We analyze properties of a nonlocal hyperbolic balance law equation that we previously proposed to model the dynamics of unstable detonation waves. We show that much of the complexity present in one-dimensional detonations can be captured by a properly forced Burgers’ equation. Furthermore we employ a combination of numerical and analytical tools to investigate the nature of the steady-state solutions, their linear stability, and nonlinear dynamics. We establish the existence of instability followed by a cascade of period-doubling bifurcations leading to chaos. Below are examples of a periodic and chaotic solution to our model equation.
L. M. Faria, A. R. Kasimov, R. R. Rosales, Study of a model equation in detonation theory, SIAM Journal on Applied Mathematics, 2014.