Mathematical Sciences: Nonlocal Bifurcations and Strange Attractors

数学科学：非局部分岔和奇异吸引子

• 批准号: 9404199
• 负责人: Shui-Nee Chow
• 金额: $ 3.93万
• 依托单位: Georgia Tech Research Corporation - GA Tech Research Institute
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1996-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9404199&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlocal; Bifurcations; Strange

9404199 Afraimovich/Chow Depending on parameters, real systems with dissipation and energy pumping can manifest slow nonessential changes as well as abrupt jumps in their dynamical behavior. In order to explain, describe, and predict phenomena of such a kind in a specific applied system, people need to know a mathematical classification of transitions from simple to complex behavior and a theory of evolution of stationary (established) motions during the changes of parameters. The theory of nonlocal bifurcations of strange attractors is an adequate mathematical tool to study changes in conduct of dissipative systems which allows one to control and, in principal, to govern their behavior. Stationary motions (regimes) of real systems correspond to attractors in their mathematical models. Simple regimes correspond to simple attractors while the complex ones correspond to so called strange attractors. Mathematical image of the onset to complex behavior is a bifurcation of a rising of a strange attractor; mathematical image of evolution of complex behavior is a scenarium (or a chain of bifurcations) of evolution of a strange attractor. In the proposed work we are going to study bifurcations leading to appearance of strange attractors and to describe scenaria of their evolution. We propose to apply the developed mathematical technique to such systems as coupled oscillators, laser systems, circuit systems of electrical engineering and others. In particular, we want to study the problem of stochastic synchronization, i.e., similar behavior of dissipatively coupled dissipative individual subsystems. The phenomenon of stochastic synchronization is interesting, for example, for the problem of secure communications. It also plays a fundamental role in the explanation of deterministic behavior of nonequilibrium media. We are going to describe mechanisms of the occurrence of stochastic synchronization in the language of nonlocal bifurcation theory and strange attractors. The problem of appearance and evolution of strange attractors during the changes of parameters in one-parametrical families of smooth vector fields is very important from the mathematical point of view. It also plays a fundamental role in studying of specific dissipative systems from applications. In the proposed work we are going to study the nonlocal codimension one bifurcations on the boundary of the Morse-Smale systems which may lead to the birth of strange attractors and, also scenaria (i.e., chains of bifurcations) which can be responsible for characteristics of strange attractors. In the first problem, we propose to classify behavior of homoclinic and heteroclinic trajectories at the bifurcation moment, single out situations related to appearance of strange attractors and describe the arising attractors in terms of symbolic dynamics. In the second problem, we want to study scenaria of appearance of new positive Lyapunov exponents in strange attractors and investigate their crises which are related to nontransversal intersections of stable and unstable manifolds. We are going to apply expected results to investigate some specific systems (coupled oscillators, laser systems and others) in the form of dissipatively coupled dissipative individual subsystems. We propose to describe mechanisms of occurrence of stochastic synchronization phenomenon in such systems. For identical subsystems, stability of spatially-homogeneous solutions implies stochastic synchronization. For different individual subsystems, a theory of stochastical synchronization based on bifurcations of strange attractors needs to be developed.

9404199 Afraimovich/Chow 根据参数的不同，具有耗散和能量泵送的真实系统可能会表现出缓慢的非本质变化以及动态行为的突然跳跃。 为了解释、描述和预测特定应用系统中的此类现象，人们需要了解从简单行为到复杂行为转变的数学分类以及参数变化期间静止（既定）运动的演化理论。 奇异吸引子的非局部分岔理论是研究耗散系统行为变化的适当数学工具，它允许人们控制并原则上控制它们的行为。 真实系统的静止运动（状态）对应于其数学模型中的吸引子。 简单的体系对应于简单吸引子，而复杂的体系对应于所谓的奇异吸引子。 复杂行为开始的数学形象是奇怪吸引子上升的分叉； 复杂行为进化的数学形象是一个奇怪吸引子进化的场景（或分叉链）。 在拟议的工作中，我们将研究导致奇怪吸引子出现的分叉，并描述它们的演化场景。 我们建议将开发的数学技术应用于耦合振荡器、激光系统、电气工程电路系统等系统。 特别是，我们想研究随机同步问题，即耗散耦合耗散单个子系统的类似行为。 例如，对于安全通信问题，随机同步现象很有趣。 它在解释非平衡介质的确定性行为方面也发挥着基础作用。 我们将用非局部分岔理论和奇怪吸引子的语言来描述随机同步的发生机制。 从数学角度来看，光滑矢量场一参数族中参数变化过程中奇异吸引子的出现和演化问题非常重要。 它还在研究应用中的特定耗散系统方面发挥着基础作用。 在拟议的工作中，我们将研究莫尔斯-斯梅尔系统边界上的非局部余维一分叉，这可能导致奇异吸引子的诞生，以及可能导致奇异吸引子特征的场景（即分叉链）。 在第一个问题中，我们建议对分叉时刻的同宿和异宿轨迹的行为进行分类，挑出与奇怪吸引子出现相关的情况，并用符号动力学来描述出现的吸引子。 在第二个问题中，我们要研究奇怪吸引子中出现新的正李雅普诺夫指数的情况，并研究它们与稳定流形和不稳定流形的非横向交相关的危机。 我们将应用预期结果以耗散耦合耗散单个子系统的形式研究一些特定系统（耦合振荡器、激光系统等）。 我们建议描述此类系统中随机同步现象的发生机制。 对于相同的子系统，空间均匀解的稳定性意味着随机同步。 对于不同的个体子系统，需要开发一种基于奇异吸引子分叉的随机同步理论。