Research of applied analysis toward the global theory for nonlinear systems

非线性系统全局理论的应用分析研究

• 批准号: 17340027
• 负责人: NISHIDA Takaaki
• 金额: $ 6.98万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17340027/
• 关键词: Nonlinear Partial Differential Equation; Dynamical Systems; Global structure of solution space; Heat convection problems; Computer assisted proof; Free surface problems; 非線形波動

Analysis for the heat convectin problems for the system of Oberbeck-Boussinesq equations in the horizontal strip domain. The existence of bifurcation curve of roll-type solutions is proved for the ten times Rayleigh number of critical Rayleigh number by a computer assisted proof . The second bifurcation point of stationary roll-type solutions is determined by a computer assisted proof. The hexagonal-type and rectangle-type solutions of 3-dimensional problems are also proved for the existence by a computer assisted proof at least for rather small Rayleigh numbers.The cocoon bifurcation for the Michelson system is analyzed and proved for the existence bf infinitely many bifurcations of heteroclinic orbits to the saddle-node periodic orbit by a topological method and a computer assisted proof.The driven-cavity problem of 2-dimensional Navier-Stokes equation is solved for rather large Reynolds numbers compared to the existing verified result by a Newton type computer assited proof.

水平条域上Oberbeck-Boussinesq方程组的对流换热问题分析用计算机辅助证明了当临界Rayleigh数为十倍时，滚动型解的分支曲线的存在性。通过计算机辅助证明确定了定常滚动解的第二分歧点。用计算机辅助证明了三维问题的六角形和矩形解的存在性；用拓扑学方法和计算机辅助证明了迈克尔逊系统的茧分支，并证明了鞍结点周期轨道存在无穷多个异宿轨道分支；用牛顿型计算机辅助证明在雷诺数较大时求解了二维Navier-Stokes方程的驱动腔问题。

项目成果

Hopf bifurcation in viscous incompressible flow down an inclined plane

斜面粘性不可压缩流中的 Hopf 分岔

• 发表时间: 2005
• 期刊: J.Math.Fluid Mechanics 7
• 作者: Nishida;T.
• 通讯作者: T.

Improved convergence theorems of Newton's method designed for the numerical verification for solutions of differential equations

为微分方程解的数值验证而设计的改进牛顿法收敛定理

• 发表时间: 2007
• 期刊: J. Comput. Appl. Math. 199
• 作者: HAYASHI;Tadayuki;T.Kawanago
• 通讯作者: T.Kawanago

Rigorous verification of the cocoon bifurcation in the Michelson system

迈克尔逊系统茧分叉的严格验证

• 发表时间: 2007
• 期刊: Nonlinearity 20
• 作者: H.Kokubu;D.Wilczak;P.Zgliczynski
• 通讯作者: P.Zgliczynski

Cocoon bifurcation in three-dimensional reversible vector fields

• DOI: 10.1088/0951-7715/19/2/004
• 发表时间: 2006-02
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: F. Dumortier;S. Ibáñez;H. Kokubu

Bifurcation problems of heat convection systems

热对流系统的分岔问题

• 发表时间: 2007
• 作者: K.Tanaka;P.Felmer;S.Martinez;Takaaki Nishida
• 通讯作者: Takaaki Nishida