Mathematical Sciences: Dynamics and Kinetics of Spatially Extended Systems

本文研究的是空间扩展系统的动力学问题。所研究的系统包括非线性双曲偏微分方程、晶格动力系统（特别是耦合映射晶格）、晶格气体细胞自动机和相互作用粒子系统。将研究模式形成问题以及有序-混沌转换问题。将要解决的大多数问题都起源于物理学和流体动力学，并在过去的十五年中由首席研究员和合作者转化为数学形式。我们将采用一种广泛的方法，结合遍历理论、非均匀双曲动力系统理论、无限维动力系统稳定性理论和分岔理论。最近在理解随时间演化的系统（动力系统）的复杂（混沌）动力学方面的突破与非扩展（点）动力系统的研究有关，这些系统的动力学可以通过在实（物理）空间的单点进行的测量来完全描述。扩展系统的演化，如流体和气体的流动，电子中相互作用元素的大型系统（如约瑟夫森结）需要同时在空间和时间上进行研究。本研究的目的是寻求适当的方法来研究这种时空演化，并开发算法来评估表征时空动力学的数量。另一个新奇之处是对具有随机和确定性成分的动力学系统的研究（但不是确定性系统的小的随机扰动，反之亦然）。应用中的大多数系统都是这种类型的，例如化学，人工智能，仓库管理等。

Abstract Bunimovich The proposed research deals with the dynamics of spatially extended systems. The systems under study include nonlinear hyperbolic partial differential equations, lattice dynamical systems (in particular, coupled map lattices), lattice gas cellular automata and the systems of interacting particles. The pattern formation problem will be studied as well as order-chaos transitions. Most of the questions that will be addressed have their origin in physics and hydrodynamics, and were transformed into the mathematical form by the principal investigator with collaborators during the last fifteen years. A broad approach will be used that combines methods of the ergodic theory, the theory of nonuniformly hyperbolic dynamical systems, stability theory of infinite dimensional dynamical systemq and the theory of bifurcations. The recent breakthrough in the understanding of a complex (chaotic) dynamics of systems evolving in time (dynamical systems) was connected with the studies of nonextended (point) dynamical systems whose dynamics can be completely described by measurements performed at a single point of a real (physical) space. Evolution of extended systems such as the flow of fluids and gases, large systems of interacting elements in electronics (e.g. Josephson junctions) needs to be studied in space and time simultaneously. The purpose of this research is to pursue the adequate approaches to study such space-time evolution and to develop algorithms to evaluate quantities that characterize space-time dynamics. The other novelty is the study of systems with dynamics which have both random and deterministic components (but not small, random perturbations of deterministic systems or vise versa). Most of systems in applications are of this type, e.g. in chemistry, artificial intelligence, in the management of warehouses etc.