Covariant Lyapunov analysis of solutions of the Navier-Stokes equations

采用协变Lyapunov分析方法，研究了不可压缩流体Navier-Stokes方程下二维平面环面（R/2πZ）2上Kolmogorov流动的混沌态。得到了解的分岔图，进行了传统的Lyapunov分析，发现第一个Lyapunov数在Re/Rc~18处变为正，第二个Lyapunov数在Re/Rc~23处变为正。基于这些数据，我们采用Ginelli et al.（2007）的方法计算了协变Lyapunov向量，通过观察轨道稳定/不稳定空间之间夹角的分布，发现解轨道在混沌跃变后是双曲的，但在Re/Rc~23处变为非双曲的。在双曲/非双曲过渡点，涡度时间相关函数的函数形式由振荡型变为非振荡型。

We studied chaotic states of Kolmogorov flows on a 2D flat torus (R/2πZ)2governed by the Navier-Stokes equations of incompressible fluids, by using the covariant Lyapunov analysis. Obtaining the bifurcation diagram of solutions, we performed the traditional Lyapunov analysis, and found that the first Lyapunov number becomes positive at Re/Rc～18, and the second one does at Re/Rc～23. Based on these data, we calculated the covariant Lyapunov vectors by the method of Ginelli et al. (2007), and found that the solution orbit is hyperbolic just after the chaotic transition, but becomes nonhyperbolic at Re/Rc～23, by observing the distribution of the angle between the stable/unstabletangent spaces of the orbit. At the hyperbolic/nonhyperbolic transition point, we found that the fuctional form of the time correlation function of the vorticity changes from oscillatory to non-oscillatory.