Chaotic combustion of solids and high-density fluids near points of strong resonance

Abstract

The loss of stability of steady, planar combustion waves propagating through solids and high-density fluids occurs via Hopf bifurcation for sufficiently large values of a parameter related to the activation energy of the reaction. Near multiple eigenvalues corresponding to points of strong resonance, the dynamical behaviour of an asymptotic model derived from the original set of conservation equations is governed by an ordinary dynamical system for the amplitudes of the eigenmodes of the linearized problem. An analysis of these amplitude equations then completely determines the local bifurcation structure. It is shown that in certain parameter régimes, these equations admit finite-amplitude, chaotic solutions following the loss of stability of a secondary bimodal solution branch of entrained time-periodic combustion waves. Thus, in contrast to direct numerical integration of the original system of conservation equations far above the neutral stability boundary that marks the transition from steady to non-steady burning, it is shown that multi-dimensional chaotic flame propagation can be predicted from nonlinear stability theory in the neighbourhood of a multiple bifurcation point.