论文标题
可集成系统中奇异性的隐藏曲线对称性和结构稳定性
Hidden toric symmetry and structural stability of singularities in integrable systems
论文作者
论文摘要
本文的目的是开发一种系统的方法来研究（也许是堕落的）整合系统及其结构稳定性。作为主要工具，我们使用“隐藏”系统保护的圆环动作在单数轨道附近。我们为存在这种行动提供了足够的条件，并表明它们在可整合的扰动下持续存在。我们发现了几个无限串联奇点的曲折对称性，并证明了Kalashnikov的抛物线轨道的结构稳定性，并在现实分析的情况下具有共鸣。我们还将所有汉密尔顿$ k $ -torus的动作分类在符号歧管$ m^{2n} $上（或在其复杂化上）上的单数轨道附近，并证明这些动作的正常形式在小扰动下是持久的。作为副产品，我们证明了Vey定理（1978）的近似版本，介绍了局部象征性正常形式的非排定奇异性。
The goal of the paper is to develop a systematic approach to the study of (perhaps degenerate) singularities of integrable systems and their structural stability. As the main tool, we use "hidden" system-preserving torus actions near singular orbits. We give sufficient conditions for the existence of such actions and show that they are persistent under integrable perturbations. We find toric symmetries for several infinite series of singularities and prove, as an application, structural stability of Kalashnikov's parabolic orbits with resonances in the real-analytic case. We also classify all Hamiltonian $k$-torus actions near a singular orbit on a symplectic manifold $M^{2n}$ (or on its complexification) and prove that the normal forms of these actions are persistent under small perturbations. As a by-product, we prove an equivariant version of the Vey theorem (1978) about local symplectic normal form of nondegenerate singularities.
