Nonlinear Dynamics

点击翻译按钮获取中文摘要

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.