Period Doubling Gained, Period Doubling Lost

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 pu rations are sinusoidal for $\mu $ near $\mu _1 $, and then develop into relaxation oscillations as $\mu $ is increased. A transition to a doublyperiodic (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.