Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva

This paper is devoted to the construction of an energy function, i.e. a smooth Lyapunov function, whose set of critical points coincides with the chain-recurrent set of a dynamical system — for a cascade that is a direct product of two systems. One of the multipliers is a structurally stable diffeomorphism given on a two-dimensional torus, whose non-wandering set consists of a zero-dimensional non-trivial basic set without pairs of conjugated points and without fixed source and sink, and the second one is an identical mapping on a real axis. It was previously proved that if a non-wandering set of a dynamical system contains a zero-dimensional basic set, as the diffeomorphism under consideration has, then such a system does not have an energy function, namely, any Lyapunov function will have critical points outside the chain-recurrent set. For an identical mapping, the energy function is a constant on the entire real line. In this paper, it is shown that the absence of an energy function for one of the multipliers is not a sufficient condition for the absence of such a function for the direct product of dynamical systems, that is, in some cases it is possible to select the second cascade in such a way that the direct product will have an energy function.

The purpose of this study is to single out a class of Morse-Smale cascades (diffeomorphisms) with a three-dimensional phase space that allow topological classification using combinatorial invariants. In the general case, an obstacle to such a classification is the possibility of wild embedding of separatrix closures in the ambient manifold, which leads to a countable set of topologically nonequivalent systems. To solve the problem, we study the orbit space of a cascade. The ambient manifold of a diffeomorphism can be represented as a union of three pairwise disjoint sets: a connected attractor and a repeller whose dimension does not exceed one, and their complement consisting of wandering points of a cascade called the characteristic set. It is known that the topology of the orbit space of the restriction of the Morse-Smale diffeomorphism to the characteristic set and the embedding of the projections of two-dimensional separatrices into it is a complete topological invariant for Morse-Smale cascades on three-dimensional manifolds. Moreover, a criterion for the inclusion of Morse-Smale cascades in the topological flow was obtained earlier.These results are used in this paper to show that the topological conjugacy classes of Morse-Smale cascades that are included in a topological flow and do not have heteroclinic curves admit a combinatorial description. More exactly, the class of Morse-Smale diffeomorphisms without heteroclinic intersections, defined on closed three-dimensional manifolds included in topological flows and not having heteroclinic curves, is considered. Each cascade from this class is associated with a two-color graph describing the mutual arrangement of two-dimensional separatrices of saddle periodic points. It is proved that the existence of an isomorphism of two-color graphs that preserves the color of edges is a necessary and sufficient condition for the topological conjugacy of cascades. It is shown that the speed of the algorithm that distinguishes two-color graphs depends polynomially on the number of its vertices. An algorithm for constructing a representative of each topological conjugacy class is described.

In this paper we consider a class G(Mⁿ) of gradient-like flows on connected closed manifolds of dimension n ≥ 4 such that for any flow f^t ∈ G(Mⁿ) stable and unstable invariant manifolds of saddle equilibria do not intersect invariant manifolds of other saddle equilibria. It is known that the ambient manifold of any flow g_f^t from the class G(Mⁿ) can be splitted into connected summ of the sphere Sⁿ, g_f^t  ≥ 0 copies of direct products S^n-1 x S¹,and a simply connected manifold which is not homeomorphic to the sphere. The number g_f^t is determined only by the number of nodal equilibria and the number of saddle equilibria such that one of their invariant manifolds has the dimension (n-1) (we call such equilibria trivial saddles). A simply connected manifold which is not homeomorphic to the sphere presents in the splitting if and only if the set of saddle equilibria contains points with unstable manifolds of dimension {2,...,n-2} (we call such equilibria non-trivial saddles). Moreover, the complete topological classification was obtained for flows from the class G(Mⁿ) without non-trivial saddles. In this paper we prove that for any flow f^t ∈ G(Mⁿ) the carrier manifold can be splitted into a connected sum along pairwise disjoint smoothly embedded spheres (separating spheres) that do not contain equilibrium states of the flow f^t and transversally intersect its trajectories. The restriction of the flow f^t to the complements to these spheres uniquely (up to topological equivalence and numbering) defines a finite set of flows f^t₁,...,f^t_l defined on the components of a connected sum. Moreover, for any j ∈ {1,...,l}, the set of saddle equilibria of the flow f^t_j onsists either only of trivial saddles or only of of non-trivial ones and then the flow f^t_j is polar. We introduce the notion of consistent topological equivalence for flows j ∈ {1,...,l} and show that flows f^t , f'^t∈ G(Mⁿ) are topologically equivalent if and only if for each of these flows the set of separating spheres exists that defines consistently topologically equivalent flows on the components of the connected sum.