Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526, USA.
Chaos. 2015 Sep;25(9):097602. doi: 10.1063/1.4915831.
To characterize transport in a deterministic dynamical system is to compute exit time distributions from regions or transition time distributions between regions in phase space. This paper surveys the considerable progress on this problem over the past thirty years. Primary measures of transport for volume-preserving maps include the exiting and incoming fluxes to a region. For area-preserving maps, transport is impeded by curves formed from invariant manifolds that form partial barriers, e.g., stable and unstable manifolds bounding a resonance zone or cantori, the remnants of destroyed invariant tori. When the map is exact volume preserving, a Lagrangian differential form can be used to reduce the computation of fluxes to finding a difference between the actions of certain key orbits, such as homoclinic orbits to a saddle or to a cantorus. Given a partition of phase space into regions bounded by partial barriers, a Markov tree model of transport explains key observations, such as the algebraic decay of exit and recurrence distributions.
刻画确定性动力系统中的输运特性，就是要计算相空间中区域的出口时间分布或区域之间的转移时间分布。本文综述了过去三十年来在这个问题上取得的相当大的进展。对体积保持映射的主要输运量度包括到一个区域的出口和输入通量。对于面积保持映射，输运受到由不变流形形成的部分障碍物的阻碍，例如，稳定和不稳定流形限制共振区或Cantori，即破坏的不变环面的残余物。当映射是精确的体积保持时，可以使用拉格朗日微分形式来减少通量的计算，从而找到某些关键轨道（如向鞍点或 Cantorus 的同宿轨道）的作用之间的差异。对于相空间的分区，由部分障碍物边界的区域，可以使用马尔可夫树模型来解释关键观测，例如出口和重现分布的代数衰减。
