Asymptotic analysis of two reduction methods for systems of chemical reactions

Abstract

This article concerns two methods for reducing large systems of chemical kinetics equations, namely, the method of intrinsic low-dimensional manifolds (ILDMs) due to Maas and Pope and an iterative method due to Fraser and further developed by Roussel and Fraser. Both methods exploit the separation of fast and slow reaction time scales to find low-dimensional manifolds in the space of species concentrations where the long-term dynamics are played out. The analysis is carried out in the context of systems of ordinary differential equations with multiple time scales and geometric singular perturbation theory. A small parameter $\eps$ measures the separation of time scales. The underlying assumptions are that the system of equations has an asymptotically stable slow manifold ${\mathcal{M}}_0$ in the limit as $\eps \downarrow 0$ and that there exists a slow manifold $\mathcal{M}_\eps$ for all sufficiently small positive $\eps$, which is asymptotically close to ${\mathcal{M}}_0$. It is shown that the ILDM method yields a low-dimensional manifold whose asymptotic expansion agrees with the asymptotic expansion of $\mathcal{M}_\eps$ up to and including terms of $\mathcal{O} (\eps)$. The error at $\mathcal{O} (\eps^2)$ is proportional to the local curvature of ${\mathcal{M}}_0$. The iterative method generates, term by term, the asymptotic expansion of the slow manifold $\mathcal{M}_\eps$. The analytical results are illustrated in two examples.

39 pages, 78 references

Cited by in corpus (20)

• Constructive Methods of Invariant Manifolds for Kinetic Problems
• Reduction of dynamical biochemical reaction networks in computational biology
• Invariant grids for reaction kinetics
• Analysis of the accuracy and convergence of equation-free projection to a slow manifold
• Quasi Equilibrium Grid Algorithm: geometric construction for model reduction
• Properties of the Michaelis-Menten Mechanism in Phase Space
• Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learning
• Coordinate-independent singular perturbation reduction for systems with three time scales
• Phase Transitions for the Brusselator Model
• Analysis of the CSP Reduction Method for Chemical Kinetics
• Exponential estimates of symplectic slow manifolds
• Effective reduction of a three-dimensional circadian oscillator model
• On fundamental unifying concepts for trajectory-based slow invariant attracting manifold computation in multiscale models of chemical kinetics
• Projecting to a Slow Manifold: Singularly Perturbed Systems and Legacy Codes
• On the quasi-steady-state approximation in an open Michaelis--Menten reaction mechanism
• An iterative method for the approximation of fibers in slow-fast systems
• Asymptotics of conduction velocity restitution in models of electrical excitation in the heart
• Evaluation of a measure on the quasi-steady state assumption of Collisional Radiative Models via Intrinsic Low Dimensional Manifold Technique
• Fast-slow vector fields of reaction-diffusion systems
• Computation of saddle type slow manifolds using iterative methods