Mathematical Sciences: Structure of Attractors

本项目的主要目的是在参数空间中推广最近发现的一维吸引子在同宿分叉瞬间所表现出的自相似形式。PI证明了当一维不稳定流形具有非退化同宿切(且特征值满足非共振条件)时，在不稳定流形的闭包的每个点的每个邻域中，(在不稳定流形的闭包中)存在某个不可数连续体集的每个成员的同胚副本。我们的目的是证明，在经历同宿分支的典型的单参数族中，这种极端的局部野性发生在一组正测量的参数上。不稳定流形闭合的次连续体的无处不在和多样性是不稳定流形中任意小折叠模式反复出现的结果。在参数空间中扩展这些模式的出现的主要方法将是有选择地删除参数的小的开放集，对于这些参数，“临界轨道”没有适当地重现，留下一个正的康托集。这一方法已经被许多研究人员成功地用来证明具有良好度量的传递吸引子的存在性。这里的目标更多是拓扑性的，而不是分析性的，估计应该不那么严格。这个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；平面连续体上的扩张同胚；以及不变平面连续体上的旋转动力学。在物理过程的数学模型中，经常出现包含对模型长期行为的描述的“吸引子”。吸引子的结构(拓扑)反映了这一行为的定性属性。在模型是混沌的情况下，吸引子的结构极其复杂。混沌吸引子一般有两种类型：双曲型和非双曲型。人们对双曲吸引子的理解相对较好。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。该项目的成功将使人们首次对非双曲吸引子的拓扑结构有一个连贯的一瞥。

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.