Nonlinear Dynamics

对于一类广义的二维耗散微分同态族，研究了带有Sinai-Ruelle-Bowen测度的吸引子存在的充分条件。并对这些测度的遍历性质和各自吸引子的几何结构进行了研究。该项目是基于与雅各布森的联合研究。其中一个主要的工具是具有无界导数的二维映射的畸变估计新技术。基本的例子是在有限数量的矩形上定义的一组分段光滑变换，除了一个矩形是抛物线映射外，其他矩形都是双曲映射。这个项目不同于以前的工作，因为没有假设使用一维地图的扰动。该技术是基于一个系统的初始条件，可以数值检查，如膨胀，收缩，初始扭曲，以及这些量对参数的依赖，可以应用于各种模型。这个项目涉及确定性系统中的随机振荡现象。最近发现，这种振荡出现在物理、化学和生物学的各种数学模型中。对于许多科学应用来说，理解导致这种振荡的机制并能够用简单的数学术语来描述它们是很重要的。通常，给定的系统取决于外部参数，如温度、压力或各种电磁量。典型的例子是大气中的气流，外部温度变化下流体的运动，或心脏中血液在各种生理压力下的运动。对于某些参数，系统可能表现出平稳或周期性的解决方案，但是，随着参数的变化，运动可以开始随机行为，并增加概率。这可以在水的沸腾中看到，它最初表现为简单的运动，随着温度的升高而变得动荡。它也发生在心律失常中，通常周期性的系统在某些力的作用下开始剧烈振荡。在某些情况下，人们希望系统表现出随机运动。在另一些地方，人们想要避免这样的运动。无论如何，理解这种运动的本质是很重要的。这项工作的目标是给出暗示随机行为的严格数学条件。这些条件可以在数值上进行检查，这为许多感兴趣的实际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.