Research in Dynamical Systems

弗兰克斯教授将继续他对地表地图动力学的研究，特别是对保留区域地图的研究。本文将讨论保面积二维离散动力系统中周期行为的存在性问题。夏教授的研究方向主要集中在动力学系统的三个方面：(1)牛顿n体问题的动力学和天体力学；(2)全球动力学的双曲性和分岔性；(3)哈密顿动力学与辛微分同态。表面映射是二维变换；对这种变换的研究，特别是对区域保存变换的研究，可以追溯到庞加莱和g.d.伯克霍夫。这一领域的结果在经典力学和更现代的混沌动力学中有许多应用。在物理方面，夏教授提出的研究方向是：物体受引力作用的动力学；系统的动力学随着参数的变化而变化；以及能量守恒系统的动力学。

Professor Franks will continue his investigation of the dynamics of surface maps, especially area preserving ones. Questions concerning the existence of periodic behavior in area preserving two-dimensional discrete dynamical systems will be addressed. Professor Xia's proposed research concentrates on the following three aspects of dynamical systems: (1) the dynamics of the Newtonian n-body problem and celestial mechanics; (2) hyperbolicity and bifurcations in global dynamics; and (3) Hamiltonian dynamics and symplectic diffeomorphisms. Surface maps are two-dimensional transformations; the study of such transformations, especially area preserving ones, has a long history going back to Poincare and G. D. Birkhoff. There are numerous applications of results in this area to classical mechanics as well as to more modern chaotic dynamics. In physical terms, Professor Xia's proposed research is on: the dynamics of bodies being acted on by gravitation; the way the dynamics of a system changes as parameters in it vary; and the dynamics of systems in which energy is conserved.