| Project/Area Number |
12440026
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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 | Osaka University |
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Principal Investigator |
OGAWA Toshiyuki Osaka University, Graduate school of Engineering Science, Associate Professor, 基礎工学研究科, 助教授 (80211811)
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| Co-Investigator(Kenkyū-buntansha) |
KUWAMURA Masataka Kobe University, Faculty of Human Development, Associate Professor, 発達科学部, 助教授 (30270333)
SUZUKI Hiromasa Shiga Univeisity, Faculty of Education, Lecturer, 教育学部, 講師 (60280450)
KAMETAKA Yoshinori Osaka University, Graduate school of Engineering Science, Professor, 基礎工学研究科, 教授 (00047218)
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| Project Period (FY) |
2000 – 2002
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| Project Status |
Completed (Fiscal Year 2002)
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| Budget Amount *help |
¥4,700,000 (Direct Cost: ¥4,700,000)
Fiscal Year 2002: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2001: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2000: ¥1,700,000 (Direct Cost: ¥1,700,000)
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| Keywords | periodic traveling wave / mode interaction / modulated wave / Eckhaus instability / gradient / skew gradient system / global bifurcation / numerical verification |
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| Research Abstract |
Structures of periodic patterns (periodic stationary solutions or traveling wave solution whose wave profile is periodic) are studied. More precisely, we study bifurcations of these solutions, stability of bifiucated solutions secondary bifurcation and the dynamics around them. First we study the behavior of periodic solutions to a perturbed integrable systems which is originally a physical problem that describe wave motion on a liquid layer over an inclined plane. Nest, these method turns out to be applicable to more general nonlinear wave phenomena, such as the Swift-Hohenberg equation which is a simple model of thermal convection. By the mathematical rigorous normal form analysis, sometimes referred to as "weak nonlinear analisys", we could show the existence of non-trivial stable mixed mode solutions to these equations and this corresponds to the modulated wave solutions. In the case of Swift-Hohenberg equation we found a localized patterns of pulses as well as the mixed mode solution by using the numerical simulation. And later we prove the existence of these solutions by numerical verification technique.
On the other hand we study the stability of periodic patterns in the context of gradient/skew gradient systems. As a result, we could show that activator-inhibitor system and the Swift-Hohenberg equations are skew gradient and also the dynamics of periodic roll patterns in these system are controlled by the Eckhaus instability and Zigzag instability.
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