Jose Tapia: Center Manifold Theorem and Applications

Abstract: The center manifold theorem is a model reduction technique for determining the local asymptotic stability of an equilibrium of a dynamical system when its linear part is not a hyperbolic system. The overall system is asymptotically stable if and only if the center manifold dynamics is asymptotically stable. This allows for a substantial reduction in the dimension of the system whose asymptotic stability must be checked. Moreover, the center manifold and its dynamics need not be computed exactly; frequently, a low degree approximation is sufficient to determine its stability.
In this paper we describe the relevant portions of the theory of dynamical systems which are repeatedly used in discussion of the Center Manifold Theorem and give some examples of calculation of center manifold for concrete system of differential equations.​