Asymptotic behavior of solutions of a system of nonlinear differential equations with small parameter

Moscow State University of Civil Engineering, Yaroslavskoye shosse, 26, 129337 Moscow, Russia

^* Corresponding author: verapetelina51@gmail.com

Abstract

The present paper addresses a qualitative pattern of the behavior of solutions of a system of ordinary differential equations when small parameter tends to zero at a finite amount of time where slow variable passes through a certain point that corresponds to a bifurcation in the system of fast motions: stable limit cycle merges with unstable one and disappears. The problem of generation of an asymptotic approximation of the solution of a perturbed differential equation system is considered in the case where a bifurcation occurs in the “fast motions” equation when the parameter changes: two equilibrium positions merge, followed by a change in stability. The results of the article are used to determine second-order perturbations in rectangular coordinates and components of the velocity of the body under study. The coefficients at the projections of the perturbing acceleration are integral functions of the independent regularizing variable. Singular points are used to reduce the degree of approximating polynomials and to choose regularizing variables.