Generic Properties of Smooth Dynamical Systems

Pi：Vadim Kaloshin，CAL TechDMS-0300229拟议的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(微分同胚)的一般性质。我们利用牛顿插值技巧研究了二维微分同胚的同宿切分支，特别是无穷多个共存槽的纽豪斯现象。同宿相切已被证明是二维动力学最有趣的特征。其次是研究一般哈密顿系统的不稳定性，主要是利用数学理论，用常微分方程组来描述系统在自然和工艺过程中的数量。因此，了解某种意义上的一般常微分方程解的长时间性态和轨道的稳定性是非常重要的。该项目的第一部分是为了帮助理解混沌低维系统的复杂行为。平面运动、彗星运动和天体动力学一般都用哈密顿方程来描述。该项目的第二部分致力于使用许多人，特别是J.马瑟发展的变分方法来研究这种动力学。

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.