# High-frequency asymptotics of time-periodic solutions to differential equations systems in a critical case

## TL;DR

This paper investigates the existence, uniqueness, and asymptotic behavior of time-periodic solutions in linear differential systems with rapidly oscillating coefficients, focusing on critical degeneracy cases.

## Contribution

It provides new results on the existence, uniqueness, and asymptotic expansions of solutions for both ODE and PDE systems with oscillatory coefficients in degenerate cases.

## Key findings

- Existence and uniqueness of solutions established.
- Asymptotic expansions constructed and proved.
- Convergence and stability analyzed for ODE systems.

## Abstract

For two linear evolution differential equations systems - a normal ordinary differential equations system and a partial differential equations system with Stokes operator in a main part - with rapidly oscillating by time coefficients in a case of an averaged stationary operator multiple degeneracy time-periodic solutions problem was considered. The results about their existence and uniqueness were proved and their asymptotic expansions were constructed and proved. Moreover, for normal ordinary differential equations system the convergence of the asymptotic expansion was proved in the ordinary meaning and Lyapunov's stability questions were investigated.