Synchronization of Chaotic Systems with Time Delay

Abstract

In many physical, industrial, and engineering systems, delays occur due to the finite capabilities of information processing and data transmission among various parts of the system. Delays could arise as well from inherent physical phenomena like mass transport flow or recycling. Also, they could be by-products of computational delays or could intentionally be introduced for some design considerations. Such delays could be constant or time varying, known or unknown, deterministic or stochastic depending on the system under consideration. In all of these cases, the time-delay factors have counteracting effects on the system behavior and most of the time lead to poor performance. Therefore, the subject of time-delay systems has been investigated as functional differential equations over the past three decades. In this chapter, we study how to synchronize chaotic systems when time delay exists and the synchronized systems have different structures. We first develop synchronization methods for a class of delayed chaotic systems when the drive system and the response system have the same structure but different parameters. After that, the problem of synchronizing different chaotic systems is studied. Some concrete examples are presented to show how to design the controller. Based on that, a more general case, synchronizing two different delayed chaotic neural networks with known and unknown parameters, is considered.