Graph topological equivalence criterion for $\Omega$-stable flows on surfaces

Study of flow dynamics on surfaces by dividing a phase space into cells with the same limit behavior of trajectories within the cell goes back to the classical works of A.A. Andronov, L.S. Pontryagin, E.A. Leontovich and A.G. Mayer. Types of cells (which are finite in number) and their adjacency to each other fully determine the class of topological equivalence of a flow with finite number of singular trajectories. If we select one trajectory in each cell of the rough stream without periodic orbits, the cells break up into so-called triangular regions which have the same single type. Combinatorial description of such a partition leads to three-colored graph of A.A. Oshemkov and V.V. Sharko; the vertices of that graph correspond to the triangular areas and edges correspond to separatrices that link them. These scientists demonstrated that two such flowsare topologically equivalent if and only if their three-colored graphs are isomorphic. In the same paper they gave full topological classification of the Morse-Smale flows in terms of atoms and molecules. In present paper the dynamics of $\Omega$-stable flows on surfaces is described with help of special oriented graphs and four-colored graphs.