1998 Fiscal Year Final Research Report Summary
Global Bifurcation Structure of Nonlinear Dynamics of Domain Motion
| Project/Area Number |
09640303
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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 |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Ryukoku University |
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Principal Investigator |
IKEDA Tsutomu Ryukoku Univ., Faculty of Science and Technology, Professor, 理工学部, 教授 (50151296)
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| Co-Investigator(Kenkyū-buntansha) |
SAKAI Kazushige Ryukoku Univ., Faculty of Science and Technology, Research Assistant, 理工学部, 助手 (00288664)
OKA Hiroe Ryukoku Univ., Faculty of Science and Technology, Professor, 理工学部, 教授 (20215221)
MORITA Yoshihisa Ryukoku Univ., Faculty of Science and Technology, Professor, 理工学部, 教授 (10192783)
YOTSUTANI Shoji Ryukoku Univ., Faculty of Science and Technology, Professor, 理工学部, 教授 (60128361)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Bistable systems / Piecewise-linear nonlinearity / Internal transition layers / Standing pulse solutions / Traveling pulse solutions / In-phase breathers / Traveling breathers / Hopf bifurcation |
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| Research Abstract |
In this research project, we deal with the dynamics of domain formed by pulse solutions of the fllowing bistable system of reaction-diffusion equations :
epsilontauu_1=epsilon^2u_<xx>+f(u, v), v=v_<xx>+g(u, v), where 0 < epsilon << 1 is a layer parameter and 0 < tau a relaxation one.
In a joint work with Prof. Hideo Ikeda (Toyama Univ.), we have studied the bifurcation phenomena of standing pulse solutions of the above system by adopting tau as a controllable parameter. It is shown that there exist two types of destabilization of standing pulse solutions when tau decreases. One is the appearance of traveling pulse solutions through the static bifurcation and the other is that of in-phase brealbers via the Hopf bifurcation. The order of these two destabilization is discussed for piecewise-linear nonlinearitics f and g.
Another joint work with Prof. Hideo Ikeda and Prof. Masayasu Mimura (Hiroshima Univ.) is devoted to the study of global bifurcation structure of standing and traveling pulse solutions of the above system. The simplification of piecewise-linear nonlinearity enables us to reveal the global branch of traveling pulse solutions. Using the singular limit analysis as epsilon * 0, the appearance of the Hopf bifurcation point on the branch is shown, from which stable propagating pulse solutions with oscillating layers (travelling breathers) arise numerically. Moreover, we have shown that
(1)All traveling pulse solutions unstably bifurcate from standing pulse solutions,
(2)All traveling pulse solutions bifurcated from standing pulse solutions recover their stability through the saddle-node bifurcation and the Hopf bifurcation, and they become stable for sufficiently small tau> 0.
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