Smooth Dynamical Systems

平滑动力系统

• 批准号: 0405985
• 负责人: Sheldon Newhouse
• 金额: $ 3.5万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-11-15 至 2006-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405985&HistoricalAwards=false
• 关键词: Smooth; Dynamical; Systems

We propose to study the dynamics of smooth diffeomorphisms of low dimensional manifolds. A significant theme here is the study of the abundance of homoclinic tangencies and its effects. It has been known for some time that homoclinic tangencies produce interesting phenomena. They form an obstruction to structural stability, and have a profound influence on bifurcations phenomena. While most of the information we have about homoclinic tangencies may be considered negative in regard to our ability to understand the underlying dynamics, recently we obtained a positive result: generically they lead to topologically transitive sets of maximal Hausdorff dimension. One part of the current proposal involves understanding the relation between these maximal dimension sets and Lebesgue asymptotic measures. These are measures obtained by taking weak limits of the averages of the iterates of measures which are absolutely continuous with respect to Lebesgue measure. Recently we have shown that in many dissipative cases SRB measures on surfaces only exist on uniformly hyperbolic attractors. We wish to extend this to area preserving cases and to make progress on the conjecture that generically in highly smooth systems a non-Anosov area preserving diffeomorphism of a surface has metric entropy zero. Another question we will consider is whether the Henon family has no SRB measure for a residual set of parameters. It is a celebrated result of Benedicks-Carleson and Benedicks-Young that SRB measures exist for a positive Lebesgue measure set of parameters. In addition to the above, we will study various questions on the existence of symbolic extensions in smooth systems. We propose to study aspects of the orbit structure of non-linear smooth dynamical systems. Our particular emphasis will be on three dimensional systems whose orbits return infinitely often to an embedded transverse surface. These systems occur in models which arise in many fields of science from Biology to Physics and even Economics. For instance, they include the problems of general forced oscillations and the Newtoninan motion of three bodies in a plane. Considering the so-called first return map to the embedded surface we are led to study the iterations of smooth transformations (mappings) of a surface to itself. There are special motions called homoclinic motions (first discovered and named by Poincare in the three body problem) which are known to produce a rich, interesting, and complicated orbit structure. In particular, typical homoclinic motions imply the existence of infinitely many unstable periodic orbits and other orbits which behave in an erratic and unpredictable way. It is now known that in many cases such systems can be studied by comparing them to statistical objects such as the random flipping of a weighted coin. This gives rise to so-called "symbolic models" whose orbit structure can be understood. Our research concerns the types of symbolic systems which can occur in modeling various smooth systems. including the estimation of certain numerical quantities called "entropy" and "dimension" which can be used to quantify different levels of complicated motion.

本文研究了低维流形上光滑反同态的动力学性质。 这里的一个重要主题是研究同宿切线的丰度及其影响。人们知道同宿切线产生有趣的现象已经有一段时间了。它们对结构的稳定性形成障碍，并对分叉现象产生深远的影响。虽然大多数的信息，我们有关于同宿切线可能被认为是负面的，我们的能力，以了解潜在的动力学，最近我们得到了一个积极的结果：一般来说，他们导致拓扑传递集的最大豪斯多夫维数。目前的建议的一部分，涉及到了解这些最大尺寸集和勒贝格渐近措施之间的关系。这些措施得到的措施采取弱限制的平均迭代的措施是绝对连续的勒贝格措施。 最近我们证明了在许多耗散情形下，曲面上的SRB测度只存在于一致双曲吸引子上。 我们希望将其扩展到面积保持的情况下，并取得进展的猜想，一般在高度光滑的系统，非Anosov面积保持一个表面的仿射度量熵为零。我们要考虑的另一个问题是， 对于剩余参数集没有SRB测量。Benedicks-Carleson和Benedicks-Young的一个著名结果是：对于正的Lebesgue测度集，SRB测度是存在的. 除此之外，我们还将研究光滑系统中符号扩张存在性的各种问题。 我们建议研究方面的轨道结构的非线性光滑动力系统。 我们将特别强调三维系统，其轨道返回无限经常嵌入横向表面。 这些系统出现在从生物学到物理学甚至经济学的许多科学领域的模型中。例如，它们包括一般的强迫振动和平面上三个物体的牛顿运动问题。 考虑到所谓的第一个返回映射到嵌入的表面，我们导致研究的光滑变换（映射）的曲面本身的迭代。 有一种特殊的运动叫做同宿运动（最早由庞加莱在三体问题中发现并命名），它可以产生丰富、有趣和复杂的轨道结构。 特别是，典型的同宿运动意味着存在无穷多个不稳定的周期轨道和其他轨道，这些轨道的行为是不稳定和不可预测的。 现在我们知道，在许多情况下，这样的系统可以通过将它们与统计对象（如随机投掷一枚加权硬币）进行比较来研究。 这就产生了所谓的“符号模型”，其轨道结构可以理解。 我们的研究涉及的符号系统的类型，可以发生在各种光滑系统建模。 包括被称为“熵”和“维数”的某些数值量的估计，这些数值量可用于量化复杂运动的不同级别。