New Routes to Quasi-Periodic Combustion of Solids and High-Density Fluids Near Resonant Hopf Bifurcation Points

Abstract

Steady, planar combustion waves propagating through solids and high-density fluids can lose stability to a time-periodic mode of burning via Hopf bifurcation as an activation-energy parameter increases past a critical value. However, if the combustion wave is confined in a long channel, there exist critical values of the transverse channel dimensions such that stability is simultaneously lost to two (or more) nonsteady modes of burning at a multiple eigenvalue of this parameter. In the present work, the case of bifurcation near a point of strong resonance from which multiple primary branches corresponding to a single pulsation frequency (but different spatial structures) emanate from the basic solution is considered. In contrast to the multiple-frequency case, the resulting amplitude equations contain transcendental terms which can lead to both secondary and isolated branches of multimodal time-periodic motions. In addition, the higher-order bifurcation of quasi-periodic waves can arise either through a classical Hopf bifurcation from a multimodal branch, or through a new type of nonlinear mechanism from a single-mode primary branch.