The thesis develops two characterizations of spatio-temporal complex patterns. While these are developed for the patterns of fluid flow in experiments on Rayleigh-Benard Convection(RBC), they are adaptable to a wide range of spatially extended systems. The characterizations may be especially useful in cases where one does not have good models describing the dynamics, making numerical and analytic studies difficult. In Spiral Defect Chaos(SDC), a weakly turbulent regime of RBC, the convective rolls exhibit complex spatial and temporal dynamics. We study the dynamics of SDC through local defect formations between convective rolls as well as the topological rearrangements of these rolls at a global scale. A laser based thermal actuation system is developed to reproducibly impose initial states for the fluid flow and construct ensembles of trajectories in the neighborhood of defect nucleation. This is used to extract the modes and their growth rates, characterizing the linear manifold corresponding to defect nucleation. The linear manifold corresponding to instabilities resulting in defect formation is key to building efficient schemes to control the dynamics exhibited. We also develop the use of computational homology as a tool to study spatially extended dynamical systems. A quantitative measure of the topological features of patterns is shown to provide insights into the underlying dynamics not easily uncovered otherwise. In the case of RBC, the homology of the patterns is seen to indicate asymmetries between hot and cold regions of the flow, stochastic evolution at a global scale as well as bifurcations occurring well into the turbulent regime of the flow.