Anticontrol of Chaos for a Class of Delay Difference Equations Based on Heteroclinic Cycles Connecting Repellers

Zongcheng Li

doi:10.1155/2014/260150

Abstract

This paper is concerned with anticontrol of chaos for a class of delay difference equations via the feedback control technique. The controlled system is first reformulated into a high-dimensional discrete dynamical system. Then, a chaotification theorem based on the heteroclinic cycles connecting repellers for maps is established. The controlled system is proved to be chaotic in the sense of both Devaney and Li-Yorke. An illustrative example is provided with computer simulations.

Highlights

Anticontrol of chaos is a process that makes a nonchaotic system chaotic or enhances a chaotic system to produce a stronger or different type of chaos
There are many delay discrete dynamical systems which have more than two fixed points
We present an example of chaotification for the delay difference equation (1) with computer simulations

Summary

Introduction

Anticontrol of chaos (or called chaotification) is a process that makes a nonchaotic system chaotic or enhances a chaotic system to produce a stronger or different type of chaos. Many chaotification schemes appeared for discrete dynamical systems based on the feedback control approach. It has been shown that introducing delays to an undelayed system can be beneficial, especially for chaotic systems This is the delayed feedback control method, which is widely used in chaos control. Motivated by the delayed feedback control method, we studied the chaotification problem for a class of delay difference equations with at least two fixed points. We succeeded in using the sine function as a controller to chaotify linear delay difference equations in [16] This motivates us to use the sine function as the controller and employ a feedback control approach to study the chaotification problem for a class of delay difference equations.

Preliminaries

A Chaotification Scheme

An Example

Conclusion