Two perspectives on reduction of ordinary differential equations

This article is concerned with general nonlinear evolution equations x ′ = g (x ) in RN involving multiple time scales, where fast dynamics take the orbits close to an invariant low-dimensional manifold and slow dynamics take over as the state approaches the manifold. Reduction techniques offer a systematic way to identify the slow manifold and reduce the original equation to an autonomous equation on the slow manifold. The focus in this article is on two particular reduction techniques, namely, computational singular perturbation (CSP) proposed by Lam and Goussis [Twenty-Second Symposium (International) on Combustion, The University of Washington, Seattle, Washington, August 14–19, 1988 (The Combustion Institute, Pittsburgh, 1988), pp. 931–941] and the zero-derivative principle (ZDP) proposed recently by Gear and Kevrekidis [Constraint-defined manifolds: A legacy-code approach to low-dimensional computation, SIAM J. Sci. Comput., to appear]. It is shown that the tangent bundle to the state space offers a unifying framework for CSP and ZDP. Both techniques generate coordinate systems in the tangent bundle that are natural for the approximation of the slow manifold. Viewed from this more general perspective, both CSP and ZDP generate, at each iteration, approximate normal forms for the system under examination. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)