Sinai-Ruelle-Bowen Measures for Non-Hyperbolic Attractors

非双曲吸引子的 Sinai-Ruelle-Bowen 测度

• 批准号: 9803635
• 负责人: Michael Jakobson
• 金额: $ 4.68万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-15 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803635&HistoricalAwards=false
• 关键词: Sinai; Ruelle; Bowen; Measures; Non

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 proposed. The project is based on joint research with S. Newhouse. 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. No underlying one-dimensional maps are assumed. 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 investigation of the phenomenon of random oscillations in deterministic systems. This type of oscillation arises in a variety of mathematical models in physics, chemistry, and biology. The behavior of such systems depends on exterior parameters. For some parameters, systems exhibit stationary or periodic solutions. However for other parameters, the same systems behave randomly. More precisely, random solutions may appear with higher probability than periodic ones. The goal of this work is to give strict mathematical conditions which imply random behavior. Sufficient conditions can be checked numerically, which provides straightforward applications to the real systems.

对于一大类二维耗散微分同胚族，将研究携带 Sinai-Ruelle-Bowen 测度的吸引子存在的充分条件。 提出了对这些测量的遍历特性和各个吸引子的几何结构的研究。 该项目基于与 S. Newhouse 的联合研究。 主要工具之一是对具有无界导数的二维地图进行失真估计的新技术。 基本示例是在有限数量的矩形上定义的一系列分段平滑变换，除了一个以抛物线映射之外，所有矩形都以双曲线映射。 不假设底层一维地图。 该技术基于初始条件系统，可以进行数值检查，例如膨胀、收缩、初始变形以及这些量对参数的依赖性，并且可以应用于各种模型。 该项目涉及确定性系统中随机振荡现象的研究。 这种类型的振荡出现在物理、化学和生物学的各种数学模型中。 此类系统的行为取决于外部参数。 对于某些参数，系统呈现稳态或周期解。然而，对于其他参数，相同的系统表现是随机的。 更准确地说，随机解出现的概率可能高于周期性解。 这项工作的目标是给出暗示随机行为的严格数学条件。 可以通过数字方式检查充分条件，这为实际系统提供了直接的应用。