Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems
Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems
复制标题
慢-快哈密顿系统中分界线交叉域的稳定性岛
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
D. Treschev
中科院分区：
文献类型：
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作者：
A. Vasiliev;A. Neishtadt;C. Simó;D. Treschev
我们考虑一个二自由度哈密顿系统，其中一个自由度对应于快速运动，另一个自由度对应于慢速运动。慢变量和快变量的典型变化速度之比是问题的小参数ɛ。在慢变量的冻结值处，快变量的相平面上存在分界线，并且相空间中存在一个区域（分界线交叉域），在慢变量的演化过程中，相点在快变量平面上的投影反复与分界线相交。在一定的对称性条件下，我们证明了分界线交叉域中存在许多（1/ɛ阶）稳定的周期轨迹。每条轨迹都被一个稳定岛包围，其测量值是从下面以 ɛ 量级的值进行估计的。因此，稳定性岛的总测量值是通过与 ɛ 无关的值从下面估计的。该证明基于对相应庞加莱图的渐近公式的分析。
We consider a two-degrees-of-freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of typical velocities of changes of the slow and fast variables is the small parameter ɛ of the problem. At frozen values of the slow variables, there is a separatrix on the phase plane of the fast variables, and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition, we prove the existence of many (of order 1/ɛ) stable periodic trajectories in the domain of separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order ɛ. So, the total measure of the stability islands is estimated from below by a value independent of ɛ. The proof is based on an analysis of asymptotic formulas for the corresponding Poincaré map.