Asymptotics of the solution to a system of four nonlinear ordinary differential equations

Abstract

This paper investigates the solution of a singularly perturbed system of differential equations with an arbitrary degree of accuracy in a particularly critical case. The Cauchy initial value problem is considered for a singularly perturbed system of four nonlinear ordinary differential equations in the case of a stability change. The matrix of the nonlinear system possesses complex conjugate multiple eigenvalues. The problem is studied for a system of fourth-order nonlinear singularly perturbed equations in which the eigenvalues governing the stability change in an unbounded domain of the complex plane have zeros only at infinitely distant points. The proximity of the solutions of the perturbed and unperturbed problems is proved, and a corresponding estimate is derived.