Studies on global bifurcations for dynamical systems

动力系统全局分岔研究

• 批准号: 09640261
• 负责人: YOSHIHARA Hideaki
• 金额: $ 1.98万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640261/
• 关键词: dynamical system; bifurcation; global structure; structural stability; hyperbolicity; singular perturbation; topological index; pattern; 数値的検証

The following results have been obtained in this research project :1. Study of non-hyperbolic dynamical systems, in particular, a result toward Palis conjecture that aims at describing dynamical structures for a dense set of dynamical systems in C^1 topology (Hayashi)2. Complex dynamical systems and hyper functions in relation to ergodic theory, in particular the Ruelle operator (Ushiki)3. Existence and statistical properties of absolutely continuous invariant measures in two dimensional real analytic and higher dimensional piecewise linear expanding maps (Tsujii)4. Conley index theory for singularly perturbed vector fields of slow-fast type in the case of one-dimensional slow dynamics (Oka)5. Monotonicity of topological entropy for piecewise-linear symmetric bimodal maps (Oka)6. Analytic and computer assisted approach to global bifurcation phenomena in the Rayleigh-Benard problem, in particular a new method for proving the existence of solutions for various different parametres using the validated numerical computation (Nishida and Yoshihara)7. Blow-up of solutions in the Proudman-Johnson equation (Okamoto)8. Analytical and dynamical investigation of self-replicating patterns which describe new spatial patterns observed in real chemical and biological experiments (Nishiura)

本研究取得了以下成果：1.研究非双曲动力系统，特别是Palis猜想的结果，该猜想旨在描述C^1拓扑中稠密动力系统的动力结构（Hayashi）2。复杂的动力系统和超函数与遍历理论，特别是Ruelle算子（Ushiki）3。二维真实的解析映射和高维分段线性扩张映射中绝对连续不变测度的存在性和统计性质（Tsujii）4.在一维慢动力学的情况下，慢-快型奇摄动向量场的康利指数理论（OKA）5。分段线性对称双峰映射拓扑熵的单调性（Oka）6.分析和计算机辅助方法的全球分歧现象的瑞利-贝纳德问题，特别是一种新的方法证明存在的解决方案，为各种不同的参数使用验证的数值计算（西田和Yoshihara）7。Proudman-Johnson方程解的爆破（Okamoto）8.对描述在真实的化学和生物实验中观察到的新空间模式的自我复制模式进行分析和动态研究（西浦）

项目成果

Shuhei Hayashi: "Hyperbolicity, stability, and the creation of homoclinic points." Proc. Int. Congress Math. Vol.II. 789-796 (1998)

Shuhei Hayashi：“双曲性、稳定性和同宿点的创建。”

西浦康政: "非線形問題I-パターン形成の数理-" 岩波書店,岩波講座現代数学の展開7, 279 (1999)

西浦康正：“非线性问题I-模式形成的数学-”岩波书店，岩波现代数学课程发展7, 279 (1999)

Shigehiro Ushiki: "Complex Ruelle operator and hyperbolic complex dynamical systems" 京都大学数理解析研究所講究録. 1072. 50-61 (1998)

Shigehiro Ushiki：“复数 Ruelle 算子和双曲复动力系统”京都大学数学科学研究所 Kokyuroku。1072. 50-61 (1998)。

H.Oka他2名: "Directional transition matrix" “Conley Index Theory"Banach Center Publication. (発表予定).

H.Oka 等 2 人：“方向转移矩阵”“康利指数理论”巴纳赫中心出版（待出版）。

H.Kokubu: "Connecting orbit structure of monotone in solutions in the shadow system" Journal of Differential Equations. 140. 309-364 (1997)

H.Kokubu：“影子系统解中单调的连接轨道结构”微分方程杂志。