Slow-fast system in Rosales-Majda combustion model with fractional order kinetics

We consider traveling wave solutions of a one-dimensional model for detonation waves derived by Rosales and Majda, when the reaction order $\alpha$ belongs to $[0,1)$. The chemical kinetics is a simplified Arrhenius law or a Heaviside function. The model in the reaction zone is a slow-fast dynamical system for a vector representing temperature and mass fraction, which depends on the velocity $c$ and small viscosity $\beta$. Our goal in this paper is to study the bifurcation diagram in the $(\beta,c)$ parameter space and identify the nature of the trajectories corresponding to viscous shock waves. The demonstrations are based on a variety of techniques including the Poincar\'e-Bendixson theorem and the Fenichel theory. Theoretical results are confirmed by numerical computations.