# Word-series high-order averaging of highly oscillatory differential equations with delay

## TL;DR

This paper introduces a word-series based method to derive high-order averaged equations for highly oscillatory delay differential equations, enabling better analysis and simulation of such systems.

## Contribution

It extends word-series averaging techniques to delay differential equations, providing a new approach for high-order averaging in systems with delay and oscillations.

## Key findings

- High-order averaged equations can be derived for delay differential equations.
- The method simplifies the analysis of oscillatory systems with delay.
- Potential applications in control and signal processing with delays.

## Abstract

We show that, for appropriate combinations of the values of the delay and the forcing frequency, it is possible to obtain easily high-order averaged versions of periodically forced systems of delay differential equations with constant delay. Our approach is based on the use of word-series techniques to obtain high-order averaged equations for differential equations without delay.