TOPOLOGICAL METHODS IN THE INSTABILITY PROBLEM OF HAMILTONIAN SYSTEMS
TOPOLOGICAL METHODS IN THE INSTABILITY PROBLEM OF HAMILTONIAN SYSTEMS
复制标题
解决哈密尔顿系统不稳定性问题的拓扑方法
DOI：
10.3934/dcds.2006.14.295
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发表时间：
2005
影响因子：
1.1
通讯作者：
R. Llave
中科院分区：
文献类型：
--
作者：
M. Gidea;R. Llave
我们使用拓扑学方法研究了最近提出的一些
哈密顿不稳定性(Arnol‘d扩散)机制
系统。
在这些机制中，异位连接链
胡须环面的构造是基于
正常双曲流形$\Lambda$，因此：(A)流形
传递圆环相当密集地覆盖了$\Lambda$(可能
不同拓扑)，(B)流形$W^\S_\Lambda$，
$W^\u_\λ$横交，(C)系统满足
一些显式的非简并假设，它们普遍成立。
在本文中，我们使用正确对齐窗口的方法来
证明在假设(A)、(B)、(C)下，存在轨道
这是一个相当大的变动。
事实上，这里给出的方法不需要
圆环是完全不变的，只是它们是
近似不变的。因此，与之前的
论文中，我们不需要使用KAM理论。这降低了
关于可微性的假设。
此外，这里介绍的方法允许我们生产混凝土
对搬家时间的估计，没有在
以前的论文。
We use topological methods to investigate some recently proposed
mechanisms of instability (Arnol'd diffusion) in Hamiltonian
systems.
In these mechanisms, chains of heteroclinic connections between
whiskered tori are constructed, based on the existence of a
normally hyperbolic manifold $\Lambda$, so that: (a) the manifold
$\Lambda$ is covered rather densely by transitive tori (possibly
of different topology), (b) the manifolds $W^\s_\Lambda$,
$W^\u_\Lambda$ intersect transversally, (c) the systems satisfies
some explicit non-degeneracy assumptions, which hold generically.
In this paper we use the method of correctly aligned windows to
show that, under the assumptions (a), (b), (c), there are orbits
that move a significant amount.
As a matter of fact, the method presented here does not require
that the tori are exactly invariant, only that they are
approximately invariant. Hence, compared with the previous
papers, we do not need to use KAM theory. This lowers the
assumptions on differentiability.
Also, the method presented here allows us to produce concrete
estimates on the time to move, which were not considered in the
previous papers.