Classification of Morse-Smale systems and topological structure of the underlying manifolds

Morse-Smale systems arise naturally in applications for mathematical modelling of processes with regular dynamics (for example, in chains of coupled maps describing diffusion reactions, or in the study of the topology of magnetic fields in a conducting medium, in particular, in the study of the question of existence of separators in magnetic fields of highly conducting media). Since mathematical models in the form of Morse-Smale systems appear in the description of processes of various nature, the first step in the study of such models is to distinguish properties independent of the physical context but determining a partition of the phase space into trajectories. The relation preserving the partition into trajectories up to a homeomorphism is called topological equivalence, and the relation preserving also the time of motion along trajectories (continuous in the case of flows, and discrete in the case of cascades) is called topological conjugacy. The problem of topological classification of dynamical systems consists in finding invariants that uniquely determine the equivalence class or the conjugacy class for a given system. The present survey is devoted to a description of results on topological classification of Morse-Smale systems on closed manifolds, including results recently obtained by the authors. Also presented are recent results of the authors concerning the interconnections between the global dynamics of such systems and the topological structure of the underlying manifolds.