The interaction of topology and dynamics has attracted a great deal of attention from
numerous mathematicians. This thesis is devoted to the study of dynamical systems
on low-dimensional manifolds.
In the order of dimensions, we first look at the case of two-manifolds (surfaces) and
derive explicit differential equations for dynamical systems defined on generic surfaces
by applying elliptic and automorphic function theory to uniformise the surfaces in
the upper half of the complex plane with the hyperbolic metric. By modifying the
definition of the standard theta series, we will determine general meromorphic systems
on a fundamental domain in the upper half plane, the solution trajectories of which
'roll up' onto an appropriate surface of any given genus. Meanwhile, we will show
that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p
surface contains one or more invariant sets which act as attractors. Moreover, we shall
generalize a result in [Martins, 2004] and give conditions under which these invariant
sets are not homeomorphic to a circle individually, which implies the existence of
chaotic behaviour. This is achieved by analyzing the topology of inversely unstable
solutions contained within each invariant set.
Then the thesis concerns a study of three-dimensional systems. We give an explicit
construction of dynamical systems (defined within a solid torus) containing any knot
(or link) and arbitrarily knotted chaos. The first is achieved by expressing the knots
in terms of braids, defining a system containing the braids and extending periodically
to obtain a system naturally defined on a torus and which contains the given knotted
trajectories. To get explicit differential equations for dynamical systems containing
the braids, we will use a certain function to define a tubular neighbourhood of the
braid. The second one, generating chaotic systems, is realized by modelling the Smale
horseshoe.
Moreover, we shall consider the analytical and topological structure of systems
on 2- and 3- manifolds. By considering surgery operations, such as Dehn surgery,
Heegaard splittings and connected sums, we shall show that it is possible to obtain
systems with 'arbitrarily strange' behaviour, Le., arbitrary numbers of chaotic regimes
which are knotted and linked in arbitrary ways.
We will also consider diffeomorphisms which are defined on closed 3-manifolds
and contain generalized Smale solenoids as the non-wandering sets. Motivated by the
result in [Jiang, Ni and Wang, 2004], we will investigate the possibility of generating
dynamical systems containing an arbitrary number of solenoids on any closed, orientable
3-manifold. This shall also include the study of branched coverings and Reeb
foliations.
Based on the intense development from four-manifold theory recently, we shall
consider four-dimensional dynamical systems at the end. However, this part of the
thesis will be mainly speculative.