广义椭圆型Sitnikov(N+1)体问题的混沌动力学研究  

Chaotic Dynamics of Generalized Elliptical Sitnikov(N+1)-body Problem

作　　者：程旭华 王涌泉 Xu Hua CHENG;Yong Quan WANG(School of Science,Hebei University of Technology,Tianjin 300401,P.R.China)

机构地区：[1]河北工业大学理学院,天津300401

出　　处：《数学学报（中文版）》2023年第1期1-14,共14页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金青年项目(11601257);河北省自然科学基金面上项目(A2019202342)。

摘　　要：本文研究了广义椭圆型Sitnikov(N+1)体问题中存在的混沌行为.首先,在可积哈密顿系统扰动理论的基础上,把广义的椭圆型Sitnikov(N+1)体问题看作是广义圆型Sitnikov(N+1)体问题的扰动;其次,通过计算Melnikov积分函数存在简单零点,证明了广义椭圆型Sitnikov(N+1)体问题中存在横截同宿轨道.然而,由于平衡点的退化性导致了标准的Smale-Birkhoff定理不能直接用来证明系统中存在Smale马蹄.因此,本文在非线性Poincare映射的基础上定义可逆映射f,通过证明f是一个Smale马蹄映射,解析地证明了广义椭圆型Sitnikov(N+1)体问题中存在Smale马蹄意义下的混沌行为.In this paper,the chaotic behavior of the generalized elliptic Sitnikov(N+1)-body problems is analytically studied.First,based on the perturbation method of integrable Hamiltonian systems,the generalized elliptic Sitnikov(N+1)-body problem is regarded as the perturbation of the generalized circular Sitnikov(N+1)-body problem.Then,we prove that the Melnikov integral function has a simple zero,arriving at the existence of transversal homoclinic orbits.Moreover,since the equilibrium point is a degenerate hyperbolic saddle,the standard Smale-Birkhoff theorem cannot be used directly to prove the existence of Smale horseshoes.We alternatively construct an invertible map f and check that f satisfies the Conley-Moser condition,which shows that the generalized elliptic Sitnikov(N+1)-body problem possess chaotic behaviour of Smale horseshoe type.

关 键 词：Sitnikov(N+1)体问题 MELNIKOV方法 横截同宿轨道 Conley-Moser条件 SMALE马蹄 

分 类 号：O175.1[理学—数学]