pattern / pattern formation / reaction diffusion system / pulse interaction / bifurcation / chaos / rugged landscape / transient dynamics / 自己複製パターン / パターン形式 / 反応拡散系 / パルス相互作用 / 分岐 / カオス / 多谷構造 / 遷移ダイナミクス / 反応拡散方程式 / 極限点の整列階層構造 / 界面ダイナミクス / グレインバウンダリー / 分岐解追跡ソフトウエアー 
Research Abstract 
(a) A mechanism of selfreplicating 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 saddlenode points drives SRP. (b) A new geometrical understanding for the spatiotemporal chaos arising in the GrayScott 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 saddlenode points. At the onset point there exists a generalized heteroclinic cycle on the whole line and spatiotemporal chaos emerges by unfolding this cycle. (c) Pulse interaction approach was employed to prove the existence of invariant manifold for the initiation of selfreplicatio
… More
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 saddlenode point. Combining this with natural assumptions, we are able to describe the whole process of selfreplication. (d) AUTO software is a wellknown 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 blockcopolymer dynamics. Spectral comparison theorem between the fourth and the second order equations was also shown. (f) Localized chaotic pulse of some reactiondiffusion system on a lattice was studied numerically. It is wellknown 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., timeperiodic multipulse. Such chaotic pulses are born from usual standing or traveling pulses as parameters are changed appropriately. Less
