三维二次映射的反可积性
Anti-Integrability for 3-Dimensional Quadratic Maps
AI总结：
本文研究三维二次微分同胚的反可积极限，通过二次对应、符号动力学与压缩论证确定 AI 状态及其延续参数域，并结合数值延拓揭示分岔、类马蹄与自相似轨道。
摘要：
我们利用三十年前为凝聚态物理中的 Frenkel-Kontorova 模型首次引入的概念——反可积（AI）极限，研究三维二次微分同胚的动力学。在传统 AI 极限下，映射的轨道退化为符号序列，动力学被约化为移位算子，即一种纯粹形式的混沌。在非退化条件下，压缩映射论证可以表明，无穷多个 AI 状态会延续为确定性映射的轨道。对于三维二次映射，我们研究的 AI 极限是一种二次对应，其分支是一对一维映射，并引入两个符号上的符号动力学。然而，AI 状态是该对应的非平凡轨道。这些轨道的性质取决于二次曲线呈现为椭圆、双曲线还是一对直线。利用压缩论证，我们为每种情形找到了参数域，使每个符号序列都对应唯一的 AI 状态。在某些参数域中，随后又找到了充分条件，使每个这样的 AI 状态能够离开该极限延续，成为原始三维映射的轨道。数值延拓方法扩展了这些结果，使计算分岔成为可能，并获得具有类马蹄结构和引人入胜的自相似性的轨道。我们猜想，在鞍结或倍周期分岔中成对出现的周期轨道，其符号序列恰好在一个位置上不同。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
