The parameterization method for invariant manifolds I: Manifolds associated to non-resonant subspaces

Abstract

We introduce a method to prove existence of invariant manifolds and, at the same time to find simple polynomial maps which are conjugated to the dynamics on them. As a first application, we consider the dynamical system given by a C r map F in a Banach space X close to a fixed point: F (x) = Ax + N (x), A linear, N (0) = 0, DN (0) = 0. We show that if X 1 is an invariant subspace of A and A satisfies certain spectral properties, then there exists a unique C r manifold which is invariant under F and tangent to X 1. When X 1 corresponds to spectral subspaces associated to sets of the spectrum contained in disks around the origin or their complement, we recover the classical (strong) (un)stable manifold theorems. Our theorems, however, apply to other invariant spaces. Indeed, we do not require X 1 to be an spectral subspace or even to have a complement invariant under A.

Analysis of the Computational Singular Perturbation Reduction Method for Chemical Kinetics

“…In such cases, the system of kinetic equations can be reduced to a much smaller system for the evolution of only the slow variables, and the fast variables can be determined simply by table look-ups or by direct computation. Over the years, a large number of reduction methods have been proposed and implemented in computer codes; references can be found in our earlier article [12], and additional references are [1], [7], [22].…”

“…), the asymptotic expansions of the CSPM and M ε agree up to and including terms of O(ε q ); they differ in general at O(ε q+1 ). Also, the qth application of the CSP algorithm leaves the terms at O (1) through O(ε q−1 ) invariant. (This observation is important because the lower-order terms have already been determined correctly in the preceding applications.)…”

“…Therefore, we need only to show that the third term equals −(Dh 0 )g 1,0 in order to prove the theorem for q = 1. According to the definitions (4.13) and (3.11) with q = 0, we havẽ 27) where (0) = B (0) (Dg)A (0) , according to the definition in (3.10).…”

“…In this section, we carry out the first iteration of the full CSP method and determine the resulting approximation K (1) ε of the slow manifold. We then compare K (1) ε andK (1) ε .…”

“…We then compare K (1) ε andK (1) ε . The update quantities U (0) and L (0) follow from the definition (3.11),…”