Related papers: Knot as a complete invariant of a Morse-Smale 3-di…
Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the…
Topological classification of even the simplest Morse-Smale diffeomorphisms on 3-manifolds does not fit into the concept of singling out a skeleton consisting of stable and unstable manifolds of periodic orbits. The reason for this lies…
In this paper, we consider a class of Morse-Smale diffeomorphisms defined on a closed 3-manifold (non-necessarily orientable) under the assumption that all their saddle points have the same dimension of the unstable manifolds. The simplest…
In the present paper we consider class $G$ of orientation preserving Morse-Smale diffeomorphisms $f$, which defined on closed 3-manifold $M^3$, and whose non-wandering set consist of four fixed points with pairwise different Morse indices.…
We solve the problem of topological classification for smooth structurally stable flows on closed four-dimensional manifolds, the non-wandering set of which contains exactly two saddle equilibria, and the wandering set contains isolated…
We show that, up to topological conjugation, the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on 3-manifold is completely defined by an em- bedding of two-dimensional stable and unstable heteroclinic…
Topological equivalence of Morse-Smale flows without fixed points (NMS-flows) under assumptions of different generalities was studied in a number of papers. In some cases when the number of periodic orbits is small, it is possible to give…
We study a topological structure of a closed $n$-manifold $M^n$ ($n\geq 3$) which admits a Morse-Smale diffeomorphism such that codimension one separatrices of saddles periodic points have no heteroclinic intersections different from…
A topological classification of many classes of dynamical systems with regular dynamics in low dimensions is often reduced to combinatorial invariants. In dimension 3 combinatorial invariants are proved to be insufficient even for simplest…
In this paper we consider non-singular Morse-Smale flows on closed orientable 3-manifolds, under the assumption that among the periodic orbits of the flow there is only one saddle orbit and it is twisted. It is found that any manifold…
The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the form of a "source-sink", that means identifying an attractor-repeller pair, which are attractor-repellent sets for all other…
Let $M$ be either $n$-sphere $\mathbb{S}^{n}$ or a connected sum of finitely many copies of $\mathbb{S}^{n-1}\times \mathbb{S}^{1}$, $n\geq4$. A flow $f^t$ on $M$ is called gradient-like whenever its non-wandering set consists of finitely…
In the present paper, non-singular Morse-Smale flows on closed orientable 3-manifolds under the assumption that among the periodic orbits of the flow there is only one saddle one and it is twisted are considered. An exhaustive description…
This paper studies regular topological flows $f^t$ defined on closed {topological} manifolds $M^n$. The chain recurrent set of such a flow consists of a finite number of topologically hyperbolic fixed points and periodic orbits. Like their…
This paper is a step towards the complete topological classification of {\Omega}-stable diffeomorphisms on an orientable closed surface, aiming to give necessary and sufficient conditions for two such diffeomorphisms to be topologically…
In this paper we prove the breakdown of the two-dimensional stable and unstable manifolds associated to two saddle-focus points which appear in the unfoldings of the Hopf-zero singulariry. The method consists in obtaining an asymptotic…
We consider a class $G(S^n)$ of orientation preserving Morse-Smale diffeomorphisms of the sphere $S^{n}$ of dimension $n>3$ in assumption that invariant manifolds of different saddle periodic points have no intersection. We put in a…
We describe all possible topological structures of typical one-parameter bifurcations of gradient flows on the 2-sphere with holes in the case that the number of singular point of flows is at most six. To describe structures, we separatrix…
We consider general Morse-Smale diffeomorphisms on a closed orientable two-dimentional surface. In this paper it is proved that the complete topological invariant of Morse-Smale diffeomorphisms is finite, the algorithm of the construction…
We describe the topological structure of closed manifolds of dimension no less than four which admit Morse-Smale diffeomorphisms such that its non-wandering set contains any number of sink periodic points, and any number of source periodic…