Summary: Pulsating solutions to a model of gasless condensed phase combustion are studied numerically as a function of a parameter \(\mu\), which is proportional to the nondimensional activation energy. Due to the exothermic reaction, a combustion wave propagates into the fresh fuel mixture. Below a critical value \(\mu_ 1\) the wave propagates at a uniform velocity. For \(\mu >\mu_ 1\), periodic pulsations occur of period \(T=T(\mu)\). That is, the velocity of the combustion wave increases and decreases periodically. The pulsations are sinusoidal for \(\mu\) near \(\mu_ 1\), and then develop into relaxation oscillations as \(\mu\) is increased. A transition to a doubly periodic (period 2T) solution branch is found at a value \(\mu_ 2>\mu_ 1\). Stable doubly periodic solutions can no longer be computed beyond a third critical point \(\mu_ 3>\mu_ 2\). For \(\mu >\mu_ 3\), there is a return to the T periodic solution branch, with an interval of bistability \((\mu^*<\mu <\mu_ 3\), with \(\mu_ 2<\mu^*<\mu_ 3)\) in which both singly and doubly periodic solutions stably coexist, each with its own domain of attraction.