Magnetic charging topology explains many energy processes (flares, prominences, etc.) in the solar corona by changing the domain structure associated with the appearance or disappearance of the separators. It is known that at most of the nulls of the magnetic field are prone. In this paper it is proved that a topology of the domains of a field with the prone nulls is completely described by a multi-color graph. In addition, we give an efficient algorithm for distinguishing of these graphs.

Starting from dimension 4, so-called non-smoothed manifolds, manifolds that do not allow triangulation and other obstacles that prevent the use of the technique of smooth manifolds for the study of multidimensional manifolds appear. In addition, all methods for studying smooth dynamical systems on multidimensional manifolds are based on the approximation of all subsets by piecewise linear or topological objects. In this regard, the idea of consideration of dynamical systems on multidimensional manifolds that do not use the concept of smoothness in their definition is completely natural. So homeomorphisms and topological Morse-Smale flows, which are also firmly connected with the topology of the ambient manifold, as well as their smooth analogues, have already entered into scientific usage. In the present paper we investigate general dynamical properties of homeomorphisms and topological flows with a finite hyperbolic chain recurrent set.

In this paper we construct an omega-stable diffeomorphism $f$ on closed 3-manifold $M$ so that non-wandering set of $f$ consists of exactly one-dimensional attractor and repeller. All known examples were constructed by Ch. Bonatti, N. Guilman, Sh. Yi. We suggest a new model of the construction of such diffeomorphism.

Manifolds admitting a Morse function with three critical points are called projective-like, by analogy with the projective plane. Eells and Kuiper showed that the dimension n of such manifolds takes on the values 2, 4, 8, and 16, and the critical points of the Morse function have indices 0, n / 2, and n. Zhuzhoma and Medvedev obtained a topological classification of gradient flows of such a function on manifolds of dimension 4. It is not difficult to construct a Morse function (and gradient-like flow) on a projective-like manifold, the set of critical points (equilibrium states) of which is not limited to three points with the listed indices. In this paper, we study the question of the relationship between the numbers of equilibrium states of different indices of a gradient-like flow, given on a projective-like manifold of dimension four, and give an algorithm for realization of such a flow for the given numbers of equilibrium states of indices 0,1,2,3,4.

В настоящей работе рассматриваются диффеоморфизмы Морса-Смейла, заданные на неодносвязном замкнутом многообразии M^n,n>3. Для таких систем вводится понятие тривиальной (нетривиальной) связанности их периодических орбит. Устанавливается, что изотопные тривиальные и нетривиальные диффеоморфизмы не могут быть соединены дугой с бифуркациями коразмерности один. Построены примеры таких каскадов Морса-Смейла на многообразии S^(n-1)xS^1.

The class of C^1-smooth gradient-like flows (Morse flows) on closed surface is the subclass of the Morse-Smale flows class, which are rough. Their non-wandering set consists of a finite number of hyperbolic fixed points and a finite number of hyperbolic limit cycles, and they does not have trajectories connecting saddle points. It is well known that the topological equivalence class of a Morse- Smale flow on a surface can be described combinatorially, for example, by the directed Peixoto graph, or by the Oshemkov-Sharko molecule. However, the description of the class of the topological conjugacy of such a system already requires the introduction of continuous invariants (moduli), corresponding to the periods of limit cycles at least. Thus, one class of the equivalence contains continuum classes of the topological conjugacy. Gradient-like flows are Morse-Smale flows without limit cycles. In this paper we prove that gradient-like flows on a closed surface are topologically conjugate iff they are topologically equivalent.

In this paper, a class of gradient-like diffeomorphisms $f$ on a closed orientable surface is considered, under the assumption that all non-wandering points of $f$ are fixed and have a positive orientation type. The main result is a construction of a stable arc joining two such diffeomorphisms. The diffeomorphisms under the consideration are Palis diffeomorphisms, who highlights their as only surface diffeomorphisms included in topological flows. By S. Newhouse, M. Peixoto, and J. Fleitas result, all Morse-Smale flows on a given manifold are joined by a stable arc. However, this fact cannot be used directly to construct an arc between cascades, since Palis diffeomorphisms are included only in the topological flow. An idea of a stable arc construction between Palis diffeomorphisms is based on the construction of a bifurcation-free arc joining a Palis diffeomorphism with a diffeomorphism that is a one-time shift of a generic gradient flow of a Morse function.

We obtain topological classification of 3 dimensional manifolds admitting gradient - like flows whose non-wandering set belongs to attracting and repelling closed invariant surfaces. We show that such manifolds are mapping tori (that are factor spaces of direct product of a surface $\mathbb{S}_g $ and the interval $[0,1]$ via equivalence relation $(z,1)\sim (\tau,0)$, where $\tau\colon \mathbb{S}_g\to \mathbb{S}_g$ is a homeomorphism). We obtain sufficient conditions for $\tau$ to be isotopic to periodic map.

We consider a class $G$ of Morse-Smale diffeomorphisms on the sphere $S^n$ of dimension $n\geq 4$ such that invariant manifolds of different saddle periodic points of any diffeomorphisms from $G$ have no intersection. Dynamics of an arbitrary diffeomorphism $f\in G$ can be represented as ``sink-source'' dynamics where the ``sink'' $A_f$ (the ``source'' $R_f$) is the connected unions of one- and zero-dimensional unstable (stable) manifolds. We study a structure of the space $V_f=S^n\setminus (A_f\cup R_f)$ and the topology of embedding in $V_f$ of separatrices of dimension $(n-1)$. We prove that the orbit space $\widehat{V}_f=V_f/_f$ is homeomorphic to the direct product $\mathbb{S}^{n-1}\times \mathbb{S}^1$, and the projection $l_\sigma\subset \widehat{V}_f$ of $(n-1)$-dimensional separatrix of a saddle periodic point $\sigma$ is either homeomorphic to the direct product $\mathbb{S}^{n-2}\times \mathbb{S}^1$ and bounds in $\widehat{V}_f$ a manifold homeomorphic to $\mathbb{B}^{n-1}\times \mathbb{S}^1$, or homeomorphic to a non-oriented locally-trivial fiber bundle under the circle $\mathbb{S}^1$ with the fiber $\mathbb{S}^{n-2}$, and such the manifold may by only one amount all projection of the separatrices.

The paper is devoted to the study of local bifurcations of symmetry breaking which arise under reversible perturbations of conservative reversible systems. We chose a perturbed conservative cubic diffeomorphism of a plane as an example of the model on which such bifurcations were investigated. It is shown that the main symmetry breaking bifurcations here are the so-called reversible pitch-fork bifurcations due to which a symmetric elliptic point becomes a symmetric saddle point and a pair of asymptotically stable and completely unstable points (one point is symmetric to another) appears in its neighborhood. The mechanism of destruction of conservative dynamics is demonstrated on the example of 1:3 and 1:4 resonances, which appear near the elliptic point of conservative cubic Henon map. In addition, in this paper we present an algorithm for constractig reversible perurbations. which break down the conservative dynamics in two-dimension maps.