Dynamics and Kinetics

PI：Leonid Bunimovich提案编号：0140165 本文研究的是有限维和无限维动力系统以及混合（介于动力系统和随机系统之间）系统，这类混合系统是由随机环境中的确定性游动形成的。最近已经表明，这样的系统（在刚性环境中行走）的一个大子类在1D中是完全可解的。建议的研究涉及变刚度环境的情况。建议的另一部分涉及最近构建的第一个自然例子的哈密顿系统与划分相空间的进一步发展。现在的目标是证明一个相当普遍的类这样的台球在任何（有限）维具有正度量熵。所考虑的统计力学问题处理严格推导公式的动力学系数和逃逸率在开放系统。在处理无限维系统的问题中，有格子动力系统中的混沌阶跃和非线性波动方程中的混沌。其他问题涉及各种系统，从遍历理论（本福德定律）到一些简单的大脑动力学和物流模型。 传统上，真实的系统的数学模型要么是纯确定性的，要么是纯随机的。这两类模型都有丰富的理论，研究人员对它们的行为有很好的直觉，也就是说，他们基本上知道会发生什么。这种直觉本质上是基于丰富的完全可解（即完全理解）模型的集合。然而，大多数真实的系统既不是纯确定性的，也不是纯随机的。相反，它们具有两种（确定性和随机性）特征。本课题研究了在通信理论、统计物理、化学动力学、人工智能理论等领域中独立引入的一大类这类模型。

PI: Leonid BunimovichProposal Number: 0140165 ABSTRACTThe proposed research deals with the studies of finite- andinfinite-dimensional dynamical systems as well as of somehybrid (intermediate between dynamical and stochastic) systems.The class of hybrid systems which will be specifically addressedis formed by deterministic walks in random environments. It has been shownrecently that a large subclass of such systems (walksin rigid environments) is completely solvable in 1D. The proposedstudy deals with the case of environments with variable rigidity.Another part of the proposal deals with the farther development ofrecently constructed first natural examples of Hamiltonian systemswith divided phase space. The goal now is to show that a rather general class of such billiards has positive metric entropy in any (finite) dimension. The considered problems of statistical mechanics deal with the rigorous derivation of formulas for kinetic coefficients and for escape rate in open systems. Among the problems which deal with infinite-dymensional systems are chaos-order transitions in Lattice Dynamical Systems and chaos in nonlinear wave equations. Other problems deal with various systems ranging from the Ergodic Theory (Benford's law)to some simple models of brain dynamics and logistics. Traditionally the mathematical models of real systems are either purely deterministic or purely stochastic. These two classes of models enjoy having a rich theory, and researchers have quite good intuition on their behavior, i.e. they basically know what to expect. This intuition is essentially based on a rich collection of completely solvable (i.e. completely understood) models. However, a majority of real systems are neither purely deterministic nor purely stochastic. Instead they have both (deterministic and stochastic) features. This project deals with a big class of such models which were independently introduced in communication theory, statistical physics,chemical kinetics, theory of artificiall intellect, etc.