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 μ, 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 μ ₁ the wave propagates at a uniform velocity. For μ > μ ₁, periodic pulsations occur of period T = T( μ ). That is, the velocity of the combustion wave increases and decreases periodically. The pu rations are sinusoidal for μ near μ ₁, and then develop into relaxation oscillations as μ is increased. A transition to a doublyperiodic (period $2T$) solution branch is found at a value μ ₂ > μ ₁. Stable doubly periodic solutions can no longer be computed beyond a third critical point μ ₃ > μ ₂. For μ > μ ₃, there is a return to the T periodic solution branch, with an interval of bistability (μ^* < μ < μ ₃, with μ ₂ < μ ^ * < μ ₃) in which both singly and doubly periodic solutions stably coexist, each with its own domain of attraction.