On asymptotic expansions of oscillating solutions of quasilinear ordinary differential equations systems

TL;DR

This paper investigates the asymptotic behavior of oscillating solutions in quasilinear ordinary differential equations systems, providing formal solutions and analyzing their nature under specific eigenvalue conditions.

Contribution

It proves the existence of formal oscillating solutions and analyzes their asymptotic properties based on eigenvalue conditions of the linear part.

Findings

Existence of formal oscillating solutions established.

Asymptotic nature characterized for different eigenvalue cases.

Results depend on eigenvalues being non-pure imaginary or simple.

Abstract

The existence of a formal particular solution (family of solutions) of oscillating type under certain conditions has been proved for the quasi-linear ordinary differential equations system. The asymptotic nature of this solution (the family of solutions) is investigated in two individual cases when all the eigenvalues of the matrix of the linear homogeneous part of the shorten system of differential equations 1) are not pure imaginary, 2) are simple and under some additional assumptions.