UDC 517.938We study the class G(M n ) of orientation-preserving Morse-Smale diffeomorfisms on a connected closed smooth manifold M n of dimension n 4 which is defined by the following condition: for any f ∈ G(M n ) the invariant manifolds of saddle periodic points have dimension 1 and (n − 1) and contain no heteroclinic intersections. For diffeomorfisms in G(M n ) we establish the topoligical type of the supporting manifold which is determined by the relation between the numbers of saddle and node periodic orbits and obtain necessary and sufficient conditions for topological conjugacy. Bibliography: 14 titles.
In this paper we partly solve the problem of existence of separators of a magnetic field in plasma. We single out in plasma a 3-body with a boundary in which the movement of plasma is of special kind which we call an (a-d)-motion. We prove that if the body is the 3-annulus or the "fat" orientable surface with two holes then the magnetic field necessarily has a heteroclinic separator. The statement of the problem and the suggested method for its solution lead to some theoretical problems from Dynamical Systems Theory which are of interest of their own.
In this paper we consider a class of structurally stable diffeomorphisms with two-dimensional basic sets given on a closed 3-manifold. We prove that each such diffeomorphism is a locally direct product of a hyperbolic automorphism of the 2-torus and a rough diffeomorphism of the circle. We find algebraic criteria for topological conjugacy of the systems.
According to Pixton [8] there are Morse-Smale diffeomorphisms of S 3 which have no energy function, that is a Lyapunov function whose critical points are all periodic points of the diffeomorphism. We introduce the concept of quasi-energy function for a MorseSmale diffeomorphism as a Lyapunov function with the least number of critical points and construct a quasi-energy function for any diffeomorphism from some class of Morse-Smale diffeomorphisms on S 3 .Mathematics Subject Classification: 37B25, 37D15, 57M30.
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