Mathematical modeling and techniques from the field of dynamical systems can be used to study a variety of phenomena. The applications included in this dissertation span from social to physical sciences.The first chapter presents a conceptual model of voter turnout. Using numerical and analytical techniques, we explore a variety of network structures and identify the conditions that facilitate minority factions winning elections. The remainder of the dissertation focuses on mathematical models in atmospheric science, specifically one-dimensional and two-dimensional models of exoplanet atmospheres. We compare two one-dimensional energy balance models with explicit dependence on obliquity to study the likelihood of different stable ice configurations. We compare the results of models with different methods of heat transport and different insolation distributions and show that stable partial ice cover is possible for any obliquity, provided the insolation distribution is sufficiently accurate. In the final chapters of this dissertation, we present a two-dimensional shallow-water model, SWAMPE (Shallow-Water Atmospheric Model in Python for Exoplanets). Two-dimensional shallow-water models fill the gap between minimal-complexity models which lack longitudinal variation and high-complexity computationally expensive three-dimensional models. Our model can accurately and rapidly explore a vast parameter space. The code is flexible, modular, built fully in Python, and could be easily adapted to model a variety of dissimilar space objects, from Brown Dwarfs to smaller, terrestrial planets. The code can produce the thermal maps necessary for the generation of phase curves and secondary eclipse maps, which will help constrain and make predictions for observations. Finally, we apply SWAMPE to explore the circulation regimes of synchronously rotating sub-Neptunes as we consider the interactions between planetary rotation period and radiative forcing from the host star.