Anti-integrability for Three-Dimensional Quadratic Maps
Anti-integrability for Three-Dimensional Quadratic Maps
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DOI：
10.1137/21m1433289
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发表时间：
2021-07
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影响因子：
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通讯作者：
Amanda E Hampton;J. Meiss
中科院分区：
文献类型：
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作者：
Amanda E Hampton;J. Meiss
利用三十年前首次为凝聚态物理的Frenkel-Kontorova模型引入的一个概念：反可积极限，我们研究了三维二次微分同胚的动力学。在传统的人工智能极限下，映射的轨道退化为符号序列，动力学简化为移位算子，这是一种纯粹的混沌形式。在非退化条件下，一个压缩映射引理可以证明无限多个人工智能状态继续沿着确定性映射的轨道运行。对于3D二次映射，我们研究的AI极限是一个二次对应，它的分支，一对一维映射，引入了关于两个符号的符号动力学。然而，AI态是这种对应的非平凡轨道。这些轨道的性质取决于二次曲线是椭圆、双曲线还是一对直线的形式。使用压缩变元，我们找到了每种情况的参数域，使得每个符号序列对应于唯一的AI状态。然后，在一些参数域中，找到使每个这样的AI状态继续远离极限成为原始3D地图的轨道的充分条件。数值延拓方法扩展了这些结果，允许计算分支，并获得具有马蹄形结构和有趣的自相似的轨道。我们猜想，在鞍结点或倍周期分叉中的周期轨道对具有恰好在一个位置不同的符号序列。
We study the dynamics of the three-dimensional quadratic diffeomorphism using a concept first introduced thirty years ago for the Frenkel-Kontorova model of condensed matter physics: the anti-integrable (AI) limit. At the traditional AI limit, orbits of a map degenerate to sequences of symbols and the dynamics is reduced to the shift operator, a pure form of chaos. Under nondegeneracy conditions, a contraction mapping argument can show that infinitely many AI states continue to orbits of the deterministic map. For the 3D quadratic map, the AI limit that we study is a quadratic correspondence whose branches, a pair of one-dimensional maps, introduce symbolic dynamics on two symbols. The AI states, however, are nontrivial orbits of this correspondence. The character of these orbits depends on whether the quadratic takes the form of an ellipse, a hyperbola, or a pair of lines. Using contraction arguments, we find parameter domains for each case such that each symbol sequence corresponds to a unique AI state. In some parameter domains, sufficient conditions are then found for each such AI state to continue away from the limit to become an orbit of the original 3D map. Numerical continuation methods extend these results, allowing computation of bifurcations and obtaining orbits with horseshoe-like structures and intriguing self-similarity. We conjecture that pairs of periodic orbits in saddle-node or period doubling bifurcations have symbol sequences that differ in exactly one position.