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
沿着不变 Tori 和 Aubry-mather 集的转移链的扩散
DOI：
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发表时间：
2011
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影响因子：
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通讯作者：
Aubry
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作者：
Aubry
我们描述了动力系统中扩散轨道存在的一种拓扑机制，满足以下假设：（i）相空间包含一个与二维环同构的常双曲不变流形，（ii）动力学对环的约束是一个保面积单调扭曲映射，（iii）环包含形成过渡链的不变环面序列，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交叉，（iv）环面的过渡链散布有由共振产生的间隙，（v）在每个间隙内存在指定的Aubry-Mather集的有限集合。在这些假设下，存在沿着过渡链、穿过间隙并在每个间隙内以任何规定的顺序遵循Aubry-Mather集的轨迹。这一机制与哈密顿系统中的Arnold扩散问题有关。特别地，我们证明了哈密顿系统的大间隙问题中扩散轨道的存在性。这个论证是拓扑性的和建设性的。
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 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 a designated, 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 prescribed order. This mechanism is related to the Arnold diffusion problem in Hamiltonian system. In particular, we prove the existence of diffusing trajectories in the large gap problem of Hamiltonian systems. The argument is topological and constructive.