Nonlinear Dynamics

对于一个广泛的2维耗散类同态的家庭，吸引子携带Sinai-Ruelle-Bowen措施存在的充分条件将被调查。 遍历性的研究，这些措施和各自的吸引子的几何结构也提出了。 该项目是基于与M.雅各布森 主要工具之一是新技术的失真估计的二维地图无界衍生物。 基本的例子是一个家庭的分段光滑变换定义的有限数量的矩形，所有映射双曲线，除了一个，这是映射抛物线。这个项目不同于早期的工作，因为没有假设使用一维映射的扰动。 该技术是基于一个系统的初始条件，可以进行数值检查，如膨胀，收缩，初始扭曲，这些量的参数的依赖性，并可以应用于各种模型。该项目涉及确定性系统中的随机振荡现象。 最近发现，这种类型的振荡出现在物理学、化学和生物学中的各种数学模型中。 对于许多科学应用，重要的是要了解导致这种振荡的机制，并能够用简单的数学术语描述它们。 通常，给定的系统将取决于外部参数，如温度，压力或各种电磁量。典型的例子是大气中的气流，外部温度变化下的流体运动，或各种生理压力下心脏中血液的运动。对于某些参数，系统可能会表现出稳态或周期解，但是，随着参数的变化，运动可能会以越来越大的概率开始随机行为。 这在水的沸腾中可以看到，最初显示简单的运动，随着温度的升高而变得湍流。 它也发生在心律失常的心脏，其中一个系统，这是正常的周期性开始疯狂振荡下一定的力量。 在某些情况下，人们希望系统表现出随机运动。 在其他情况下，人们希望避免这种运动。 在任何情况下，重要的是要了解这种运动的性质。 这项工作的目标是给严格的数学条件，意味着随机行为。 这些条件可以用数值方法来检验，这为许多感兴趣的真实的系统提供了直接的应用。

For a broad class of families of 2-dimensional dissipative diffeomorphisms, sufficient conditions for the existence of attractors carrying Sinai-Ruelle-Bowen measures will be investigated. The study of ergodic properties of these measures and of geometric structure of the respective attractors is also proposed. The project is based on joint research with M. Jakobson. One of the main tools is the new technique of distortion estimates for two-dimensional maps with unbounded derivatives. The basic example is a family of piecewise smooth transformations defined on a finite number of rectangles which are all mapped hyperbolically except for one, which is mapped parabolically. This project differs from earlier work in that there is no assumption that perturbations of one-dimensional maps are used. The technique is based on a system of initial conditions which can be numerically checked, such as expansion, contraction, initial distortions, and dependence of these quantities on the parameters and can be applied to a variety of models. This project involves the phenomenon of random oscillations in deterministic systems. It has been recently discovered that this type of oscillation arises in a variety of mathematical models in physics, chemistry, and biology. For many applications to science, it is important to understand the mechanisms leading to such oscillations and to be able to describe them in simple mathematical terms. Usually a given system will depend on external parameters such as temperature, pressure, or various electromagnetic quantities. Typical examples are airflow in the atmosphere, the motion of fluids under changes in external temperature, or the motion of the blood in one's heart under various physiological stresses. For some parameters, systems may exhibit stationary or periodic solutions, but, as the parameters change, the motion can start to behave randomly with increasing probability. This is seen in the boiling of water which initially shows simple motion and becomes turbulent with increasing temperature. It also occurs in arrhythmia of the heart, where a system which is normally periodic starts to oscillate wildly under certain forces. In some cases, one wants to have systems show random motion. In others, one wants to avoid such motion. In any case, it is important to understand the nature of such motion. The goal of this work is to give strict mathematical conditions which imply random behavior. These conditions can be checked numerically, which provides straightforward applications to many real systems of interest.