In recent years, actually the last two decades, an increasing amount of interest has been addressed to the fact that, in many different domains of science, systems with a similar strange behavior are frequently encountered. These systems display a so-called chaotic time evolution. Indeed, the irregular behavior of a larger and larger class of phenomena can be described with a relatively small class of mathematical objects, each of them specifying an iteration procedure that, despite its deterministic nature, produces time evolutions which are really unpredictable. Let us make this more precise by saying that the nonlinearities present in the problem can produce an extreme sensitivity with respect to the initial state of the system, so that even a very small error in its knowledge will be exponentially amplified as the time goes on. Then, a fundamental question is whether it is possible to extract useful information on the motion from experimental observations.

The present review will lead us through a progressive discussion of those theoretical and experimental aspects which constitute, at present time, the main and successful ideas on the statistical analysis of time series for deterministic nonlinear systems. By time series we mean sequences of data representing the time evolution of one or more observables of the system, monitored at fixed time intervals. On one hand, there are no constraints on which kinds of phenomena this time evolution is representing – it may describe situations emerging from experimental physics, economics, biology, chemistry, and so on; on the other hand, the claim is that its qualitative features are the same as those of a differentiable dynamical system.