Rather than starting from abstract mathematical definitions related to dynamical systems and the concepts used to analyse them, I prefer to start from the outset with a few familiar examples. The systems described below are related to the paradigms of deterministic chaos, some of which have indeed been the ones leading to the discovery, definition and understanding of chaotic behaviour. Instead of repeating here the so often quoted examples such as biological population growth, nonlinearly driven electrical oscillations, weather unpredictability, three-body Hamiltonian dynamics and chemical reaction oscillations and patterns, I shall attempt to motivate the reader by trying to find such examples among simplistic models of astrophysical systems. Obviously, the underlying mathematical structure of these will be very similar to the above mentioned paradigms. This only strengthens one of the primary lessons of nonlinear dynamics, namely that this is a generic, universal approach to natural phenomena.

Examples and analogies may sometimes be misleading and decide nothing, but they can make one feel more at home. This was, at least, the view of Sigmund Freud, the father of psychology, whose advice on matters didactic should not be dismissed. Indeed, as stressed before, these examples are the readers' old acquaintances from their astrophysics educational ‘home’. In the next chapter, where the basic notions characterising chaotic behaviour will be dealt with in detail, these examples will sometimes be used again for demonstrating abstract concepts.