The graph criterion for the topological equivalence of $\Omega $ – stable flows without periodic trajectories on surfaces and efficient algorithm for its application

Studying of dynamics of flows on surfaces by dividing a phase space into cells with 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 abutting each other fully determine the class of topological equivalence of a flow with finite number of singular trajectories. If in each cell of the rough stream without periodic orbits we select one trajectory, the cells fall into so-called triangular regions which has the same single type. Combinatorial description of such a partition leads to a three-colored graph of A.A. Oshemkov and V.V. Sharko. The vertices of this graph correspond to the triangular areas and edges correspond to those separatrices that link them. A.A. Oshemkov and V.V. Sharko demonstrated that two such flow topologically equivalent if and only if their three-colored graphs are isomorphic and an algorithm of distinction of three-colored graphs is described. However, their algorithm is not effective in terms of graph theory. In this work the dynamics of $ \Omega $ -stable flows without periodic trajectories on surfaces is described in terms of four-colored graphs and effective algorithm of distinction of these graphs is given.