Generic Properties of Smooth Dynamical Systems

平滑动力系统的一般性质

• 批准号: 0300229
• 负责人: Vadim Kaloshin
• 金额: $ 13.43万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-01 至 2006-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300229&HistoricalAwards=false
• 关键词: Generic; Properties; Smooth; Dynamical; Systems

PI: Vadim Kaloshin, CAL TechDMS-0300229The proposed project consists of two main parts. The first part is investigation generic properties of discrete dynamical systems (diffeomorphisms) on a compact manifold. We apply Newton Interpolation technique to investigate bifurcations of homoclinic tangency for 2-dimensional diffeomorphisms and, in particular, Newhouse phenomenon of infinitely many coexisting sinks. Homoclinic tangency has proved to be the most interesting feature of dynamics in dimension 2. The second is investigation of instabilities of generic of Hamiltonian systems, primarily using Mather theory.Enormous number of systems in the nature and technological processes are described by ordinary differential equations. Therefore, it is extremely important to understand long time behavior and stability of trajectories of in some sense generic ordinary differential equations. The first part of proposed project is an attempt to contribute to understanding of complicated behavior of chaotic low dimensional systems. Motion of plans, comets, and celestial dynamics in general are described by Hamiltonian equations. The second part of the project is devoted to investigation of such dynamics using primarily variational methods developed by many people and J. Mather in particular.

PI：Vadim Kaloshin，CAL TechDMS-0300229拟议项目包括两个主要部分。第一部分是研究紧致流形上离散动力系统（同构）的通有性质。利用Newton插值方法研究了二维单同态的同宿切分支，特别是无穷多个汇共存的纽豪斯现象。同宿相切已被证明是二维动力学中最有趣的特征。第二部分是研究一般哈密顿系统的不稳定性，主要是利用Mather理论，自然界和工艺过程中大量的系统都是用常微分方程来描述的。因此，了解一般常微分方程的长时间行为和轨迹的稳定性是非常重要的。第一部分的建议项目是试图有助于理解混沌低维系统的复杂行为。行星、彗星和天体动力学的运动一般都用哈密顿方程来描述。该项目的第二部分是专门调查这种动态使用主要是变分方法开发的许多人和J.马瑟特别。