The utility of an invariant manifold description of the evolution of a dynamical system

The long-time evolution of a physical system may be dominated by a small number of modes. An invariant manifold description of the asymptotic evolution, briefly discussed here in two simple examples, can give a powerful and useful view of the physical system. The relationship between invariant manifolds and centre manifolds is also discussed, as is the relationship between invariant manifolds and modal numerical approximations. Many classical approximations in fluid mechanics can be viewed as approximating the evolution on an approximate invariant manifold.