  Here we analyze properties of an equation that we previously proposed to
model the dynamics of unstable detonation waves [A. R. Kasimov, L. M. Faria,
and R. R. Rosales. Model for shock wave chaos. Physical Review Letters,
110(10):104104, 2013]. The equation is \[
u_{t}+\frac{1}{2}\left(u^{2}-uu\left(0_{-},t\right)\right)_{x}=f\left(x,u\left(0_{-},t\right)\right),\quad
x\le0,\quad t>0. \] It describes a detonation shock at $x=0$ with the reaction
zone in $x<0$. We investigate the nature of the steady-state solutions of this
nonlocal hyperbolic balance law, the linear stability of these solutions, and
the nonlinear dynamics. We establish the existence of instability followed by a
cascade of period-doubling bifurcations leading to chaos.
