Publications

In the present paper we survey existing graph invariants for gradient-like flows on surfaces up to the topological equivalence and develop effective algorithms for their distinction (let us recall that a flow given on a surface is called a gradient-like flow if its non-wandering set consists of a finite set of hyperbolic fixed points, and there is no trajectories connecting saddle points). Additionally, we construct a parametrized algorithm for the Fleitas’s invariant, which will be of linear time, when the number of sources is fixed. Finally, we prove that the classes of topological equivalence and topological conjugacy are coincide for gradient-like flows, so, all the proposed invariants and distinguishing algorithms works also for topological classification, taking in sense time of moving along trajectories. So, as the main result of this paper we have got multiple ways to recognize equivalence and conjugacy class of arbitrary gradient-like flow on a closed surface in a polynomial time.

A mathematically correct description is presented on the interrelations between the dynamics of divergence free vector fields on an oriented 3-dimensional manifold M and the dynamics of Hamiltonian systems. It is shown that for a given divergence free vector field Xwith a global cross-section there exist some 4-dimensional symplectic manifold M̃⊃M and a smooth Hamilton function H:M̃→R such that for some c∈R one gets M={H=c}and the Hamiltonian vector field XH restricted on this level coincides with X. For divergence free vector fields with singular points such an extension is impossible but the existence of local cross-section allows one to reduce the dynamics to the study of symplectic diffeomorphisms in some sub-domains of M. We also consider the case of a divergence free vector field X with a smooth integral having only finite number of critical levels. It is shown that such a noncritical level is always a 2-torus and restriction of X on it possesses a smooth invariant 2-form. The linearization of the flow on such a torus (i.e. the reduction to the constant vector field) is not always possible in contrast to the case of an integrable Hamiltonian system but in the analytic case (M and X are real analytic), due to the Kolmogorov theorem, such a linearization is possible for tori with Diophantine rotation numbers.

In this paper we find all possible periodic data for orientation preserving gradient-like diffeomorphisms of orientable surfaces with one saddle orbit. We also construct a system of this class for every admissible collection of periodic data.

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 primarily in the possible ``wild'' behaviour of separatrices of saddle points. Another difference between Morse-Smale diffeomorphisms in dimension 3 from their surface analogues lies in the variety of heteroclinic intersections: a connected component of such an intersection may be not only a point as in the two-dimensional case, but also a curve, compact or non-compact. The problem of a topological classification of Morse-Smale cascades on 3-manifolds either without heteroclinic points (gradient-like cascades) or without heteroclinic curves was solved in a series of papers from 2000 to 2006 by Ch. Bonatti, V. Grines, F. Laudenbach, V. Medvedev, E. Pecou, O. Pochinka. The present paper is devoted to a complete topological classification of the set $MS(M^3)$ of orientation preserving Morse-Smale diffeomorphisms $f$ given on smooth closed orientable 3-manifolds $M^3$. A complete topological invariant for a diffeomorphism $f\in MS(M^3)$ is an equivalent class of its scheme $S_f$, which contains an information on a periodic date and a topology of embedding of two-dimensional invariant manifolds of the saddle periodic points of $f$ into the ambient manifold.

We show that, up to topological conjugation,} the equivalence class of a Morse-Smale diffeomorphism without heteroclinic curves on a $3$-manifold is completely defined by an embedding of two-dimensional stable and unstable heteroclinic laminations to a characteristic space.

We investigate totally geodesic foliations (M, F) of arbitrary  codimemsion q on n-dimensional pseudo-Riemannian manifolds for which the induced metrics on leaves don't degenerate. We assume that the q-dimensional orthogonal distribution D to (M, F) is an Ehresmann connection for this foliation. Since the usual graph G(F) is not Hausdorff manifold in general, we investigate the graph G(F, D) of the foliation with an Ehresmann connection D introduced early by the author. This graph is always Hausdorff manifold. We prove that on the graph G(F, D) a pseudo-Riemannian metric is defined, with respect to which the induced foliation and the simple foliations formed by the fibers of the canonical projections are totally geodesic. It is proved that the leaves of the induced foliation on the graph are nondegenerately reducible pseudo-Riemannian manifolds and their structure is described. The application to parallel foliations on nondegenerately reducible pseudo-Riemannian manifolds is considered. It is shown that every foliation defined by the suspension of a homomorphism of the fundamental group of a pseudo-Riemannian manifold belongs to the investigated class of foliations.           

In the present paper axiom $A$ diffeomorphisms of closed 2-manifolds of genus $p \geq 2$ whose nonwandering set contains perfect spaciously situated one-dimensional attractor are considered. It is shown that such diffeomorphisms are topologically semiconjugate to pseudo-Anosov homeomorphism with the same induced automorphism of fundamental group. The main result of the paper is the following. Two diffeomorphisms from the given class are topologically conjugate on attractors if and only if corresponding pseudo-Anosov homeomorphisms are topologically conjugate by means of homeomorphism that maps a certain subset of one pseudo-Anosov map onto the certain subset of the other pseudo-Anosov map.

The review is devoted to the presentation of results, including recently obtained by the authors, on the topological classification of Morse-Smale systems and the topology of ambient manifolds.

We provide a definition of a two-colored graph of a Morse-Smale diffeomorphism  without heteroclinical intersection defined on the sphere  $S^n$, $n\geq 4$ and prove that this graph is the complete topological invariant for such diffeomorphisms.  

In this paper it is proved that every orientable surface admits an orientation-preserving Morse-Smale diffeomorphism with one saddle orbit. It is shown that these diffeomorphisms have exactly three node orbits. In addition, all possible types of periodic data for such diffeomorphisms are established.

Studying the dynamics of a flow on surfaces by partitioning the phase space into cells with the same limit behaviour of trajectories within a cell goes back to the classical papers of Andronov, Pontryagin, Leontovich and Maier. The types of cells (the number of which is finite) and how the cells adjoin one another completely determine the topological equivalence class of a flow with finitely many special trajectories. If one trajectory is chosen in every cell of a rough flow without periodic orbits, then the cells are partitioned into so-called triangular regions of the same type. A combinatorial description of such a partition gives rise to the three-colour Oshemkov-Sharko graph, the vertices of which correspond to the triangular regions, and the edges to separatrices connecting them. Oshemkov and Sharko proved that such flows are topologically equivalent if and only if the three-colour graphs of the flows are isomorphic, and described an algorithm of distinguishing three-colour graphs. But their algorithm is not efficient with respect to graph theory. In the present paper, we describe the dynamics of Ω-stable flows without periodic trajectories on surfaces in the language of four-colour graphs, present an efficient algorithm for distinguishing such graphs, and develop a realization of a flow from some abstract graph.

We present a general method of solving the Cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two, one and zero. The method is based on the Chernoff approximation procedure applied to a specially constructed shift operator. It is proven that approximations converge uniformly to the exact solution.

We study the peculiarities of chaotic dynamics in the phenomenological model of the ensemble of two FitzHugh-Nagumo elements with weak excitatory couplings. This model was recently proposed as a suitable model for describing the behaviour of two coupled neurons. A rich diversity of different types of neuron-like behaviour, including regular in-phase, anti-phase, sequential spiking activities and also chaotic activity was observed in this model. We focus on chaotic bursting and chaotic spiking neuron-like activity in this paper. We study in details bifurcation scenarios of the emergence and destruction of these types of neuron-like activity.

We suggest a new model of the fast nondissipative kinematic dynamo which describes the phenomenon of exponential growth of the magnetic eld caused by the motion of the conducting medium. This phenomenon is known to occur in the evolution of magnetic elds of astrophysical bodies. In the 1970s A.D. Sakharov and Ya.B. Zeldovich proposed a \rope" scheme of this process which in terms of the modern theory of dynamical systems can be described as Smale solenoid. The main disadvantage of this scheme is that it is non-conservative. Our model is a modication of the Sakharov-Zeldovich's model. We apply methods of the theory of dynamical systems to prove that it is free of this fault in the neighborhood of the nonwandering set.

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large “chaotic sea” with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. In the present paper (which opens a series of three of our papers), elements of the theory of pseudohyperbolic attractors of multidimensional maps and flows are presented. Such attractors, as well as hyperbolic ones, are genuine strange attractors, but they allow the existence of homoclinic tangencies. We describe two principal phenomenological scenarios for the appearance of pseudohyperbolic attractors in one-parameter families of three-dimensional diffeomorphisms, and also consider some basic examples of concrete systems in which these scenarios occur. We propagandize new methods for studying pseudohyperbolic attractors (in particular, the method of saddle charts, the modified method of Lyapunov diagrams and the socalled LMP-method for verification of pseudohyperbolicity of attractors) and test them on the above examples. We show that Lorenz-like attractors in three-dimensional generalized H´enon maps and in a nonholonomic model of Celtic stone as well as figure-eight attractors in the model of Chaplygin top are genuine (pseudohyperbolic) ones. Besides, we show an example of fourdimensional Lorenz model with a wild spiral attractor of Shilnikov–Turaev type that was found recently in [Gonchenko et al., 2018].

A parabolic partial differential equation u 0 t (t, x) = Lu(t, x) is considered, where L is a linear second-order differential operator with time-independent (but dependent on x) coefficients. We assume that the spatial coordinate x belongs to a finite- or infinitedimensional real separable Hilbert space H. The aim of the paper is to prove a formula that expresses the solution of the Cauchy problem for this equation in terms of initial condition and coefficients of the operator L. Assuming the existence of a strongly continuous resolving semigroup for this equation, we construct a representation of this semigroup using a Feynman formula (i.e. we write it in the form of a limit of a multiple integral over H as the multiplicity of the integral tends to infinity), which gives us a unique solution to the Cauchy problem in the uniform closure of the set of smooth cylindrical functions on H. This solution depends continuously on the initial condition. In the case where the coefficient of the first-derivative term in L is zero, we prove that the strongly continuous resolving semigroup indeed exists (which implies the existence of a unique solution to the Cauchy problem in the class mentioned above), and that the solution to the Cauchy problem depends continuously on the coefficients of the equation

In the paper we derive two formulas representing solutions of Cauchy problem for two Schr\"{o}dinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients that do not change over time, this general equation is solved in the paper. We construct a family of translation operators in the space of square integrable functions and then use methods of functional analysis based on Chernoff product formula to prove that this family approximates the solution-giving semigroup. This leads us to some formulas that express the solution for Cauchy problem in terms of initial condition and coefficients of the equations studied.

We study the groups of conformal transformations of 𝑛-dimensional pseudo- Riemannian orbifolds (𝒩, 𝑔) as 𝑛 > 3. We extend the Alekseevskii method for studying conformal transformation groups of Riemannian manifolds to pseudo-Riemannian orbifolds. We show that a conformal pseudo-Riemannian geometry is induced on each stratum of such orbifold. Due to this, for 𝑘 ∈ {0, 1}∪{3, . . . , 𝑛−1}, we obtain exact estimates for the dimensions of the conformal transformation groups of 𝑛-dimensional pseudo-Riemannian orbifolds admitting 𝑘-dimensional stratum with essential groups of conformal transforms. A key fact in obtaining these estimates is that each connected transformation group of an orbifold preserves every connected component of each its stratum. The influence of stratification of 𝑛-dimensional pseudo-Riemann orbifold to the similarity transformation group of this orbifold is also studied for 𝑛 > 2. We prove that the obtained estimates for the dimension of the complete essential groups of conformal transformations and the similarity transformation groups of 𝑛-dimensional pseudo-Riemann orbifolds are sharp; this is done by adducing corresponding examples of locally flat pseudo-Riemannian orbifolds.

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 heteroclinic points. Also we consider gradient like flow on $M^n$ such that codimension one separatices of saddle singularities have no intersection at all. We show that $M^n$ is either an $n$-sphere $S^n$, or the connected sum of a finite number of copies of $S^{n-1}\otimes S^1$ and a finite number of special manifolds $N^n_i$ admitting polar Morse-Smale systems. Moreover, if some $N^n_i$ contains a single saddle, then $N^n_i$ is projective-like (in particular, $n\in\{4,8,16\}$, and $N^n_i$ is a simply-connected and orientable manifold). Given input dynamical data, one constructs a supporting manifold $M^n$. We give a formula relating the number of sinks, sources and saddle periodic points to the connected sum for $M^n$. As a consequence, we obtain conditions for the existence of heteroclinic intersections for Morse-Smale diffeomorphisms and a periodic trajectory for Morse-Smale flows.

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.

This article presents a number of models that arise in physics, biology, chemistry, etc., described by a one-dimensional reaction-diffusion equation. The local dynamics of such models for various values of the parameters is described by a rough transformation of the circle. Accordingly, the control of such dynamics reduces to the consideration of a continuous family of maps of the circle. In this connection, the question of the possibility of joining two maps of the circle by an arc without bifurcation points naturally arises. In this paper it is shown that any orientation-preserving source-sink diffeomorphism on a circle is joined by such an arc. Note that such a result is not true for multidimensional spheres.