The field of differential games has broad applicability to topics of economics, engineering, business, and warfare. Given the increasing levels of autonomy implemented in man-made systems in these fields, competition-based analysis may be the best option for understanding behavioral bounds when such systems interact. Differential games are governed by the Hamilton-Jacobi-Isaacs PDE, and many solution techniques are explored before identifying a gap in the existing literature. This dissertation develops a new approach to analyzing differential games based on a saddle-point solution technique. In a 2D system, the standard algorithmic approach produces both a value function interpolation and an approximate control map. Additionally, analysis of the observation error indicates that future analysis should prefer a problem formulation with relative motion. In the Circular Restricted Three-Body Problem, the same algorithms are applied to a system with real-world implications. The value and control interpolations produce a near-optimal trajectory, but the radial basis function approach suffered from high data density and did not exactly recreate the nominal solution. A perturbation analysis indicated that any mid-flight disturbance to the game state is most likely to benefit the pursuer, extending the works of Isaacs to a new domain. Ultimately, the proposed method is demonstrated to be a valuable tool for future differential games research.