# Period Doubling Gained, Period Doubling Lost

A. Bayliss, B. J. Matkowsky, M. Minkoff

Research output: Contribution to journalArticlepeer-review

## Abstract

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.
Original language English 1047-1063 SIAM Journal on Applied Mathematics 49 https://doi.org/10.1137/0149063 Published - 1989

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