Chaotic behaviour occurs because it becomes harder and harder to predict trajectories over longer and longer time periods, given a certain precision in the known initial conditions.

A region of phase space may flow, continuously and incompressibly, into a very fractured image even though, by Liouville's theorem, both regions have exactly the same size.

Nb. By a region in phase space, we mean a space of possible systems, each characterised by initial conditions.

We can, however measure this fracturedness if we apply a course-graining of the region, represented in figure 7.3 by the number of pixels it takes to cover the region. Even though both regions have the same precise area, it takes significantly more course-grained pixels to cover the fractured region.

This can be seen as a loss of information, or of precision, and we could characterise chaotic behaviour in this way. In practise, all physical systems eventually become chaotic. This phenomenon is called the **entropy** of the system.