Unsteady fronts in an auto catalytic system

Travelling waves in a model for autocatalytic reactions have, for some parameter regimes, been suggested to have oscillatory instabilities. These instabilities are confirmed by various methods, including linear-stability analysis (exploiting Evans's function) and direct numerical simulations. The front instability sets in when the order of the reaction, m, exceeds some threshold, m_c(τ), that depends on the inverse of the Lewis number, τ. The stability boundary, m = m_c(τ), is found numerically for m order one. In the limit m ≫ 1 (in which the system becomes similar to combustion systems with Arrhenius kinetics), the method of matched asymptotic expansions is employed to find the asymptotic front speed and show that m_c ∼ (V -1)^-1 as τ rarr;1. Just beyond the stability boundary, the unstable rocking of the front saturates supercritically. If the order is increased still further, period-doubling bifurcations occur, and for small τ there is a transition to chaos through intermittency after the disappearance of a period-4 orbit.