Nonlinear Dynamics

对于一类广泛的二维耗散微分同胚族，研究了吸引子具有Sinai-Ruelle-Bowen测度的充分条件。研究了这些测度的遍历性质和相应吸引子的几何结构。该项目是基于与M·雅各布森的联合研究。其中一个主要工具是对具有无界导数的二维映射的失真估计的新技术。基本的例子是定义在有限数量的矩形上的一族分段光滑变换，除了一个以抛物线映射的矩形外，所有这些矩形都是双曲线映射的。这个项目与以前的工作不同，因为没有假设使用一维映射的扰动。该技术基于一个初始条件的系统，这些初始条件可以通过数值进行检查，例如膨胀、收缩、初始变形以及这些量对参数的依赖，并且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，在物理、化学和生物学中的各种数学模型中都会出现这种类型的振荡。对于许多科学应用来说，重要的是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。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。典型的例子有大气中的气流，在外界温度变化下的液体运动，或在各种生理压力下心脏内的血液运动。对于某些参数，系统可能会表现出平稳或周期解，但随着参数的变化，运动可以开始以增加的概率随机行为。这在沸腾的水中可以看到，它最初表现为简单的运动，随着温度的升高而变得湍急。它也会出现在心脏的心律失常中，通常是周期性的系统在一定的力的作用下开始疯狂地振荡。在某些情况下，人们希望系统显示随机运动。在另一些国家，人们希望避免这样的动议。无论如何，理解这类动议的性质是很重要的。这项工作的目标是给出严格的数学条件，暗示随机行为。这些条件可以用数字进行检查，这为许多感兴趣的真实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.