标题： 相对周期轨道构成了湍流管道流动的骨干 显示英文标题

标题： Relative periodic orbits form the backbone of turbulent pipe flow

摘要： 低维系统的混沌动力学，如Lorenz或Rössler流动，由其奇异吸引子中嵌入的无限周期轨道所引导。 Navier--Stokes方程的无限维动力学是否也如此，长期以来一直存在猜测，并且是持续研究的主题。 周期性和相对周期解已被证明参与了向湍流的转变。 它们对湍流动力学的相关性——特别是周期轨道在高维非线性系统（如Navier--Stokes方程）中是否起到与低维系统相同的作用——是当前研究的重点。 我们在这里对能量和平均耗散接近湍流值的管道流动相对周期轨道进行了详细研究。 我们概述了几种减少系统平移对称性的方法。 我们在一个最小的计算单元中研究了管道流动，并报告了一个通过切片方法找到的不变解库。 对这些解中一部分的不稳定流形的详细研究与相对周期轨道嵌入在混沌鞍中并引导湍流动力学的图景一致。 显示英文摘要

摘要： Chaotic dynamics of low-dimensional systems, such as Lorenz or R\"ossler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this also be the case for the infinite-dimensional dynamics of Navier--Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics---specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier--Stokes equations as they do in lower-dimensional systems---is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.