论文标题
机器辅助发现可集成的符号映射
Machine-assisted discovery of integrable symplectic mappings
论文作者
论文摘要
我们提出了一种新的自动化方法，用于查找平面的可集成符号图。这些动力学系统具有与保守量的存在相关的隐藏对称性，即运动积分。该算法的核心思想是基于以下知识：相位空间中可集成系统的演变仅限于较低维度的亚策略。我们将自己限制在多边形运动中，我们分析了单个轨迹的形状，因此成功地将可集成的运动与混乱的情况区分开。例如，我们的方法重新发现了一些著名的McMillan-Suris可集成映射和离散的Painlevé方程。总共提出和分析了100多个新的综合家庭；其中一些是在参数空间中隔离的，其中一些是一个参数（或参数比率）是连续或离散的家族。在本文的末尾，我们建议新发现的地图如何通过引入离散扰动理论的引入一般的2D符号图与一般的2D符号图相关，并提出了一种方法，即如何基于与多边形映射的映射来构建光滑的近乎整合的动力系统。
We present a new automated method for finding integrable symplectic maps of the plane. These dynamical systems possess a hidden symmetry associated with an existence of conserved quantities, i.e. integrals of motion. The core idea of the algorithm is based on the knowledge that the evolution of an integrable system in the phase space is restricted to a lower-dimensional submanifold. Limiting ourselves to polygon invariants of motion, we analyze the shape of individual trajectories thus successfully distinguishing integrable motion from chaotic cases. For example, our method rediscovers some of the famous McMillan-Suris integrable mappings and discrete Painlevé equations. In total, over 100 new integrable families are presented and analyzed; some of them are isolated in the space of parameters, and some of them are families with one parameter (or the ratio of parameters) being continuous or discrete. At the end of the paper, we suggest how newly discovered maps are related to a general 2D symplectic map via an introduction of discrete perturbation theory and propose a method on how to construct smooth near-integrable dynamical systems based on mappings with polygon invariants.
