Publications

The purpose of this article is to review the author's results on the existence and structure of minimal sets and attractors of conformal foliations. Results on strong transversal equivalence of conformal foliations are also presented. Connections with works of other authors are indicated. Examples of conformal foliations with exceptional, exotic and regular minimal sets which are attractors are constructed.

A method of windowed spatiotemporal spectral filtering is proposed to segregate different nonlinear wave components and to calculate the surface of free waves. The dynamic kurtosis (i.e., produced by the free wave component) is shown able to contribute essentially to the abnormally large values of the surface displacement kurtosis, according to the direct numerical simulations of realistic sea waves. In this situation the free wave stochastic dynamics is strongly non-Gaussian, and the kinetic equation for sea surface waves fails. Traces of coherent wave patterns are found in the Fourier transform of the directional irregular sea waves; they may form “jets” in the Fourier domain which strongly violate the classic dispersion relation. 

In the present paper we consider preserving orientation Morse-Smale diffeomorphisms on surfaces. Using the methods of factorization and linearizing neighborhoods we prove that such diffeomorphisms have a finite number of orientable heteroclinic orbits.

The nonlinear stage of the modulational (Benjamin–Feir) instability of unidirectional deep-water surface gravity waves is simulated numerically by the fifth-order nonlinear envelope equations. The conditions of steep and breaking waves are concerned. The results are compared with the solution of the full potential Euler equations and with the lower-order envelope models (the 3-order nonlinear Schrödinger equation and the standard 4-order Dysthe equations). The generalized Dysthe model is shown to exhibit the tendency to re-stabilization of steep waves with respect to long perturbations.

 In the present paper we construct an example of 4-dimensional flows on $S^3\times S^1$ whose saddle periodic orbit has a wildly embedded 2-dimensional unstable manifold. We prove that such a property has every suspension under a non-trivial Pixton's diffeomorphism. Moreover we give a complete topological classification of these suspensions.

In the present paper we prove that frames of one-dimensional separatrices in basins of sources of structurally stable 3-diffeomorhisms with two-dimensional expanding attractor are trivially embedded. This result plays an important part in the classification of such systems. The classification was given by V. Grines and E. Zhuzhoma with assumption that all one-dimensional separatrices are trivially embedded into the ambient manifold but the proof of the assumption was never given. Thus, the present paper is a necessary and nontrivial element of the classification of structurally stable diffeomorphisms with codimension one expanding attractors.

Currently, a complete topological classification has been obtained with respect to the topological equivalence of Morse-Smale flows, [9,7], as well as their generalizations of Ω-stable flows on closed surfaces, [4]. Some results on topological conjugacy classification for such systems are also known. In particular, the coincidence of the classes of topological equivalence and conjugacy of gradient-like flows (Morse-Smale flows without periodic orbits) was established in [3]. In the classical paper [8], it was proved that in the presence of connections (coincidence of saddle separatrices), the topological equivalence class of a Ω-stable flow splits into a continuum of topological conjugacy classes (has moduli). Obviously, each periodic orbit also generates at least one modulus associated with the period of that orbit. In the present work, it was established that the presence of a cell in a flow bounded by two limit cycles leads to the existence of an infinitely many stability moduli. In addition, a criterion for the topological conjugation of flows on such cells was found.

The Smale surgery on the three-dimensional torus allows one to obtain a so-called DA diffeomorphism from the Anosov automorphism. The nonwandering set of a DA diffeomorphism consists of a single two-dimensional expanding attractor and a finite number of source periodic orbits. As shown by V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, the dynamics of an arbitrary structurally stable 3-diffeomorphism with a two-dimensional expanding attractor generalizes the dynamics of a DA diffeomorphism: such a 3-diffeomorphism exists only on the three-dimensional torus, and the two-dimensional attractor is its unique nontrivial basic set, but its nonwandering set may contain isolated saddle periodic orbits together with source periodic orbits. In the present study, we describe a scenario of a simple transition (through elementary bifurcations) from a structurally stable diffeomorphism of the three-dimensional torus with a two-dimensional expanding attractor to a DA diffeomorphism. A key moment in the construction of the arc is the proof that the closure of the separatrices of boundary periodic points of a nontrivial attractor and of isolated saddle periodic points are tamely embedded. This result demonstrates the fundamental difference of the dynamics of such diffeomorphisms from the dynamics of three-dimensional Morse–Smale diffeomorphisms, in which the closure of the separatrices of saddle periodic points may be wildly embedded.

We consider the problem of topological classification of arrangements in the real projective plane of the union of nonsingular curves of degrees 2 and 6 under certain conditions of maximality and general position. We list admissible topological models of such arrangements to be studied by using the Orevkov method based on the theory of braids and links and prove that most of these models cannot be realized by curves of degree 8.

In the present paper, we consider the class of orientation-preserving Morse-Smale diffeomorphisms f defined on an orientable surface M2. The work of A. A. Beznezhennykh and V. Z. Grines showed that such diffeomorphisms have a finite number of heteroclinic orbits. In addition, the classification problem of the diffeomorphisms under consideration is reduced to the problem of distinguishing orientable graphs with permutations describing the geometry of a heteroclinic intersection. However, such graphs generally do not admit polynomial distinguishing algorithms. This article proposes a new approach to the classification of cascade data. For this, each diffeomorphism f under consideration is assigned a graph whose embeddability in the ambient surface makes it possible to construct an effective algorithm for distinguishing such graphs.

In this article, we study gradient-like flows without heteroclinic intersections on an n-dimensional sphere, n>2, in sense of topological conjugacy. We prove that the class of topological conjugacy of such a flow is completely determined by the two-color tree corresponding to the skeleton formed by separatrices of codimension 1. In addition, we show that for such flows the concepts of topological equivalence and topological conjugacy coincide (which is completely wrong in the case of presence of limit cycles and connections)

This review presents the results of recent years on solving of the J. Palis's problem  on finding  necessary and sufficient conditions for the embedding of  Morse – Smale cascades in  topological flows. To date, the problem has been solved by Palis for Morse-Smale diffeomophisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that  leads to additional obstacles for  Morse-Smale diffeomorphisms to embed in  topological flows. The progress achieved in solving of Palis's problem in dimension three is associated with   recently  obtaining the  complete topological classification of Morse-Smale diffeomorphisms on three-dimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

In this article, we consider a class of gradient-like diffeomorphisms that have an attractor and repeller separated by a circle on a 2-sphere. For any diffeomorphism of this class, a stable arc is constructed connecting it with the source-sink system.

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

The problem of the existence of an arc with no more than a countable (finite) number of bifurcations connecting structurally stable systems (Morse-Smale systems) on manifolds is included in the list of fifty Palis-Pugh problems at number 33. This article describes a solution to this problem for gradient-like diffeomorphisms of a two-dimensional sphere.

The problem of topological classification of locations in the real projective plane of the union of nonsingular curves of degrees 2 and 6 is considered under some conditions of maximality and general position. After listing the permissible topological models of such locations to be investigated using the Orevkov method, based on the theory of braides and links, it is proved that most of these models cannot be realized by curves of degree 8.

In the paper the topological classification of gradient-like flows on mapping tori is obtained. Such flows naturally appear in the modelling of processes with at least on cyclic coordinate.