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

TL;DR

This paper analyzes traveling wave solutions in a one-dimensional detonation model with fractional order kinetics, focusing on bifurcations and viscous shock wave trajectories using dynamical systems theory and numerical methods.

Contribution

It introduces a detailed bifurcation analysis of a slow-fast dynamical system in the Rosales-Majda model with fractional reaction order, combining theoretical and numerical approaches.

Findings

Bifurcation diagram in the (β,c) parameter space identified.

Trajectories corresponding to viscous shock waves characterized.

Theoretical results confirmed through numerical simulations.

Abstract

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.