We propose a phase-field model to describe reaction front propagation in activated transitions obeying Arrhenius kinetics. The model is applicable, for example, to the explosive crystallization of amorphous films. Two coupled fields interact during the reaction, a temperature field T(x,t) and a field C(x,t) describing the amorphous/crystal transition, which are continuous functions of space x and time t. Unlike previous work, our model incorporates a nonzero front width _ in a natural way, corresponding to that region in space where T and C undergo rapid variation. In the limit of __0, our model reduces to the sharp interface approach of others. Treating the background temperature of the reacting sample as a control parameter, periodic solutions in C and T can be found which go through a series of period doubling bifurcations. We find that the substrate temperature marking the onset of period doubling bifurcations decreases with increasing concentration diffusion. Furthermore, it is shown that period doubling bifurcations of C-T solutions of period greater than 2 are generated by dynamics isomorphic to those of the one-dimensional logistic map, for all values of concentration diffusion studied.