Publication
Pattern formation is ubiquitous in nature and is often governed by complex interactions across a multitude of disciplines. Understanding the origin and the underlying mechanisms of many of these natural patterns remains challenging. Similar patterns also emerge in many physically relevant systems of practical interest. These systems are governed by known partial differential equations, such as various wall-bounded flows. To characterise patterns in these systems, theoretical and numerical analysis can be developed, and mathematical tools like equivariant bifurcation theory may offer additional insights if the pattern exhibits specific symmetries. In this thesis, (inclined) thermal convection is studied, which is a non-linear dynamical system exhibiting a variety of complex flow patterns above the onset of linear instability. We numerically identify unstable invariant solutions capturing spatio-temporal flow patterns observed in simulations and experiments, and continue these solutions in the parameter space to reveal their bifurcation-theoretic origins. Specifically, 3 different configurations in terms of the inclination angle are studied in the thesis: Rayleigh-Bénard convection where the laminar shear force is zero; vertical convection where the laminar shear force is the maximum; and beyond where the convection cell is actually heated from above. Intricate patterns are observed in each case. Linear instability in convection systems leads to straight convection rolls that have the symmetry groups Dn and O(2). These symmetries have particular properties, and may dictate possible bifurcation scenarios and consequently possible patterns. Among these symmetry groups, special emphasis will be given to the group D4, and in particular pitchfork and Hopf bifurcations from D4-symmetric steady states. We discuss in detail different manifestations of D4-branching scenarios, its related physical interpretations, as well as its correspondence with the equivariant bifurcation theory. In addition to time-independent states, a large number of time-periodic solution branches are also identified. These periodic orbits show rich spatio-temporal symmetries, and their bifurcation diagram contains a variety of local and global bifurcations. Beyond pattern formation, since periodic orbits are expected to be dense within the chaotic attractor, these orbits can also serve as a basis for predicting statistics of transitional turbulence. Even though this aspect is not covered in this thesis, a logical and potential next step is to use these orbits to quantitatively reproduce the statistics of a three-dimensional flow. While the Newton-based shooting methods are used for converging invariant solutions, these methods have small radii of convergence as they rely on time-marching the chaotic dynamics. An alternative is the recently developed adjoint-based variational methods. Besides advantages over shooting methods in providing improved convergence properties, the variation