| Co-Investigator(Kenkyū-buntansha) |
HOSAKA Tetsuya Utsunomiya Univ., Dept. of Education, Lecturer, 教育学部, 講師 (50344908)
KITAGAWA Yoshihisa Utsunomiya Univ., Dept. of Education, Professor, 教育学部, 教授 (20144917)
FUJIHIRA Hideyuki Utsunomiya Univ., Dept. of Education, Professor, 教育学部, 教授 (70114171)
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| Research Abstract |
The purpose of this research project is to analyze the bifurcation phenomena of a 1-parameter family, which contains a vector field possessing the shadowing property (abbr.SP), composed of differentiate vector fields.
Denote by SP the set of vector fields possessing SP endowed with the C^1-topology, and let intSP be the C^1-interior. At the first stage of this research, it was indispensable to characterize the vector fields in intSP by making use of the stability theory of dynamical systems. In the end of 2001, however, we could not get a complete characterization for the vector fields.
In 2002, we continuously proceeded to characterize the above vector fields, and to analyze the bifurcation phenomena of such 1-parameter family. The main obstacle to the characterization is the existence of singularities of X ∈ intSP. We intimately analyzed the behavior of the orbits of the integrated flow X_t in the neighborhood of singularities, and we have found a special (but very natural) property of the flow being displayed in the neighborhood. Let us call the property (*) for simplicity. Very recently, it has been proved that X ∈ intSP satisfies (*) if and only if X satisfies both Axiom A and the strong transversality condition. This result is a partial answer to the problem we aimed to solve in 2001 at the first stage. Nowadays, using the above result we are analyzing the bifurcation phenomena of a 1-parameter family which contains a vector field possessing SP. In our opinion, the achieve percentage of this research project might be evaluated 70 %.
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