for such invariant manifolds, say , satisfies the differential equation for the system to residuals as , then the invariant manifold is approximated by to an error of the same order, namely . Center manifolds of infinite-D and/or of non-autonomous systems However, some applications, such as to dispersion in tubes or channels, require an infinite-dimensional center manifold. The most general and powerful theory was developed by Aulbach and Wanner. They addressed non-autonomous dynamical systems in infinite dimensions, with potentially infinite dimensional stable, unstable and center manifolds. Further, they usefully generalised the definition of the manifolds so that the center manifold is associated with eigenvalues such that , the stable manifold with eigenvalues , and unstable manifold with eigenvalues . They proved existence of these manifolds, and the emergence of a center manifold, via nonlinear coordinate transforms. Potzsche and Rasmussen established a corresponding approximation theorem for such infinite dimensional, non-autonomous systems. Alternative backwards theory All the extant theory mentioned above seeks to establish invariant manifold properties of a specific given problem. In particular, one constructs a manifold that approximates an invariant manifold of the given system. An alternative approach is to construct exact invariant manifolds for a system that approximates the given system---called a backwards theory. The aim is to usefully apply theory to a wider range of systems, and to estimate errors and sizes of domain of validity. This approach is cognate to the well-established backward error analysis in numerical modeling. Center manifold and the analysis of nonlinear systems As the stability of the equilibrium correlates with the "stability" of its manifolds, the existence of a center manifold brings up the question about the dynamics on the center manifold. This is analyzed by the center manifold reduction, which, in combination with some system parameter μ, leads to the concepts of bifurcations. Correspondingly, two web services currently undertake the necessary computer algebra to construct just the center manifold for a wide range of finite-dimensional systems (provided they are in multinomial form). One web service constructs slow manifolds for systems which are linearly diagonalised, but which may be non-autonomous or stochastic. Another web service constructs center manifolds for systems with general linearisation, but only for autonomous systems. Examples The Wikipedia entry on slow manifolds gives more examples. A simple example Consider the system The unstable manifold at the origin is the y axis, and the stable manifold is the trivial set {(0,