  Vertical thermal convection is a non-equilibrium system in which both
buoyancy and shear forces play a role in driving the convective flow. Beyond
the onset of convection, the driven dissipative system exhibits chaotic
dynamics and turbulence. In a three-dimensional domain extended in both the
vertical and the transverse dimensions, Gao et al. (2018) have observed a
variety of convection patterns which are not described by linear stability
analysis. We investigate the fully non-linear dynamics of vertical convection
using a dynamical-systems approach based on the Oberbeck-Boussinesq equations.
We compute the invariant solutions of these equations and the bifurcations that
are responsible for the creation and termination of various branches. We map
out a sequence of local bifurcations from the laminar base state, including
simultaneous bifurcations involving patterned steady states with different
symmetries. This atypical phenomenon of multiple branches simultaneously
bifurcating from a single parent branch is explained by the role of D4
symmetry. In addition, two global bifurcations are identified: first, a
homoclinic cycle from modulated transverse rolls and second, a robust
heteroclinic cycle linking two symmetry-related diamond-roll patterns. These
are confirmed by phase space projections as well as the functional form of the
divergence of the period close to the bifurcation points. The intricacy of this
bifurcation diagram highlights the essential role played by dynamical systems
theory and computation in hydrodynamic configurations.
