Natural convection in a vertical channel. Part 3. Bifurcations of many (additional) unstable equilibria and periodic orbits

Abstract: Vertical thermal convection system exhibits weak turbulence and spatio-temporally chaotic behavior. 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. (2024) 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 flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects in rolls. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions are discussed; the bifurcation scenarios include Hopf, pitchfork, saddle-node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. Given this large number of unstable orbits, our results may pave the way to quantitatively describing transitional fluid turbulence using periodic orbit theory.