Title: Natural convection in a vertical channel. Part 3. Bifurcations of many (additional) unstable periodic orbits and their dynamical relevance Show Chinese title

Title: 自然对流在垂直通道中。 第三部分。 许多（额外的）不稳定周期轨道的分岔及其动力学意义

Abstract: Vertical thermal convection system exhibits weak turbulence and spatio-temporally chaotic behaviour. In this system, we report seven equilibria and 26 periodic orbits, all new and linearly unstable. These orbits, together with four previously studied in Zheng et al. (2024b) bring the number of periodic orbit branches computed so far to 30, all solutions to the fully non-linear three-dimensional Navier--Stokes equations. These new invariant solutions capture intricate spatio-temporal flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects in rolls. These interesting and important fluid mechanical processes in a small flow unit are shown to appear locally and instantaneously in a chaotic simulation in a large domain. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions is discussed; the bifurcation scenarios include Hopf, pitchfork, saddle--node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. These orbits are shown to be able to reconstruct statistically the core part of the attractor, and these results may pave the way to quantitatively describing transitional fluid turbulence using periodic orbit theory. Show Chinese abstract

Abstract: 垂直热对流系统表现出弱湍流和时空混沌行为。 在这个系统中，我们报告了七个平衡态和26个周期轨道，这些都是新的且线性不稳定。 这些轨道，加上Zheng等人在(2024b)中之前研究的四个，使得目前已计算的周期轨道分支数量达到30个，所有解都是完全非线性的三维Navier--Stokes方程的解。 这些新的不变解捕捉到了包括直线、斜向、波状、倾斜和扭曲的对流环以及环中的爆发和缺陷在内的复杂时空流动模式。 这些在小流动单元中有趣且重要的流体力学过程被发现在大域的混沌模拟中局部且瞬时出现。 大部分解分支显示出丰富的空间和/或时空对称性。 讨论了这些解的分岔理论组织；分岔情景包括霍普夫分岔、叉式分岔、鞍点-节点分岔、周期倍增、周期减半、全局同宿和异宿分岔以及孤立分支。 这些轨道被证明能够统计地重建吸引子的核心部分，这些结果可能为使用周期轨道理论定量描述过渡流体湍流铺平道路。