Southwest Regional Workshop on New Directions in Dynamical Systems

9619881该奖项是为了支持北得克萨斯大学数学系的动力系统会议/研讨会。研讨会的演讲者包括Katok教授(宾夕法尼亚州立大学)、Keane教授(CWI，阿姆斯特丹)、Przytycki教授(IM PAN，华沙)和Strein教授(阿姆斯特丹大学)。研讨会的参与者将包括来自美国西南部和墨西哥北部的学者。动力系统理论描述了随着时间的推移而发展的过程。特别是，动力系统出现在物理学(洛伦茨图)、生物学和环境研究(种群动力学)和化学(例如，对Belousov-Zhabotinskii反应建模)中。系统的发展取决于它的初始状态和来自“环境”的参数。非常重要的是，我们要了解系统在其初始状态的“大多数”时可能表现出的未来行为；这与本项目第一部分致力于的所谓吸引子的描述相对应。另外，在建议的第一部分，我们研究了一些重要的动力系统的性质如何依赖于“环境”参数。有时，一个系统表现出的现象是相互关联的，关于其中一些现象的信息允许人们对其他现象的存在做出判断；换句话说，现象是共存的。这里的一个很好的例子是周期性，即系统中相同状态的循环发生。结果表明，对于某一初始状态，给定周期的循环过程的存在保证了对于不同的初始状态和相同的环境，可以实现另一个不同周期的循环过程。在项目的第二部分，我们计划深入研究同一系统中不同周期过程之间的这种关系。

9619881 This award is for support of a conference/workshop in dynamical systems at the Department of Mathematics of the University of North Texas. Speakers at the workshop will include Professors Katok (Penn State), Keane (CWI, Amsterdam), Przytycki (IM PAN, Warsaw) and Strein (University of Amsterdam). Participants in the workshop will include scholars form the Southwestern US and Northern Mexico. Dynamical systems theory describes processes that develop over time. In particular, dynamical systems arise in physics (Lorenz map), biology and environmental studies (population dynamics) and chemistry (e.g. modeling the Belousov-Zhabotinskii reaction). The development of the system depends on its initial state and parameters coming from the "environment." It is of great importance that we understand the future behavior which may be exhibited by the system for the "majority" of its initial states; this corresponds to the description of so-called attractors to which the first part of the project is devoted. Also in the first part of the proposal we study how properties of some important dynamical systems depend on "environmental" parameters. Sometimes phenomena exhibited by a system are related to one another and information about some of them allows one to make a judgment about the existence of others; in other words phenomena coexist. A good example here is periodicity, i.e. cyclic occurrence of the same states in the system. It turns out that the existence of a cyclic process with a given period for some initial state guarantees that another cyclic process with a different period can be realized for a different initial state and the same environment. In the second part of the project we plan to thoroughly study this relationship between different periods of cyclic processes in the same system.