Generic Properties of Smooth Dynamical Systems

PI: Vadim Kaloshin， CAL technologies ms -0300229提议的项目包括两个主要部分。第一部分研究紧流形上离散动力系统（微分同态）的一般性质。本文应用牛顿插值技术研究了二维微分同态的同斜切线分岔问题，特别是研究了无穷多个共存槽的Newhouse现象。同斜切已被证明是二维动力学中最有趣的特征。第二部分主要利用马瑟理论研究了一般哈密顿系统的不稳定性。自然界和技术过程中大量的系统都是用常微分方程来描述的。因此，了解某种意义上的一般常微分方程轨迹的长时间行为和稳定性是极其重要的。计划的第一部分是尝试对混沌低维系统的复杂行为的理解作出贡献。行星的运动、彗星和天体动力学一般都是用哈密顿方程来描述的。该项目的第二部分致力于使用主要由许多人（特别是J. Mather）开发的变分方法来研究这种动态。

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.