A Type Ia supernova explosion may occur if a subsonic nuclear flame in a white dwarf star is wrinkled by turbulence produced by the Rayleigh-Taylor instability and becomes supersonic. Our research investigates the plausibility of this theory by simulating a very simple case- a model flame in 2D. We performed a parameter study in which we changed only the non-dimensional gravitational force, G. The overarching goal of the thesis was to figure out how changing the strength of the Rayleigh-Taylor instability affects the flame surface and therefore the flame speed. At low values of G, this is a transition-to-turbulence type problem and we approached it from the dynamical systems point of view. Specifically, we measured various observables in our simulations and used them to search for simple bifurcation models of the flame behavior. For instance, the initial vortex shedding instability in the wake behind the flame front can be described by a Hopf bifurcation. Overall, simple temporal bifurcations are sufficient to describe the flame for low G. For high values of G, the simple dynamical systems approach breaks down. The area just behind the flame becomes fully turbulent and this turbulence wrinkles the flame front. Because the wrinkling takes place on all scales between the integral scale and the viscous scale, the flame assumes a fractal shape. We measured the fractal dimension of the flame front to assess the importance of this effect. For very high G, it turns out that large-scale Rayleigh-Taylor stretching is responsible for creating a larger part of flame surface than the turbulent wrinkling. This suggests that the flame speed is mostly determined by large-scale stretching driven by the Rayleigh-Taylor instability, not by the secondary interaction of turbulence with the flame front. The flame speed predicted for this situation is much too small for the flame to become supersonic, casting some doubt on the Rayleigh-Taylor wrinkling mechanism for Type Ia explosions.