We consider a system of reaction diffusion equations which describe gasless combustion of condensed systems. To analytically describe recent experimental results, we show that a solution exhibiting a periodically pulsating, propagating reaction front arises as a Hopf bifurcation from a solution describing a uniformly propagating front. The bifurcation parameter is the product of a nondimensional activation energy and a factor which is a measure of the difference between the nondimensionalized temperatures of unburned propellant and the combustion products. We show that the uniformly propagating plant front is stable for parameter values below the critical value. Above the critical value the plane front becomes unstable and perturbations of the system evolve to the bifurcated state, i.e., to the pulsating propagating state. In our nonlinear analysis we calculate the amplitude, frequency and velocity of the propagating pulsating front. In addition we demonstrate analytically that the mean velocity of the oscillatory front is less than the velocity of the uniformly propagating plane front. diffusion equations for temperature and concentration, with the diffusion coefficient of the combustible component which limits the chemical reaction taken to be zero. In an interesting recent paper, Merzhanov, Filonenko and Borovinskaya (2) report on various new phenomena in the combustion of condensed systems. Among these phenomena they describe what they refer to as autooscillatory combustion for a mixture of 3Nb + 2B. They find that the velocity of propagation of the reaction front exhibits periodic pulsations. They also noted that the burned samples have a layered structure, normal to the front, with the number of layers equal to the number of pulsations. This work is described in the recent monographs of Novozhilov (3), and Zeldo- vich, Leypunsky and Librovich (4) in which they attribute the cause of these pulsations to the absence of diffusion. The existence of pulsating combustion fronts was first described theoretically by Shkadinsky, Khaikin and Merzhanov (5) who obtained