Research in Dynamical Systems

弗兰克斯教授将继续他对地形图动态的研究，特别是保存面积的地形图。本文将讨论二维离散保面积动力系统周期行为的存在性问题。夏教授提出的研究集中在动力系统的三个方面：(1)牛顿n体问题和天体力学的动力学；(2)全球动力学中的双曲性和分叉；(3)哈密顿动力学和辛微分同态。曲面地图是二维变换；对这种变换，特别是面积保持变换的研究，可以追溯到Poincare和G.D.Birkhoff。这一领域的结果在经典力学以及更现代的混沌动力学中有许多应用。在物理方面，夏教授提议的研究是：物体受引力作用的动力学；系统动力学随着其中参数的变化而变化的方式；以及能量守恒的系统动力学。

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.