Boxing-in of a blender in a Hénon-like family

IF 1.3 Q3 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Frontiers in Applied Mathematics and Statistics Pub Date : 2023-03-30 DOI:10.3389/fams.2023.1086240
Stefanie Hittmeyer, B. Krauskopf, H. Osinga, Katsutoshi Shinohara
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引用次数: 1

Abstract

Introduction The extension of the Smale horseshoe construction for diffeomorphisms in the plane to those in spaces of at least dimension three may result in a hyperbolic invariant set referred to as a blender. The defining property of a blender is that it has a stable or unstable invariant manifold that appears to have a dimension larger than expected. In this study, we consider a Hénon-like family in ℝ3, which is the only explicitly given example of a system known to feature a blender in a certain range of a parameter (corresponding to an expansion or contraction rate). More specifically, as part of its hyperbolic set, this family has a pair of saddle fixed points with one-dimensional stable or unstable manifolds. When there is a blender, the closure of these manifolds cannot be avoided by one-dimensional curves coming from an appropriate direction. This property has been checked for the Hénon-like family by the method of computing extremely long pieces of global one-dimensional manifolds to determine the parameter range over which a blender exists. Methods In this study, we take the complimentary and local point of view of constructing an actual three-dimensional box (a parallelopiped) that acts as an outer cover of the hyperbolic set. The successive forward or backward images of this box form a nested sequence of sub-boxes that contains the hyperbolic set, as well as its respective local invariant manifold. Results This constitutes a three-dimensional horseshoe that, in contrast to the idealized affine construction, is quite general and features sub-boxes with curved edges. The initial box is defined in a parameter-dependent way, and this allows us to characterize properties of the hyperbolic set intuitively. Discussion In particular, we trace relevant edges of sub-boxes as a function of the parameter to provide additional geometric insight into when the hyperbolic set may or may not be a blender.
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在一个类似hsamnon的家庭里装一个搅拌机
将平面上的微同态的小马蹄形构造扩展到至少三维空间中的微同态,可以得到一个称为混合的双曲不变量集。混合器的定义性质是它具有稳定或不稳定的不变量流形,其维数似乎比预期的要大。在本研究中,我们考虑了一个hsam非类族,它是已知系统在某一参数范围内(对应于膨胀率或收缩率)具有搅拌器特征的唯一明确给出的例子。更具体地说,作为它的双曲集的一部分,这个族有一对鞍形不动点,它们具有一维稳定或不稳定流形。当存在混合器时,这些流形的闭合不能通过来自适当方向的一维曲线来避免。通过计算极长的全局一维流形来确定搅拌机存在的参数范围的方法,对hsamnon -like族进行了这一性质的检查。方法在本研究中,我们采取互补和局部的观点,构建一个实际的三维盒子(平行六面体),作为双曲集的外层覆盖。该框的连续向前或向后图像形成了包含双曲集及其各自的局部不变流形的嵌套子框序列。结果这构成了一个三维马蹄形,与理想的仿射构造相反,它是相当普遍的,并且具有弯曲边缘的子盒。初始框以参数相关的方式定义,这使我们能够直观地表征双曲集的性质。特别是,我们跟踪子框的相关边作为参数的函数,以提供额外的几何洞察力,当双曲集可能是或可能不是一个混合。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Frontiers in Applied Mathematics and Statistics
Frontiers in Applied Mathematics and Statistics Mathematics-Statistics and Probability
CiteScore
1.90
自引率
7.10%
发文量
117
审稿时长
14 weeks
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