Mathematical Sciences: Nonlocal Bifurcations and Strange Attractors

9404199 Afraimovich/Chow根据参数的不同，具有耗散和能量泵浦的真实系统可以表现出缓慢的非本质变化以及动力学行为的突然跳跃。为了在特定的应用系统中解释、描述和预测这类现象，人们需要知道从简单行为到复杂行为的转变的数学分类，以及在参数变化过程中静止(已建立的)运动的演化理论。奇异吸引子的非局部分叉理论是研究耗散系统行为变化的一个适当的数学工具，它允许人们控制并在原则上控制它们的行为。真实系统的静止运动(区域)对应于其数学模型中的吸引子。简单的区域对应于简单的吸引子，而复杂的区域对应于所谓的奇怪吸引子。从开始到复杂行为的数学图像是一个奇怪吸引子上升的分支；复杂行为演化的数学图像是一个奇怪吸引子演化的场景(或一系列分支)。在这项拟议的工作中，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。我们建议将发展的数学方法应用于耦合振荡器、激光系统、电工电路系统等系统。特别地，我们想要研究随机同步问题，即耗散耦合的耗散单个子系统的相似行为。随机同步现象很有趣，例如，对于安全通信问题。它在解释非平衡介质的确定性行为中也起着基础性的作用。我们将用非局部分叉理论和奇异吸引子的语言来描述随机同步的发生机制。从数学角度看，光滑向量场的单参数族参数变化过程中奇异吸引子的出现和演化问题是一个非常重要的问题。对于从应用上研究特定的耗散系统也起到了基础性作用。在所提出的工作中，我们将研究Morse-Spale系统边界上的非局部余维1分支，它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情景(即分支链)。在第一个问题中，我们建议对同宿轨和异宿轨在分叉时刻的行为进行分类，挑出与奇异吸引子出现有关的情况，并用符号动力学的形式描述产生的吸引子。在第二个问题中，我们想要研究奇异吸引子中新的正Lyapunov指数出现的情景，并研究它们与稳定流形和不稳定流形的非横截相交有关的危机。我们将把预期的结果应用于一些具体的系统(耦合振子、激光系统等)，以耗散耦合的单个子系统的形式来研究。我们建议描述这类系统中随机同步现象的发生机制。对于相同的子系统，空间齐次解的稳定性意味着随机同步。对于不同的子系统，需要发展一种基于奇异吸引子分叉的随机同步理论。

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.