11.   We have noted that chaotic behaviour can arise only for systems governed by non-linear equations.

12.   Then chaotic behaviour recurs, with the phase space trajectories being initially confined to bands and subsequently filling a whole region.

13.   However, there are also quite different ways in which flows exhibit chaotic behaviour -- with the connection to the general theory of chaos being perhaps more definitely established.

14.   Secondly the standard routes to chaos imply the existence of a well-defined onset of chaotic behaviour.

15.   The existence of expanding and contracting directions is an essential prerequisite for chaotic behaviour in dissipative systems of this sort.