Algebraic and Analytic Structures in Dynamics

Rychlik将研究以下问题：非线性差分方程和常微分方程对应于同斜和异斜连接的特殊类解的存在性。一类辛映射的代数积分的存在性。台球系统中周期轨迹的测量。偏微分方程的行波解和呼吸解的存在性。开发和实现处理应用中经常出现的多变量多参数多项式方程组的算法。为这个项目选择的问题通常需要使用代数几何和复杂分析的技术。当前项目的主要目标是表明这些领域的先进技术在解决动力学问题方面是有用的。这项研究的结果将是一套代数和复分析方法，足以应用于其他问题。

Abstract Rychlik Rychlik will investigate the following problems: The existence of special classes of solutions in non-linear difference equations and ordinary differential equations which correspond to homoclinic and heteroclinic connections. The existence of algebraic integrals for a class of symplectic maps. The measure of periodic trajectories in billiard systems. The existence of traveling wave and breather solutions to PDEs. Development and implementation of algorithms for dealing with systems of multi-variate polynomial equations with many parameters that often arise in applications. The problems selected for this project generally invite the use of techniques from algebraic geometry and complex analysis. The main objective of the current project is to show that advanced techniques from these fields are useful in solving problems of dynamics. A result of this research will be a suite of algebraic and complex-analytic methods general enough to be applied to other problems.