Diffusion along transition chains of invariant tori and Aubry–Mather sets
Diffusion along transition chains of invariant tori and Aubry–Mather sets
复制标题
沿不变环面集和奥布里-马瑟集的过渡链的扩散
DOI：
--
复制
发表时间：
2012
影响因子：
0.9
通讯作者：
C. Robinson
中科院分区：
文献类型：
--
作者：
M. Gidea;C. Robinson
摘要我们描述了一个动力系统中扩散轨道存在的拓扑机制，满足以下假设：(i)相空间包含一个常双曲不变流形，与二维环空微分同构；（ii）动力学对环空的限制是一个保持面积的单调扭转映射；（iii）环空包含形成过渡链的不变一维环面序列（即，每个环面的不稳定流形与序列中下一个环面的稳定流形具有拓扑横向相交）；（iv）环面过渡链中散布着共振产生的间隙；(v)在每个间隙内都有一个限定的奥布里-马瑟集集合。在这些假设下，存在沿着过渡链，跨越间隙，并在每个间隙内以任何指定顺序遵循奥布里-马瑟集的轨迹。这一机制与哈密顿系统中的阿诺德扩散问题有关。特别地，我们证明了哈密顿系统大间隙问题中扩散轨迹的存在性。这个论点是拓扑的和建设性的。
Abstract We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional annulus; (ii) the restriction of the dynamics to the annulus is an area preserving monotone twist map; (iii) the annulus contains sequences of invariant one-dimensional tori that form transition chains (i.e., the unstable manifold of each torus has a topologically transverse intersection with the stable manifold of the next torus in the sequence); (iv) the transition chains of tori are interspersed with gaps created by resonances; (v) within each gap there is prescribed a finite collection of Aubry–Mather sets. Under these assumptions, there exist trajectories that follow the transition chains, cross over the gaps, and follow the Aubry–Mather sets within each gap, in any specified order. This mechanism is related to the Arnold diffusion problem in Hamiltonian systems. In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems. The argument is topological and constructive.