Publications

In 1976 S. Newhouse, J. Palis and F. Takens introduced a stable arc joining two structurally stable systems on a manifold. Later in 1983 they proved that all points of a regular stable arc are structurally stable diffeomorphisms except for a finite number of bifurcation diffeomorphisms which have no cycles, no heteroclinic tangencies and which have a unique nonhyperbolic periodic orbit, this orbit being the orbit of a noncritical saddle-node or a flip which unfolds generically on the arc. There are examples of Morse – Smale diffeomorphisms on manifolds of any dimension which cannot be joined by a stable arc. There naturally arises the problem of finding an invariant defining the equivalence classes of Morse – Smale diffeomorphisms with respect to connectedness by a stable arc. In the present review we present the classification results for Morse – Smale diffeomorphisms with respect to stable isotopic connectedness and obstructions to existence of stable arcs including the authors’ recent results.

Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labeling. Addressing these problems corresponds to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety of chaotic behaviors discussed in the abundant literature. Starting from the hierarchy of chaos initially proposed by one of us, we systematized the description of chaotic regimes observed in three- and four-dimensional spaces, which cover a large variety of known (and less known) examples of chaos. Starting with the spectrum of Lyapunov exponents as the first taxonomic ranks, we extended the description to higher ranks with some concepts inherited from topology (bounding torus, surface of section, first-return map, . . . ). By treating extensively the R¨ossler and the Lorenz attractors, we extended the description of branched manifold — the highest known taxonomic rank for classifying chaotic attractor — by a linking matrix (or linker) to multicomponent attractors (bounded by a torus whose genus g ≥ 3).

In this paper, we consider a class of A-diffeomorphisms given on a 3-manifold, assuming that all the basic sets of the diffeomorphisms are two dimensional. It is known that such basic sets are either attractors or repellers and they are two types only, surface or expanding (contracting). One of the results of the paper is the proof that different types of two-dimensional basic sets do not coexist in the non-wandering set of the same 3-diffeomorphism. Also, the existence of an energy function is constructively proved for systems of the class under consideration. It is illustrated by examples that the two-dimensionality of the basic sets is essential in this matter and a decrease in the dimension can lead to the absence of the energy function for a diffeomorphism.

The paper reports on application of the Gompertz model to describe the growth dynamics of COVID- 19 cases during the first wave of the pandemic in different countries. Modeling has been performed for 23 countries: Australia, Austria, Belgium, Brazil, Great Britain, Germany, Denmark, Ireland, Spain, Italy, Canada, China, the Netherlands, Norway, Serbia, Turkey, France, Czech Republic, Switzerland, South Korea, USA, Mexico, and Japan. The model parameters are determined by regression analysis based on official World Health Organization data available for these countries. The comparison of the predictions given by the Gompertz model and the simple logistic model (i.e., Verhulst model) is performed allowing to conclude on the higher accuracy of the Gompertz model.

Breathers or oscillating wave packets, along with solitons, are the most energy-carrying waves in various physical media, i.e. surface and internal waves, optical networks, and Josephson junctions. Breathers largely determine the overall wave dynamics of wave fields. In this work, the dynamics of a set of breathers or the so-called breather turbulence or breather gas is studied in the framework of the modified Korteweg–de Vries equation with positive cubic nonlinearity. Different multi-interactions of breathers occurring in a breather gas are analyzed. Numerical simulation of mixed turbulence of breathers and irregular waves is performed. Two approaches are realized: the first corresponds to the comparison of dynamics of the mixed wave fields with a fixed breather component and irregular waves with different momentum. The second corresponds to the comparison of dynamics of the wave fields with the same momentum but with different wave components, including breathers or pure wave fields of irregular waves. The statistical properties of considered wave fields are studied and analyzed. The most extreme wave fields from the point of view of freak waves are identified.

This paper considers the transformation of a sine wave in the framework of the extended modified Korteweg–de Vries equation or (2+4) KdV, which includes a combination of cubic and quintic nonlinearities. It describes the internal waves in a medium with symmetric vertical density stratification, and all the considerations in this study are produced for the reasonable combinations of the signs of the coefficients for nonlinear and dispersive terms, provided by this physical problem. The features of Riemann waves—solutions of the dispersionless limit of the model—are described in detail: The times and levels of breaking are derived in an implicit analytic form depending on the amplitude of the initial sine wave. It is demonstrated that the shock occurs at two (for small amplitudes) or four (for moderate and large amplitudes) levels per period of sine wave. Breaking at different levels occurs at different times. The symmetric (2+4) KdV equation is not integrable, but nevertheless it has stationary solutions in the form of traveling solitary waves of both polarities with a limiting amplitude. With the help of numerical calculations, the features of the processes of a sinusoidal wave evolution and formation of undular bores are demonstrated and analyzed. Qualitative features of multiple inelastic interactions of emerging soliton-like pulses are displayed.

In the present paper the exhaustive topological classification of nonsingular Morse-Smale flows of n-manifolds with two limit cycles is presented. Hyperbolicity of periodic orbits implies that among them one is attracting and another is repelling. Due to Poincare-Hopf theorem Euler characteristic of closed manifold Mn which admits the considered flows is equal to zero. Only torus and Klein bottle can be ambient manifolds for such flows in case of n=2. Authors established that there exist exactly two classes of topological equivalence of such flows of torus and three of the Klein bottle. There are no constraints for odd-dimensional manifolds which follow from the fact that Euler characteristic is zero. However, it is known that orientable 3-manifold admits a flow of considered class if and only if it is a lens space. In this paper, it is proved that up to topological equivalence each of S3 and RP3 admit one such flow and other lens spaces two flows each. Also, it is shown that the only non-orientable n-manifold (for n>2), which admits considered flows is the twisted I-bundle over (n−1)-sphere. Moreover, there are exactly two classes of topological equivalence of such flows. Among orientable n-manifolds only the product of (n−1)-sphere and the circle can be ambient manifold of a considered flow and the flows are split into two classes of topological equivalence.

—We consider reversible nonconservative perturbations of the conservative cubic H´enon maps H± 3 : ¯x = y, y¯ = −x + M1 + M2y ± y3 and study their influence on the 1:3 resonance, i. e., bifurcations of fixed points with eigenvalues e±i2π/3. It follows from [1] that this resonance is degenerate for M1 = 0, M2 = −1 when the corresponding fixed point is elliptic. We show that bifurcations of this point under reversible perturbations give rise to four 3- periodic orbits, two of them are symmetric and conservative (saddles in the case of map H+ 3 and elliptic orbits in the case of map H− 3 ), the other two orbits are nonsymmetric and they compose symmetric couples of dissipative orbits (attracting and repelling orbits in the case of map H+ 3 and saddles with the Jacobians less than 1 and greater than 1 in the case of map H− 3 ). We show that these local symmetry-breaking bifurcations can lead to mixed dynamics due to accompanying global reversible bifurcations of symmetric nontransversal homo- and heteroclinic cycles. We also generalize the results of [1] to the case of the p : q resonances with odd q and show that all of them are also degenerate for the maps H± 3 with M1 = 0.

In this paper, following J. Nielsen, we introduce a complete characteristic of orientationpreserving periodic maps on the two-dimensional torus. All admissible complete characteristics were found and realized. In particular, each of the classes of orientation-preserving periodic homeomorphisms on the 2-torus that are nonhomotopic to the identity is realized by an algebraic automorphism. Moreover, it is shown that the number of such classes is finite. According to V. Z. Grines and A.Bezdenezhnykh, any gradient-like orientation-preserving diffeomorphism of an orientable surface is represented as a superposition of the time-1 map of a gradient-like flow and some periodic homeomorphism. Thus, the results of this work are directly related to the complete topological classification of gradient-like diffeomorphisms on surfaces.

It is well-known that there is a close relationship between the dynamics of diffeomorphisms satisfying the axiom $A$ and the topology of the ambient manifold. In the given article, this statement is considered for the class $\mathbb G(M^2)$ of $A$-diffeomorphisms of closed orientable connected surfaces, the non-wandering set of each of which consists of $k_f\geq 2$ connected components of one-dimensional basic sets (attractors and repellers). We prove that the ambient surface of every diffeomorphism $f\in \mathbb G(M^2)$ is homeomorphic to the connected sum of $k_f$ closed orientable connected surfaces and $l_f$ two-dimensional tori such that the genus of each surface is determined by the dynamical properties of appropriating connected component of a basic set and $l_f$ is determined by the number and position of bunches, belonging to all connected components of basic sets. We also prove that every diffeomorphism from the class $\mathbb G(M^2)$ is $\Omega$-stable but is not structurally stable.

We describe scenarios for the emergence of Shilnikov attractors, i.e. strange attractors containing a saddle-focus with two-dimensional unstable manifold, in the case of threedimensional flows and maps. The presented results are illustrated with various specific examples

Westudy the stability of one-dimensional solitons propagating in an anisotropic medium.Wederived the Kadomtsev-Petviashvili equation for nonlinear waves propagating in an anisotropic medium. By a proper variable substitution this equation reduces either to the KPI or to the KPII equation. In the former case solitons are unstable with respect to the normal modes of transverse perturbations, and in the latter they are stable.Weonly consider the case when the solitons are unstable.Weformulated the linear stability problem. Using the Laplace–Fourier transform, we found the solution describing the evolution of an initial perturbation. Then, using Briggs’ method we studied the absolute and convective instabilities.Wefound that a soliton is convectively unstable unless it propagates at an angle smaller then critical with respect to a critical direction defined by the condition that the group velocity is parallel to the phase velocity. The critical angle is proportional to the ratio of the dispersion length to the soliton width, which is a small parameter. The coefficient of proportionality is expressed in terms of the phase speed and its second derivative with respect to the angle between the propagation direction and the critical direction. As an example we consider the stability of solitons propagating in Hall plasmas.

We study the graph G(F, Q) of a foliation F with Ehresmann connection Q. We prove that for any suspended foliation (M,F) the induced foliation on the graph G(F, Q) is also suspended. We describe the structure of the graph G(F, Q), as well as leaves of the induced foliation. We show that the induced foliation is a suspended foliation with the same transversal manifold and the same structure group as the original foliation.

We study foliations of arbitrary codimension 𝑞 on 𝑛-dimensional smooth manifolds admitting an integrable Ehresmann connection. The category of such foliations is considered, where isomorphisms preserve both foliations and their Ehresman connections. We show that this category can be considered as that of bifoliations covered by products. We introduce the notion of a canonical bifoliation and we prove that each foliation (𝑀, 𝐹) with integrable Ehresmann connection is isomorphic to some canonical bifoliation. A category of triples is constructed and we prove that it is equivalent to the category of foliations with integrable Ehresmann connection. In this way, the classification of foliations with integrable Ehresman connection is reduced to the classification of associated diagonal actions of discrete groups of diffeomorphisms of the product of manifolds. The classes of foliations with integrable Ehresmann connection are indicated. The application to 𝐺-foliations is considered.

The problem of the existence of traveling waves in inhomogeneous fluid is very important for enabling an explanation of long-distance wave propagations such as tsunamis and storm waves. The present paper discusses new solutions to the variable-coefficient wave equations describing traveling waves in fluid layers of variable depths (1D shallow-water theory). Such solutions are obtained by using the transformation methods when variable-coefficient equations can be reduced to the constant coefficient equation when the existence of traveling waves is evident. It is shown that there is a wide class of monotonic bottom profiles (discrete set) that allow the existence of traveling waves that are not reflected in a strongly inhomogeneous water medium. Their temporal shape changes with distance, mainly near the water–land boundary (shoreline). Traveling waves can transfer the wave energy over a long distance that is often observed at the transoceanic propagation of tsunami waves

The paper proves that the Morse index (dimension of an unstable manifold) of any saddle equliblrium  state  of a gradient-like flow without heteroclinic intersections, defined on a connected sum of  S^{n-1}\times S^1, n>3, equals either 1 or ( n-1).

We construct new substantive examples of nonautonomous vector fields on a 3-dimensional sphere having simple dynamics but nontrivial topology. The construction is based on two ideas: the theory of diffeomorphisms with wild separatrix embedding and the construction of a nonautonomous suspension over a diffeomorphism. As a result, we obtain periodic, almost periodic, or even nonrecurrent vector fields that have a finite number of special integral curves possessing exponential dichotomy on R such that among them there is one saddle integral curve (with a (3, 2) dichotomy type) with a wildly embedded 2-dimensional unstable separatrix and a wildly embedded 3-dimensional stable manifold. All other integral curves tend to these special integral curves as t → ±∞. We also construct other vector fields having k ⩾2 special saddle integral curves with the tamely embedded 2-dimensional unstable separatrices forming mildly wild frames in the sense of Debrunner–Fox. In the case of periodic vector fields, the corresponding specific integral curves are periodic with the period of the vector field, and are almost periodic in the case of an almost periodic vector field.

It is well known from the homotopy theory of surfaces that an ambient isotopy does not change the homotopy type of a closed curve. Using the language of dynamical systems, this means that an arc in the space of diffeomorphisms that joins two isotopic diffeomorphisms with invariant closed curves in distinct homotopy classes must go through bifurcations. A scenario is described which changes the homotopy type of the closure of the invariant manifold of a saddle point of a polar diffeomorphism of a 2-torus to any prescribed homotopically nontrivial type. The arc constructed in the process is stable and does not change the topological conjugacy class of the original diffeomorphism. The ideas that are proposed here for constructing such an arc for a 2-torus can naturally be generalized to surfaces of greater genus.

We consider the problem of topological classification of mutual dispositions in the real projective plane of two M-curves of degree 4. We studi  arrangements which are subject to the maximality condition (the oval of one of these curves has 16 pairwise different common points with the oval of the other of them) and  some combinatorial condition to select a special type of  such arrangements.  Pairwise different topological models of arrangements of this type are listed, which satisfy the known facts about the topology of nonsingular curves and the topological consequences of Bezout's theorem. There are more than 2000 such models. Examples of curves of degree 8 realizing some of these models are given, and it is proved that 1734 models cannot be realized by curves of degree 8. Proofs of non-realizability are carried out by Orevkov's method based on the theory of braids and links.

According to the results of V. Z. Grines and A. N. Bezdenezhnykh, for each gradient-like diffeomorphism of a closed orientable surface M2 there exist a gradient-like flow and a periodic diffeomorphism of this surface such that the original diffeomorphism is a superposition of a diffeomorphism that is a shift per unit time of the flow and the periodic diffeomorphism. In the case when M2 is a two-dimensional torus, there is a topological classification of periodic maps. Moreover, it is known that there is only a finite number of topological conjugacy classes of periodic diffeomorphisms that are not homotopic to identity one. Each such class contains a representative that is a periodic algebraic automorphism of a two-dimensional torus. Periodic automorphisms of a two-dimensional torus are not structurally stable maps, and, in general, it is impossible to predict the dynamics of their arbitrarily small perturbations. However, in the case when a periodic diffeomorphism is algebraic, we constructed a one-parameter family of maps consisting of the initial periodic algebraic automorphism at zero parameter value and gradient-like diffeomorphisms of a two-dimensional torus for all non-zero parameter values. Each diffeomorphism of the constructed one-parameter families inherits, in a certain sense, the dynamics of a periodic algebraic automorphism being perturbed.