Statistical analysis and design of chaotic systems

Abstract

This chapter deals with the statistical description of chaotic signals and systems. Although chaotic systems are purely deterministic they can be modelled, analysed and designed by using probability measures and statistical characteristics, such as probability density functions and correlation functions, widely used in signal theory and in engineering applications. The fundamental approach is to model the initial state of the chaotic system to be a random variable. Then the generated signal can be treated as a stochastic process. Ensembles of trajectories are regarded and modelled by the time development of their density functions. This is done in section 2 where the fundamentals of statistical analysis of chaotic systems are explained systematically. The theory is mainly developed for multidimensional systems and illustrated with numerous examples for the one-dimensional case. The section introduces and develops tools for the statistical analysis. The subsequent section 3 copes with the solution of the inverse problem, the synthesis and design of chaotic systems from prescribed statistical characteristics of the signals to be generated. Statistical analysis and design are especially simple for piecewise linear system maps. So the corresponding systems are a paradigm for chaos generator design and experiments. The practical realisation of adjustable generator structures with piecewise linear maps is regarded in section 4. It is shown how system structures for piecewise linear maps can be constructed and be implemented as electronic circuits. In section 5 a statistical design case, the synthesis of a chaotic system for information encryption, is provided. A top-down design approach leads from the design objectives to system structures from which a simple example is provided up to its realisation. The system then undergoes a crypto-analysis in order to evaluate its security performance. Section 6 treats an extended problem of statistical analysis, the performance evaluation of chaotic signal processing schemes. This requires the introduction of an extended calculus because random processes (signals and noise) interact with chaotic signals. Several chaos communication schemes are analysed and the results are presented in terms of performance criteria commonly used in communication engineering. (orig.)Preprint of chapter 2.3. in: Application of Chaotic Electronics to Telecommunications, Kennedy, M.P.; Rovatti, R.; Setti, G. (eds.), CRD-publishers 2000SIGLEAvailable from TIB Hannover: RR 7265(99,1) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman