Hopf Bifurcation Analysis and Matched Asymptotics for a Bi-Stable Flame Propagation Model

In this article, we consider a model for flame propagation dynamics influenced by both temperature and pressure. The model is based on nonlinear diffusion and incorporates elements from porous structure theory within the framework of partial differential equations. Initially, Hopf bifurcation analysis is used to identify equilibrium points where the system exhibits asymptotic stability. Subsequently, the solution profiles are derived using a combination of traveling wave solutions and geometric perturbation theory. The existence of finite propagation speed is demonstrated, and conditions for the stability of the system are established. Furthermore, through the method of matched asymptotic expansions, the inner (short-time) and outer (long-time or large-distance) solutions are connected, ensuring consistency in the asymptotic behavior across both regimes via Van Dyke’s matching principle. Numerical simulations are also used to validate the results.