  Convection in a layer inclined against gravity is a thermally driven
non-equilibrium system, in which both buoyancy and shear forces drive
spatio-temporally complex flow. As a function of the strength of thermal
driving and the angle of inclination, a multitude of convection patterns is
observed in experiments and numerical simulations. Several observed patterns
have been linked to exact invariant states of the fully nonlinear 3D
Oberbeck-Boussinesq equations. These exact equilibria, traveling waves and
periodic orbits reside in state space and, depending on their stability
properties, are transiently visited by the dynamics or act as attractors. To
explain the dependence of observed convection patterns on control parameters,
we study the parameter dependence of the state space structure. Specifically,
we identify the bifurcations that modify the existence, stability and
connectivity of invariant states. We numerically continue exact invariant
states underlying spatially periodic convection patterns at Pr=1.07 under
changing control parameters for temperature difference between the walls and
inclination angle. The resulting state branches cover various inclinations from
horizontal layer convection to vertical layer convection and beyond. The
collection of all computed branches represents an extensive bifurcation network
connecting 16 different invariant states across control parameter values.
Individual bifurcation structures are discussed in detail and related to the
observed complex dynamics of individual convection patterns. Together, the
bifurcations and associated state branches indicate at what control parameter
values which invariant states coexist. This provides a nonlinear framework to
explain the multitude of complex flow dynamics arising in inclined layer
convection.
