Abstract: We start by defining some basic notions from dynamical systems, culminating with that of an inertial manifold. The restriction of a flow to such a manifold can reduce an infinite dimensional system to a finite set of ordinary differential equations which captures all the long time behavior. We then move to the related notion of a foliation of phase space consisting of a pair manifolds at each point. Finally, we consider the case where such a reduction is not known to be possible, and instead derive an ODE in a Banach space of trajectories in which those on the attractor are readily identified.