Geometry of the Computational Singular Perturbation Method

The Computational Singular Perturbation (CSP) method, developed by Lam and Goussis [Twenty-Second Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1988, pp. 931–941], is a commonly-used method for finding approximations of slow manifolds in systems of ordinary differential equations (ODEs) with multiple time scales. The validity of the CSP method was established for fast–slow systems with a small parameter ε by the authors in [Journal of Nonlinear Science, 14 (2004), 59–91]. In this article, we consider a more general class of ODEs which lack an explicit small parameter ε, but where fast and slow variables are nevertheless separated by a spectral gap. First, we show that certain key quantities used in the CSP method are tensorial and thus invariant under coordinate changes in the state space. Second, we characterize the slow manifold in terms of these key quantities and explain how these characterizations are related to the invariance equation. The implementation of the CSP method can be either as a one-step or as a two-step procedure. The one-step CSP method aims to approximate the slow manifold; the two-step CSP method goes one step further and aims to decouple the fast and slow variables at each point in the state space. We show that, in either case, the operations of changing coordinates and performing one iteration of the CSP method commute. We use the commutativity property to give a new, concise proof of the validity of the CSP method for fast–slow systems and illustrate with an example due to Davis and Skodje.