Mathematical Sciences: Structure of Attractors

摘要：本课题的主要目标是在参数空间中扩展最近发现的由一维吸引子在同斜分岔瞬间显示的自相似形式。π证明了当一个一维不稳定流形具有非简并同斜切线（且特征值满足非共振条件）时，在该不稳定流形闭包的每一点的每一个邻域内，（在该不稳定流形的闭包中）存在某一不可数连续集合的每一个成员的同纯拷贝。目的是证明，在发生同斜分岔的典型单参数族中，对于一组正测度参数会出现这种极端局部野性。不稳定流形闭合次连续体的普遍性和多样性是不稳定流形中任意小褶皱模式重复出现的结果。扩展这些模式在参数空间中出现的主要方法将是选择性地去除“临界轨道”不适当重复出现的小开放参数集，留下康托尔正测度集。该方法已被许多研究者成功地用于证明具有良好测度的传递吸引子的存在性。这里的目标更偏向于拓扑学，而不是分析性的，并且估计应该不那么精确。该项目还包括正在进行的其他三个主题的工作：与一维和二维马尔可夫空间相关的代数不变量；平面连续体上的扩展同胚以及不变连续平面上的旋转动力学。在物理过程的数学模型中，经常出现包含模型长期行为描述的“吸引子”。吸引子的结构（拓扑）反映了这种行为的定性性质。当模型为混沌时，吸引子的结构非常复杂。混沌吸引子有两种一般类型：双曲型和非双曲型。双曲吸引子相对来说比较容易理解。理解非双曲系统的技术的发展可能是当今动力学中最大的挑战。这个项目的成功将提供对非双曲吸引子拓扑结构的第一次连贯的一瞥。

Abstract Barge The main goal of this project is to extend, in parameter space, a recently discovered form of self-similarity displayed by a one-dimensional attractor at the instant of homoclinic bifurcation. The PI has proved that when a one- dimensional unstable manifold has a nondegenerate homoclinic tangency (and the eigenvalues satisfy a nonresonance condition) then in every neighborhood of every point of the closure of the unstable manifold, there is (in the closure of the unstable manifold) a homeomorphic copy of every member of a certain uncountable collection of continua. The goal is to prove that, in typical one-parameter families undergoing homoclinic bifurcation, this extreme local wildness occurs for a set of parameters of positive measure. The ubiquity and diversity of the subcontinua of the closure of the unstable manifold is a result of the recurrence of arbitrary patterns of small folds in the unstable manifold. The main approach to extending the occurence of these patterns in parameter space will be to selectively excise small open sets of parameters for which the "critical orbit" does not recur appropriately, leaving a Cantor set of positive measure. This approach has been used successfully by a number of researchers to establish the existence of transitive attractors with nice measures. The goal here is more topological, less analytical, and the estimates should be less exacting. This project also includes ongoing work on three other topics: algebraic invariants associated with one- and two-dimensional Markov spaces; expansive homeomorphisms on plane continua; and rotational dynamics on invariant plane continua. In mathematical models of physical processes, there frequently occur "attractors" that contain the discription of the long term behavior of the model. The structure (topology) of the attractor reflects qualitative properties of this behavior. In case the model is chaotic, the structure of the attractor is extremely complicated. Chaotic attractors are of two general types: hyperbolic and non-hyperbolic. The hyperbolic attractors are relatively well understood. The developement of techniques for understanding non-hyperbolic systems is perhaps the biggest challenge in dynamics today. Success in this project would provide the first coherent glimpse into the topological structure of non-hyperbolic attractors.