# Averaging method for dynamic systems on time scales with periodicity

## TL;DR

This paper enhances the averaging method for dynamic systems on time scales, providing more precise estimates for solutions of $ riangle$-periodic and $ riangle$-quasiperiodic systems, with applications demonstrated through an example and numerical modeling.

## Contribution

It introduces $ riangle$-periodic and $ riangle$-quasiperiodic systems and extends the averaging theorem to these, improving solution accuracy and application scope.

## Key findings

- More accurate proximity estimates between original and averaged solutions.
- Introduction of $ riangle$-periodic and $ riangle$-quasiperiodic systems.
- Extension of averaging method applications to optimal control problems.

## Abstract

This paper aims to improve existing results about using averaging method for analysis of dynamic systems on time scales. We obtain a more accurate estimate for proximity between solutions of original and averaged systems regarding $\Delta$-periodic and $\Delta$-quasiperiodic systems, which are introduced for the first time. To illustrate the application of the averaging theorem for such kind of system we considered an example and conducted numerical modelling. Obtained results extend an application area for previously developed numerically-asymptotic method of solution for optimal control problems on time scales.