Lorenz System and Interfacial Dynamics

中文（简体） 陈将研究两个领域的问题：洛伦兹系统的混沌行为和材料科学中的界面现象。 对于第一个领域，陈试图获得一个完整的分岔图的所有参数值的系统的混沌行为，并显示存在一个受约束的洛伦兹吸引子。 将使用的主要工具是射击参数，同伦理论，分歧理论和混沌理论的地图。 第二个领域的目标是研究某些描述超曲面几何运动的自由边界问题，分析某些发展方程的渐近行为，并将自由边界问题的解与发展方程的渐近行为联系起来。 洛伦兹系统是最早的例子之一，刺激了混沌理论的形成和发展，后来成为许多动力系统和/或混沌理论教科书中的典型例子。 然而，除了大量的数值实验，很少有严格的分析，这个系统。 这个项目的目标就是提供这样一个坚实的基础。 界面动力学的相关研究在材料科学中具有特别重要的意义。

Abstract Chen Chen will investigate problems in two areas: chaotic behavior of the Lorenz system and the interfacial phenomena in material science. For the first area Chen seeks to obtain a complete bifurcation diagram of the chaotic behaviors of the system for all the parameter values and to show the existence of a conjectured Lorenz attractor. The main tools to be used will be shooting arguments, homotopy theory, bifurcation theory, and chaos theory on maps. For the second area the goal is to study certain free boundary problems which describe geometric motions of hypersurfaces, to analyze the asymptotic behavior of certain evolution equations, and to connect the solutions of free boundary problems with the asymptotic behaviors of the evolution equations. The Lorenz system was one of the very first examples that stimulated the formation and development of the theory of chaos and later on became a typical example in many text books of dynamical systems and/or chaos theory. Nevertheless, except for numerous numerical experiments, there are few rigorous analyses for this system. The object of this project is to provide such a rigorous foundation. The related study of interfacial dynamics is of particular importance in material science.