Synchronization of Chaotic Systems


Nonlinear system theory influences more and more engineer sciences. This is due to two reasons. 1st: The reality is mostly nonlinear. Therefore the engineer is interested in knowledge about nonlinear systems and 2nd: Nonlinear phenomena offer intersting applications.

Recently, the idea to use chaotic systems for information transmission has received much attention. For this the synchronization of chaotic systems plays an important role. This work is devoted to chaotic synchronization in general and to the inverse system principle especially.

In the second chapter we refer to conventional synchronization and reveal its relation to the stability of solutions. Then we describe two chaotic synchronization principles and their use for information transmission. We give the state of the art from a general viewpoint on stability of solutions of controlled systems. Chapter 2

In chapter three we give a comprehensive introduction to the inverse system principle. We want to elucidate what system inversion basically is. For this we need deep understanding of the internal system features which is provided by the relative degree and a special system transformation. We will classify known inverse system examples with respect to their relative degree. Chapter 3

Chapter four represents a systematical overview on methods for proving synchronization (and unique asymptotic behaviour in general) of nonlinear systems. At the end of this chapter we propose a new approach suitable for a proof of synchronization based on the matrix measure induced by a Ljapunov function. Chapter 4

In chapter five we develop a general structure capable of inversion and synchronization. It is applicable to analogue and discrete-time systems. We work out, how the qualitative feature of chaotic motion can change into unique asymptotic bahaviour by system inversion. We apply this structure in the design of new inverse system examples. For this we extend the methods of circuit inversions to two-port inversions. This implies the use of ideal operational amplifiers. We give a criterion, for the correct functioning of nonideal op-amp. realizations. Chapter 5

Chapter six is motivated by the fact that the differential equations describing an inverse circuit may involve input derivatives. We give a method which allows to detect the dimension of state space and the number of occuring input derivatives of many circuits by pure inspection of the network structure. For this we give a comprehensive introduction. This introduction is based on [18] and contains some additional aspects. It should allow to get fast into the subject in order to have a deep understanding of our method. Chapter 6

The included remarks are supposed to emphasize special properties, to clarify certain relations or to open other viewpoints. In case the remark concerns a special topic it has a title. Sometimes they serve only the purpose that we can refer to them later.