Stability transitions for periodic orbits in hamiltonian systems
Stability transitions for periodic orbits in hamiltonian systems
复制标题
哈密顿系统中周期轨道的稳定性跃迁
DOI：
--
复制
发表时间：
1980
期刊：
影响因子：
--
通讯作者：
D. Rod
中科院分区：
文献类型：
--
作者：
R. Churchill;G. Pecelli;D. Rod
本文研究了以能量h为参数的真实的两自由度解析Hamilton系统的单参数周期解族。条件，保证这个家庭将经历无限多的变化，稳定状态的h趋于某个有限值h 0。首先考虑的是哈密顿量在能量h 0处的临界点（特征值±α，±iβ，α和β>0）的情况，其性质是族限制于渐近于该点的同宿轨道。给出了这种情况的一些推广，并应用于Hénon-Heiles Hamilton等例子。我们得到一个无穷序列的不同的能量区间收敛到h 0的周期轨道是椭圆的。然后给出了椭圆轨道稳定性的要求。一个无穷序列的不同的能量区间收敛到h 0的附加条件，其轨道是双曲的，涉及到一个相关的希尔方程的“共存问题”，当相关的庞加莱映射沿着轨道计算时出现在坐标中。结果与Henrard和Devaney研究的临界点具有特征值（±α±iβ），α和β>0的情况进行了比较。
The paper considers one-parameter families of periodic solutions of real analytic Hamiltonian systems with two degrees of freedom, the parameter being the energy h. Conditions are given which guarantee that this family will undergo infinitely many changes in stability status as h tends to some finite value h0. First considered is the case of a critical point (with eigenvalues ±α, ±iβ, α and β>0) of the Hamiltonian at energy h0 with the property that the family limits to a homoclinic orbit asymptotic to this point. Some generalizations of this case are given, and applications are made to examples such as the Hénon-Heiles Hamiltonian. We obtain an infinite sequence of distinct energy intervals converging to h0 on which the periodic orbits are elliptic. Requirements for the elliptic stability of the orbits are then given. The additional conditions for an infinite sequence of distinct energy intervals converging to h0, on which the orbits are hyperbolic, involve the “coexistence problem” for an associated Hill's equation that appears when the relevant Poincaré maps along the orbits are computed in coordinates. The results are compared to the case where the critical point has eigenvalues (±α±iβ), α and β>0, investigated by Henrard and Devaney.