Local and global bifurcations to large-scale oblique patterns in inclined layer convection

In the inclined layer convection system, thermal convection in a Rayleigh--B\'enard cell tilted against gravity, the flow is subject to competing buoyancy and shear forces. For varying inclination angle ($\gamma$) and Rayleigh number ($Ra$), a variety of spatio-temporal patterns is observed. We investigate the switching diamond panes (SDP) pattern, observed at $(\gamma, Ra)\simeq(100\degree,10000)$, which exhibits large-scale oblique features and is one of the five complex tertiary patterns at Prandtl number $Pr=1.07$. First, we study the linear instability of the secondary-state transverse convection rolls and the five branches including two travelling waves and three periodic orbits, bifurcating simultaneously from it. These non-generic bifurcations arise from the breaking of the spatial $D_4$ and $O(2)$ symmetries of transverse rolls, and the resulting bifurcated solutions show large-scale diamond-shaped amplitude modulations. Second, we explore a periodic orbit that captures both the large-scale structure and small-scale defects of modulated rolls. Parametric continuation in $Ra$ reveals the global homoclinic bifurcation via which this periodic orbit emerges. Third, the boundary of the basins of attraction of two dynamically relevant periodic orbits has been characterised. Specifically, additional steady and time-periodic solutions are identified on the basin boundary and their bifurcation structures are analysed. Together, using non-linear invariant solutions and their bifurcations, we take a further step toward understanding the emergence and dynamics of SDP far from the onset of convection, where linear methods have not been applied successfully.