In this chapter an overview is given on a class of chaotic and hyperchaotic circuits which can all be represented in Lur'e form. Starting from the well-known Chua's circuit several generalizations are discussed. By introducing additional breakpoints into the nonlinear characteristic one can generate so-called n-scroll attractors. These attractors consist of multiple scrolls which are located on a line in state space. This can be extended to a family of scroll grid attractors, which consist of scrolls located in a plane or a 3D body. Hyperchaotic versions have been obtained for four-dimensional circuits and by weak coupling of chaotic cells within cellular neural networks. Finally a short overview is given about several synchronization methods which have been developed for Lur'e systems, including master-slave and mutual synchronization, robust H ? synchronization, time-delayed systems and impulsive synchronization.

8.1 Introduction

Chua's circuit is a paradigm for chaos ^[1], being a simple nonlinear electrical circuit that exhibits a rich variety of phenomena ^{[2] - [6]}. Among the generalizations that have been proposed to the Chua's circuit one basically has two categories: either one modifies the nonlinear characteristic or one tries to increase the dimensionality of the circuit ^{[1] , [7] - [9]}. Originally, by Suykens and Vandewalle ^{[10] , [11]} a family of so-called n-double scroll attractors was introduced,...