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On existence of Morse energy function for topological flow
Advances in Mathematics. 2021. Vol. 378. P. 1–15.
It is a well known fact that any smooth manifold admits a Morse function, whereas the problem of existence of a Morse function for a topological manifold stated by Marston Morse in 1959 is still open. In the present paper we prove that a topological manifold admits a continuous Morse function if it admits a topological flow with a finite hyperbolic chain recurrent set. We construct this function as a Lyapunov function whose set of the critical points coincides with the chain recurrent set of the flow.
Publication based on the results of:
Ovcharenko M., / Series arXiv "math". 2026.
We introduce an explicit class of tempered Laurent polynomials in the sense of Villegas and Doran--Kerr in n⩽4 variables including all Landau--Ginzburg models for smooth Fano threefolds with very ample anticanonical class. We check that it contains Landau--Ginzburg models for various Fano fourfolds which are complete intersections in smooth toric varieties and Grassmannians of planes, ...
Added: April 30, 2026
Domrin V. I., Malova H. V., V. Yu. Popov et al., Cosmic Research 2026 Vol. 64 No. 2 P. 238–252
During magnetospheric perturbations a relatively thin current sheet with thickness about several
proton gyroradii forms in the Earth’s magnetotail. In a framework of the kinetic model describing current
sheet thinning in the magnetotail, the processes of its formation are investigated depending on the normal
magnetic field magnitude which affects both the current sheet structure and particle dynamics within ...
Added: April 27, 2026
Tsareva O. O., Malova H. V., V. Yu. Popov et al., Plasma Physics Reports 2026 Vol. 52 No. 2 P. 179–185
The influence of asymmetry of plasma sources on the structure and spatial localization of a superthin
current sheet (STCS) supported by demagnetized electrons is studied using a self-consistent model. The
simulation takes into account the presence of a single plasma source in the northern hemisphere, which
makes the plasma flow asymmetric. It is demonstrated that the asymmetry of ...
Added: April 27, 2026
Pochinka O., Yakovlev E., Shmukler V., Russian Journal of Nonlinear Dynamics 2026
Every discrete dynamical system (cascade) generated by a homeomorphism induces a continuous
dynamic system (flow) — a suspension. However, not every flow is equivalent to a suspension
over a cascade, a necessary and sufficient condition for this is the existence of a global
section for the flow. In the case of the existence, the flow is equivalent to ...
Added: April 24, 2026
Kazaryan M., Dunin-Barkowski P., Bychkov B. et al., Selecta Mathematica, New Series 2026 Vol. 32 Article 25
We revise the notion of the blobbed topological recursion by extending it to the setting of generalized topological recursion as well as allowing blobs which do not necessarily admit topological expansion. We show that the so-called non-perturbative differentials form a special case of this revisited version of blobbed topological recursion. Furthermore, we prove the KP ...
Added: April 23, 2026
Kazaryan M., Lando S., Kodaneva N., Journal of Geometry and Physics 2026 No. 225 Article 105841
Weight systems associated to the Lie algebras 𝔤𝔩(N) for N = 1,2,... can be unified into auniversal one. The construction is based on an extension of the 𝔤𝔩(N) weight systems to permutations. This universal weight system takes values in the algebra of polynomials C[N;C1,C2,...] in infinitely many variables. We show that under the substitution Cm ...
Added: April 23, 2026
Kychkin A., Chernitsin I., Прикладная информатика 2026 Т. 21 № 1 С. 40–58
The results of the development of a software microservice embedded in atmospheric air quality monitoring systems to support the identification of industrial pollution sources are presented. The emission and subsequent spread of harmful substances in the lower layers of the atmosphere is dynamic and characterized by high uncertainty due to the specific features of technological ...
Added: April 23, 2026
IEEE, 2026.
Added: April 21, 2026
Galkin O., Galkina S., Ястребова И. Ю., Журнал Средневолжского математического общества 2026 Т. 28 № №1 С. 11–30
Polynomials of least deviation from zero play an important role in the theory and practice of numerical methods. They can be used to solve problems of optimizing the properties of various computational algorithms. Our work is devoted to the study of polynomials of least deviation from zero on a ray in the exponential norm. In ...
Added: April 20, 2026
Zlotnik Alexander, / Series arXiv "math". 2026. No. 2602.03481v1.
We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type continuous dependence of the solution $(\eta,u,\theta)$, in a norm slightly stronger than $L^{2,\infty}(Q)\times L^2(Q)\times L^2(Q)$, on the initial data $(\eta^0,u^0,e^0)$ in a ...
Added: April 18, 2026
Petrov I., Doklady Mathematics 2026 Vol. Volume 112 P. S103–S110
This paper examines games on networks with linear best responses, which allow for the analysis of how interaction structures influence agents’ strategic behavior. Special attention is given to intervention issues in such models, particularly in selecting optimal intervention strategies aimed at maximizing the central planner’s objective function. Two main control policies are analyzed: individual agent ...
Added: April 17, 2026
A. V. Pereskokov, Theoretical and Mathematical Physics 2026 Vol. 226 No. 3 P. 470–484
We consider the spectral problem for a hydrogen atom in orthogonal electric and magnetic fields with
an additional self-consistent field. We obtain an asymptotic expansion of self-consistent energy levels.
We find an asymptotic expansion of asymptotic eigenfunctions near the sphere |q| = 2. We calculate the
asymptotics of their norm in the space L2(R3). ...
Added: April 12, 2026
Kolachev N., Адамский А. И., Drozdov D. et al., Моделирование и анализ данных 2026 Т. 16 № 1 С. 157–176
Context and relevance. Despite the widespread adoption of the competency-based approach in higher education, a gap remains between the understanding of competence as a dynamic process and the tools available for its design and management. Dominant practices of learning outcomes assessment rely on static “snapshots,” which limits the possibilities for forecasting and purposeful development of competence. ...
Added: April 10, 2026
Zvereva V., Вопросы экономики 2026 № 4 С. 100–129
This paper examines the hypothesis of asymmetric responses of bank interest rates to restrictive and accommodative monetary policy conducted by the Bank of Russia across different market segments, industries, and macroregions over the period 2017—2025. Using a Markov-switching error-correction model, the paper estimates the effects of monetary policy shocks, households’ inflation expectations, firms’ price expectations, ...
Added: April 8, 2026
Petropavlovsky S., Turkel E., Journal of Computational Physics 2026 Vol. 558 Article 114880
We propose a method for the numerical computation of the 3D time-harmonic scattering about objects of complex shape. Our approach relies on the method of difference potentials combined with the lacunae-based integration of the Helmholtz equation. The former allows to handle curvilinear boundaries of scattering shapes on regular Cartesian grids with no loss of accuracy. ...
Added: April 6, 2026
Barinova M., Kolchurina O., Yakovlev E., Математический сборник 2024 Т. 215 № 9 С. 3–29
Известно, что нетривиальный аттрактор в неблуждающем множестве Ω-устойчивого 3-диффеоморфизма сосуществует с тривиальными базисными множествами тогда и только тогда, когда он либо одномерный неориентируемый, либо двумерный растягивающийся (ориентируемый или неориентируемый). Ранее были построены примеры соответствующих диффеоморфизмов, за исключением случая двумерного неориентируемого аттрактора. Настоящая работа восполняет этот пробел. Кроме того, здесь конструктивно доказывается существование энергетической функции у построенного диффеоморфизма, тем самым ...
Added: September 2, 2024
M. K. Barinova, K. Y. Tirskaya, Partial Differential Equations in Applied Mathematics 2024 Vol. 11 Article 100876
The Lyapunov function is a powerful tool for studying dynamical systems. In particular, the existence of such a function for any dynamical system given on compact manifold, established by C. Conley, is the basis for proving the Ω-stability criterion of dynamical systems. Moreover, the Lyapunov function, whose set of critical points of aligns with the chain-recurrent ...
Added: August 22, 2024
Barinova M., Grines V., Pochinka O., Труды Математического института им. В.А. Стеклова РАН 2023 Т. 321 С. 45–61
C. Conley’s fundamental theorem of the theory of dynamical systems states that every dynamical system, even a nonsmooth one (i.e., a continuous flow or a discrete dynamical system generated by a homeomorphism), admits a continuous Lyapunov function. A Lyapunov function is strictly decreasing along the trajectories of the dynamical system outside the chain recurrent set ...
Added: September 7, 2023
Barinova M., Shustova E., Журнал Средневолжского математического общества 2023 Т. 25 № 2 С. 11–21
This paper is devoted to the construction of an energy function, i.e. a smooth Lyapunov function, whose set of critical points coincides with the chain-recurrent set of a dynamical system — for a cascade that is a direct product of two systems. One of the multipliers is a structurally stable diffeomorphism given on a two-dimensional ...
Added: August 2, 2023
Barinova M., Grines V., Pochinka O., Journal of difference equations and applications 2023 Vol. 29 No. 9-12 P. 1275–1286
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 ...
Added: July 13, 2022
Barinova M., Shustova E., Журнал Средневолжского математического общества 2022 Т. 24 № 1 С. 21–30
A natural way for creating new dynamical systems is to consider direct products of already known systems. The paper studies some dynamical properties of direct products of homeomorphisms and diffeomorphisms. In particular, authors prove that a chain-recurrent set of the direct product of homeomorphisms is a direct product of the chain-recurrent sets. Another result established ...
Added: April 1, 2022
Barinova M., Lobachevskii Journal of Mathematics 2021 Vol. 42 No. 14 P. 3317–3323
If the chain recurrent set of a diffeomorphism f given on a closed n-manifold M^n is
hyperbolic (equivalently, f is an Ω-stable) then it coincides with the closure of the periodic points set
Perf and its chain recurrent components coincide with the basic sets. Due to C. Conley for such a
diffeomorphism there is a Lyapunov function which ...
Added: March 17, 2022
Pochinka O., Zinina S., Regular and Chaotic Dynamics 2021 Vol. 26 No. 4 P. 350–369
In this paper, we consider regular topological flows on closed n-manifolds. Such
flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number
of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale
flows, which are closely related to the topology of the supporting manifold. This ...
Added: July 16, 2021
Grines V., Pochinka O., Journal of Mathematical Sciences 2020 Vol. 250 P. 537–568
In this paper, we review results related to the existence of the energy function for discrete dynamical systems. Also, we consider the technique of constructing such functions for various classes of Ω-stable and structurally stable diffeomorphisms on manifolds of dimension 2 and 3. ...
Added: September 25, 2020