Asymptotic Behaviour Near a Nonlinear Sink

TL;DR

This paper develops an iterative method to analyze the detailed asymptotic behavior of solutions to nonlinear vector differential equations approaching a nonlinear sink, applicable even with resonant eigenvalues, with examples from enzyme kinetics.

Contribution

Introduces a resonance-independent iterative procedure for asymptotic analysis of nonlinear systems approaching a sink, including planar systems and biological examples.

Findings

Method effectively characterizes asymptotic behavior.

Applicable to systems with resonant eigenvalues.

Demonstrated on enzyme kinetics model.

Abstract

In this paper, we will develop an iterative procedure to determine the detailed asymptotic behaviour of solutions of a certain class of nonlinear vector differential equations which approach a nonlinear sink as time tends to infinity. This procedure is indifferent to resonance in the eigenvalues. Moreover, we will address the writing of one component of a solution in terms of the other in the case of a planar system. Examples will be given, notably the Michaelis-Menten mechanism of enzyme kinetics.