  The global behaviour of nonlinear systems is extremely important in control
and systems theory since the usual local theories will only give information
about a system in some neighbourhood of an operating point. Away from that
point, the system may have totally different behaviour and so the theory
developed for the local system will be useless for the global one.
  In this paper we shall consider the analytical and topological structure of
systems on 2- and 3- manifolds and show that it is possible to obtain systems
with 'arbitrarily strange' behaviour, i.e., arbitrary numbers of chaotic
regimes which are knotted and linked in arbitrary ways. We shall do this by
considering Heegaard Splittings of these manifolds and the resulting systems
defined on the boundaries.
