1995 Fiscal Year Final Research Report Summary
Study on global property of holomorphic vector fields
| Project/Area Number |
06640181
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
Geometry
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| Research Institution | Ryukoku University |
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Principal Investigator |
ITO Toshikazu Faculty of Economy, Ryukoku University, Professor, 経済学部, 教授 (60110178)
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| Co-Investigator(Kenkyū-buntansha) |
MORITA Yoshihisa Faculty of Science & Technology, Ryukoku Univ., Assosiate Professor, 理工学部, 助教授 (10192783)
OKA Hiroe Faculty of Science & Technology, Ryukoku Univ., Associate Professor, 理工学部, 助教授 (20215221)
YOTSUTANI Shoji Faculty of Science & Technology, Ryukoku Univ., Professor, 理工学部, 教授 (60128361)
MATSUMOTO Waichiro Faculty of Science & Technology, Ryukoku Univ., Professor, 理工学部, 教授 (40093314)
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| Project Period (FY) |
1994 – 1995
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| Keywords | singular points of holomorphic vector fields / strongly hyperbolic system / reaction diffusion system / asymptotic stability / dissipative dynamical system / inertial manifold / Lorenz attractors / piecewise linear vector fields |
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| Research Abstract |
T.Ito counted the number of compact leaves of one dimensional non-singular foliation on the 2n-1 dimensional sphere S^<2n-1> associated with holomorphic vector fields, using Poincare-Dulac polynomialization Theorem at isolated singular point.
W.Matsumoto has studied the characterization of the strongly hyperbolic systems of partial differential equations. He showed that in case where the coefficients depend only on the time variable, the strongly hyperbolic systems can be transformed to a Fuchsian with diagonal principal part at each point of the fiber space. This condition is the necessary and sufficient for the strong hyperbolicity when the dimension of x-space is one. On the other hand, in higher dimension case, he proposed a conjecture that the uniformity on the dual variables of transforming matrix might be necessary, and he showed that Petrovski's non-strongly hyperbolic system is unstable and easily change to a strongly hyperbolic system by a small perturbation.
S.Yotsutani has st
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udied the asymptotic behavior and the rate of the convergence of solutions to a practical mathematical model of an chemical interfacial reaction, which is a parabolic system with nonlinear bondary conditions. Moreover, he has also studied some class of semilinear elliptic equations, and obtained a strong classification theorem of structure of positive radial solutions. The theorem claims that structure of solution is closely related with zeros of a function determined by coefficients and power of a nonlinear term.
H.Oka showed that chaotic attractor of Lorenz-type is generated locally by an unfolding of some degenerate singularity. Also, she studied some piecewise linear vector fields and found a bifurcation including infinitely many doubling of codimension 2 homoclinic orbits which is called orbit-flip type.
Y.Morita obtained with H.Ninomiya and E.Yanagida a theorem on the existence of an inertial manifold for a reaction-diffusion equation with nonlinear boundary condition. Moreover they gave the reduced finite-dimensional equation on the manifold together with applications to specific equations. Here the inertial manifold means a finite-dimensional invariant manifold which attracts every orbit in a phase space. Y.Morita studied in a joint work with K.Mischaikow a global structure of the dissipative dynamical system for the Ginzburg-Landau equation in a bounded interval and they showed a Morese structure of the dynamics. Y.Morita with S.Jimbo obtained an instability theorem of non-constant equilibrium solutions to the Ginzburg-Landau equation in any convex domain while they constructed stable non-constant solutions in an annulus domain. Less
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