Two perspectives on reduction of ordinary differential equations

Abstract

Key words Nonlinear differential equations, kinetic equations, multiple time scales, dimension reduction, slow manifold, normal form, computational singular perturbation, zero derivative principle MSC (2000) 34C20, 34E13, 34E15, 80A30, 80A25, 92C45 Dedicated to the memory of F. V. AtkinsonThis article is concerned with general nonlinear evolution equations x = g(x) in R N 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.

Physical understanding of complex multiscale biochemical models via algorithmic simplification: Glycolysis in Saccharomyces cerevisiae

“…(18), is equivalent to differentiating Eq. (B.1) with respect to time [32]. This differentiation yields S 1 = q 3 R 3 + q 5 R 5 + q 6 R 6 + q 7 R 7 + R 8 + q 9 R 9 + q 10 R 10 + q 12 R 12 + q 15 R 15…”

Time-Scale Separation from Diffusion-Mapped Delay Coordinates

“…In particular, the local evolution of the slow variables was approximated by repeated simulations of the fast variables using legacy code [17] or stochastic simulators [28]. A closely related technique in [35] requires knowledge of the local fast and slow directions to decompose the tangent bundle to the state space. Since we have no knowledge of the system, the local factorizations in the observation space may not consistently identify the correct global fast and slow directions.…”

“…Time-scale separation involves finding the variables that are governed by these slow dynamics and ordering the variables according to the relevant time scale. Our goals of projecting onto slow dynamics have some overlap to those of [8,17,28,35], although we assume no knowledge of the equations generating the data, or availability of surrogates such as microscopic models or legacy solvers. Furthermore, in analyzing video data of experiments, we are reduced to assuming that the observations are related to the true system state in an unknown way.…”

A reduced mechanism for ethylene/methane mixtures with excessive NO enrichment

“…Such time scale analysis includes, for example, quasi steady state approximation (QSSA) [7,[23][24][25][26][27][28][29][30][31][32], partial equilibrium approximation (PEA) [33,34], computational singular perturbation (CSP) [12,[35][36][37][38], intrinsic low-dimensional manifold (ILDM) [39][40][41], pre-image curve (PIC) [42,43], and ratecontrolled constrained equilibrium (RCCE) [44][45][46][47][48].…”