Lerman L M, Trifonov K N
National Research University Higher School of Economics, 25/12 Bolshaya Pecherskaya St., Nizhny Novgorod 603155, Russia.
Chaos. 2021 Feb;31(2):023113. doi: 10.1063/5.0035534.
An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same level of a Hamiltonian, and two non-symmetric heteroclinic orbits permuted by the involution. This is a codimension one structure; therefore, it can be met generally in one-parameter families of reversible Hamiltonian systems. There exist two possible types of such connections depending on how the involution acts near the equilibrium. We prove a series of theorems that show a chaotic behavior of the system and those in its unfoldings, in particular, the existence of countable sets of transverse homoclinic orbits to the saddle periodic orbit in the critical level, transverse heteroclinic connections involving a pair of saddle periodic orbits, families of elliptic periodic orbits, homoclinic tangencies, families of homoclinic orbits to saddle-centers in the unfolding, etc. As a by-product, we get a criterion of the existence of homoclinic orbits to a saddle-center.
研究了一个具有两个自由度的解析可逆哈密顿系统，该系统在其对称异宿连接的邻域内，该对称异宿连接由一个对称鞍点中心、位于哈密顿量同一水平面上的一个对称可定向鞍周期轨道以及两条由对合变换置换的非对称异宿轨道组成。这是一个余维数为一的结构；因此，它通常可以在可逆哈密顿系统的单参数族中出现。根据对合在平衡点附近的作用方式，存在两种可能类型的这种连接。我们证明了一系列定理，这些定理表明了该系统及其展开中的混沌行为，特别是在临界水平上鞍周期轨道的可数横向同宿轨道集的存在性、涉及一对鞍周期轨道的横向异宿连接、椭圆周期轨道族、同宿相切、展开中鞍点中心的同宿轨道族等。作为一个副产品，我们得到了一个鞍点中心同宿轨道存在性的判据。
