Calculations are undertaken to study the approach to equilibrium for systems of reaction-diffusion equations on bounded domains. It is demonstrated that a number of systems approach equilibrium along attractive low-dimensional manifolds over significant ranges of parameter space. Numerical methods for generating the manifolds are adapted from methods that were developed for systems of ordinary differential equations. The truncation of the infinite spectrum of the partial differential equations makes it necessary to devise a new version of one of these methods, the well-known algorithm of Maas and Pope.