# Persistence of Li-Yorke chaos in systems with relay

## TL;DR

This paper proves that Li-Yorke chaos persists in non-smooth relay systems even under perturbations, demonstrating the existence of infinite chaotic sets and proposing a chaos control method.

## Contribution

It introduces a rigorous proof of chaos persistence in relay systems and develops a chaos control technique based on the Ott-Grebogi-Yorke algorithm.

## Key findings

- Chaotic dynamics remain stable under perturbations.
- Countably infinite chaotic solution sets are identified.
- A chaos control method effectively stabilizes almost periodic motions.

## Abstract

It is rigorously proved that the chaotic dynamics of the non-smooth system with relay function is persistent even if a chaotic perturbation is applied. We consider chaos in a modified Li-Yorke sense such that infinitely many almost periodic motions take place in its basis. It is demonstrated that the system under investigation possesses countable infinity of chaotic sets of solutions. Coupled Duffing oscillators are used to show the effectiveness of our technique, and simulations that support the theoretical results are represented. Moreover, a chaos control procedure based on the Ott-Grebogi-Yorke algorithm is proposed to stabilize the unstable almost periodic motions embedded in the chaotic attractor.