Global analysis of dynamical systems on low-dimensional manifolds.

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