  In this work we present a set-oriented path following method for the
computation of relative global attractors of parameter-dependent dynamical
systems. We start with an initial approximation of the relative global
attractor for a fixed parameter $\lambda_0$ computed by a set-oriented
subdivision method. By using previously obtained approximations of the
parameter-dependent relative global attractor we can track it with respect to a
one-dimensional parameter $\lambda > \lambda_0$ without restarting the whole
subdivision procedure. We illustrate the feasibility of the set-oriented path
following method by exploring the dynamics in models for shear flows during the
transition to turbulence.
