  This paper describes a forward algorithm and an adjoint algorithm for
computing sensitivity derivatives in chaotic dynamical systems, such as the
Lorenz attractor. The algorithms compute the derivative of long time averaged
"statistical" quantities to infinitesimal perturbations of the system
parameters. The algorithms are demonstrated on the Lorenz attractor. We show
that sensitivity derivatives of statistical quantities can be accurately
estimated using a single, short trajectory (over a time interval of 20) on the
Lorenz attractor.
