Saiki Yoshitaka, Yamada Michio
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Jan;79(1 Pt 2):015201. doi: 10.1103/PhysRevE.79.015201. Epub 2009 Jan 5.
It has recently been found in some dynamical systems in fluid dynamics that only a few unstable periodic orbits (UPOs) with low periods can give good approximations to the mean properties of turbulent (chaotic) solutions. By employing three chaotic systems described by ordinary differential equations, we compare time-averaged properties of a set of UPOs and those of a set of segments of chaotic orbits. For every chaotic system we study, the distributions of a time average of a dynamical variable along UPOs with lower and higher periods are similar to each other and the variance of the distribution is small, in contrast with that along chaotic segments. The distribution seems to converge to some limiting distribution with nonzero variance as the period of the UPO increases, although that along chaotic orbits inclines to converge to a delta -like distribution. These properties seem to lie in the background of why only a few UPOs with low periods can give good mean statistical properties in dynamical systems in fluid dynamics.
最近在流体动力学的一些动力系统中发现，只有少数低周期的不稳定周期轨道（UPO）能很好地近似湍流（混沌）解的平均特性。通过使用由常微分方程描述的三个混沌系统，我们比较了一组UPO的时间平均特性和一组混沌轨道段的时间平均特性。对于我们研究的每个混沌系统，一个动力学变量沿低周期和高周期UPO的时间平均分布彼此相似，且分布的方差较小，这与沿混沌段的分布形成对比。随着UPO周期的增加，该分布似乎收敛到某个具有非零方差的极限分布，尽管沿混沌轨道的分布倾向于收敛到类似狄拉克δ函数的分布。这些特性似乎是流体动力学动力系统中只有少数低周期UPO能给出良好平均统计特性的原因所在。
