  We present a computational analysis of a 2$\times$2 hyperbolic system of
balance laws whose solutions exhibit complex nonlinear behavior. Traveling-wave
solutions of the system are shown to undergo a series of bifurcations as a
parameter in the model is varied. Linear and nonlinear stability properties of
the traveling waves are computed numerically using accurate shock-fitting
methods. The model may be considered as a minimal hyperbolic system with
chaotic solutions and can also serve as a stringent numerical test problem for
systems of hyperbolic balance laws.
