Unsteady fronts in an autocatalytic 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 Evens's function) and direct numerical simulations. The front instability sets in when the order of the reaction, m, exceeds some threshold, m(c)(tau): that depends on the inverse of the Lewis number, tau. The stability boundary, m = m(c)(tau), is found numerically for m order one. In the limit m much greater than 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) similar to (tau-1)(-1) as tau --> 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 tau there is a transition to chaos through intermittency after the disappearance of a period-4 orbit.