Fascination of chaos

This review introduces most of the concepts used in the study of chaotic phenomena in nonlinear systems and has as its objective to summarize the current understanding of results from the theory of chaotic dynamical systems and to describe the original ideas underlying the study of deterministic chaos. The presentation relies on informal analysis, with abstract mathematical ideas visualized geometrically or by examples from physics. Hyperbolic dynamics, homoclinic trajectories, homoclinic tangencies, wild hyperbolic sets and different types of attractors which appear in dynamical systems are considered. The key aspects of ergodictheory are discussed, and the basic statistical properties of chaotic dynamical systems are described. The fundamental difference between stochastic dynamics and deterministic chaos is explained. The review concludes by the investigation of the possibility to study complex systems on the basis of the analysis of registered signals, i.e. the generated time series.