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New Numerical Invariants of an Unfolding of a Polycycle “Tears of the Heart”
Russian Journal of Mathematical Physics. 2026. Vol. 33. No. 1. P. 89–106.
In this paper, new numerical invariants of structurally unstable vector fields in the plane
are found. One of the main tools is an improved asymptotics of sparkling saddle connections that
occur when a separatrix loop of a hyperbolic saddle breaks. Another main tool is a new topological
invariant of two arithmetic progressions, both perturbed and unperturbed, on the real line. For the
pairs of the unperturbed arithmetic progressions, we give a complete topological classification.
Keywords: векторные полябифуркацииvector fields on the spherenumeric invariantsGlobal bifurcationsчисловые инварианты
Publication based on the results of:
Springer, 2026.
This book presents established and new research on the close connections between graph games and systems of logic, particularly existing and newly designed modal logics. The volume utilizes two graph games – the sabotage game and the hide-and-seek game – to demonstrate the natural interplay between designing new graph games and exploring new kinds of ...
Added: June 30, 2026
Pochinka O., Barinova M., Journal of Geometry and Physics 2026 Vol. 228 P. 1–8
In the present paper we consider an Ω-stable 3-diffeomorphism with a solid or thickened surfaced non-trivial basic set. Such basic sets include, for instance, all one-dimensional expanding attractors and those two-dimensional basic sets that are not expanding. We prove that the chain recurrent set of every such a diffeomorphism necessarily contains at least two non-trivial ...
Added: June 30, 2026
German O., Illarionov A., Известия РАН. Серия математическая 2026 Т. 90 № 3 С. 3–18
Пусть симплекс с целочисленными вершинами - содержащий ровно одну целочисленную точку, отличную от своих вершин. В работе доказывается, что если точка находится во внутренности симплекса или в относительной внутренности некоторой гиперграни симплекса, то объем симплекса ограничен величиной, зависящей только от размерности, в противном случае объем симплекса может быть сколь угодно большим. Этот результат применяется для вывода асимптотической формулы для среднего числа вершин полиэдров ...
Added: June 29, 2026
Ivchenko A., Nigmatullin R. R., Dorokhin S. V., Mathematics 2021 Vol. 9 No. 4 Article 381
n this paper, we focus on the generalization of the Hurst empirical law and suggest a set of reduced parameters for quantitative description of long-time series. These series are usually considered as a specific response of a complex system (economic, geophysical, electromagnetic and other systems), where successive fixations of external factors become impossible. We consider ...
Added: June 27, 2026
Ivchenko A., Shestoperov A. I., Fomina E. V., Microgravity Science and Technology 2025 Vol. 37 No. 19 P. 1–19
The paper is dedicated to the analysis of medico-biological data obtained during locomotor testing of astronauts. Accurate data interpretation plays a crucial role in locomotion system monitoring, prophylaxis of long-duration spaceflight negative effects and thus in the development of an autonomous medical support system for deep space expeditions. During the locomotor testing the astronaut changes ...
Added: June 26, 2026
Gadzhimirzaev S., Хельвас А. В., 2023 3rd International Conference on Innovative Research in Applied Science, Engineering and Technology (IRASET) Mohammedia, Morocco 2023 P. 1–6
The article proposes the architecture for eventdriven Emergency Operation Center with Machine Vision Component. Sources of information are analyzed and approaches to machine vision events for tactical situations detection and estimation are discussed. Messages from Machine Vision Components are converted to Common Alerting Protocol and processed by Operation Center environment for tactical situations recognition. ...
Added: June 26, 2026
Gadzhimirzaev S., Хельвас А. В., Лукьянченко П. П., Computer Research and Modeling 2023 Vol. 15 No. 1 P. 129–140
In this article we propose a new approach to the analysis of econometric industry parameters for the industry consolidation level. The research is based on the simple industry automatic control model. The state of the industry is measured by quarterly obtained econometric parameters from each industry’s company provided by the tax control regulator. An approach ...
Added: June 26, 2026
Gadzhimirzaev S., Хельвас А. В., International Frequency Sensor Association (IFSA) Publishing, 19-21 February 2025 Granada, Spain 2025 P. 172–176
The paper presents models for an innovative fully robotic warehouse for storing boxed goods. A discrete multiagent simulation of the movement of shuttles in a warehouse for a given sequence of pallet shipments has been implemented. Different strategies for placement of boxes in various areas of a warehouse are evaluated, as well as optimal routing ...
Added: June 26, 2026
Fedorov Timofey, Moscow Mathematical Journal 2026 Vol. 26 No. 1 P. 73–85
We obtain a complete list of smooth projective threefolds over C for which the dimension of the space of vanishing cycles (in H2(Y,Q) of the smooth hyperplane section Y) equals 2. We also obtain a complete list of rank 2 very ample vector bundles E on smooth projective surfaces with c2(E)=3. ...
Added: June 25, 2026
Воронеж: Издательский дом ВГУ, 2026.
В сборнике представлены материалы докладов и лекций, включенных в программу весенней математической школы. ...
Added: June 25, 2026
Dorovskiy A., / Series arXiv "math". 2026.
In this paper the structural stability of generic families of vector fields of the PC-HC class on the two-dimensional sphere is proved. A classification of these families up to moderate equivalence in neighborhoods of their large bifurcation supports is presented, based on such invariants as the configuration and the characteristic set. The realization lemma is proved. ...
Added: May 14, 2026
Kilin A. A., Ivanova T. B., Lobachevskii Journal of Mathematics 2025 No. 46 P. 1113–1138
In this paper, we address the problem of the rolling motion of a sphere with axisymmetric mass distribution on a horizontal plane. It is assumed that the sphere does not slip as it rolls in the direction of the projection of the symmetry axis onto the supporting plane. The system under consideration admits a redundant ...
Added: December 10, 2025
Kazakov A., Koryakin V., Safonov K. et al., / Series arXiv "math". 2025.
The Lorenz attractor is the first example of a robustly chaotic non-hyperbolic attractor. Each orbit of such an attractor has a positive top Lyapunov exponent, and this property persists under small perturbations despite possible bifurcations of the attractor. In this paper, we study the boundary of the Lorenz attractor existence region in the Shimizu-Morioka model. ...
Added: December 4, 2025
Никулин М. А., Попеленский Ф.Ю., Математический сборник 2025 Т. 216 № 10 С. 101–158
The paper introduces a novel integrable system within an ellipse. The interior of the ellipse is divided into subdomains by arcs of confocal quadrics, each subdomain is filled with a medium characterized by a constant coefficient of «optical» density. Upon traversing the interface between these media, the trajectory adheres to the «cosine» law of refraction. ...
Added: October 3, 2025
Karatetskaia E., Aikan Shykhmamedov, Konstantin Soldatkin et al., Regular and Chaotic Dynamics 2025 Vol. 30 No. 2 P. 306–324
We study hyperchaotic attractors characterized by three positive Lyapunov exponents in numerical experiments. In order to possess this property, periodic orbits belonging to the attractor should have a three-dimensional unstable invariant manifold. Starting with a stable fixed point we describe several bifurcation scenarios that create such periodic orbits inside the attractor. These scenarios include cascades ...
Added: May 13, 2025
http://www.shilnikov.unn.ru/en/news.html?id=20, 2020.
International Conference "ShilnikovWorkshop-2020" dedicated to the memory of the outstanding Russian mathematician Leonid Pavlovich Shilnikov (1934-2011) will be held on 17-18 December, 2020 at the Lobachevsky State University of Nizhny Novgorod. The topics of the Conference include but not restricted by the following themes of the theory of dynamical systems: bifurcations, strange attractors, conservative and ...
Added: November 1, 2021
Korotkov A., Levanova T., Zaks M. et al., Communications in Nonlinear Science and Numerical Simulation 2022 No. 104 Article 106045
A minimalistic model of the half-center oscillator is proposed. Within it, we consider dynamics of two excitable neurons interacting by means of the excitatory coupling. In the parameter space of the model, we identify the regions of dynamics, characteristic for central pattern generators: respectively, in-phase, anti-phase synchronous oscillations and quiescence, and study various bifurcation transitions ...
Added: October 25, 2021
Ilyashenko Y., Chaos 2021 Vol. 31 Article 013103
We study the geometry of the bifurcation diagrams of the families of vector fields in the plane. Countable number of pairwise non-equivalent germs of bifurcation diagrams in the two-parameter families is constructed. Previously, this effect was discovered for three parameters only. Our example is related to so-called saddle node (SN)–SN families: unfoldings of vector fields with one ...
Added: May 27, 2021
Гончарук Н. Б., Ilyashenko Y., Труды Математического института им. В.А. Стеклова РАН 2020 Т. 310 С. 86–106
Обсуждаются различные определения эквивалентности для бифуркаций векторных полей на сфере, и приводится большое количество примеров (как известных, так и новых), которые иллюстрируют достоинства и недостатки разных определений. Кроме классических определений сильной и слабой эквивалентности, рассматриваются новые понятия Sing-эквивалентности и умеренной эквивалентности. Эти определения представляются более подходящими и соответствующими интуитивному понятию эквивалентных бифуркаций. Они были введены и использованы для описания структурной неустойчивости ...
Added: May 27, 2021
Ivan Shilin, / Series math "arxiv.org". 2019. No. arXiv:1903.01933.
An orientation-preserving non-contractible separatrix loop of a hyperbolic saddle of a vector field on a two-dimensional surface may be accumulated by a separatrix of the same saddle. We study the unfolding of such loops in generic one-parameter families of vector fields as a semi-local bifurcation. As a byproduct, we construct a countable family of pairwise ...
Added: November 12, 2020
Пригожин И., В кн.: Мир человека: неопределенность как вызов.: М.: ЛЕНАНД, 2019. С. 55–62.
Nobel prize winner I. Prigogine stands for peace, against the arms race, against the use of science for destruction of man and humanity. In his opinion, in the sphere of human capabilities it is essential to change the trajectory of civilization development. At the bifurcation points, unprecedented changes are possible. Instability is not a sign ...
Added: November 21, 2018
Yu. Ilyashenko, Kudryashov Y., I. Schurov, Inventiones Mathematicae 2018 Vol. 213 No. 2 P. 461–506
We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification defined below has a numerical invariant that may take an arbitrary positive value. Here and below “families” are “families of vector fields in the two-sphere”. This result disproves an Arnold’s conjecture of 1985. Then we construct an ...
Added: February 6, 2018
Springer, 2015.
In the field of Dynamical Systems, nonlinear iterative processes play an important role. Nonlinear mappings can be found as immediate models for many systems from different scientific areas, such as engineering, economics, biology, or can also be obtained via numerical methods permitting to solve non-linear differential equations. In both cases, the understanding of specific dynamical ...
Added: September 4, 2017
Ilyashenko Y., , in: Mathematical Sciences with Multidisciplinary Applications. In Honor of Professor Christiane Rousseau. And In Recognition of the Mathematics for Planet Earth InitiativeVol. 157.: NY: Springer, 2016. Ch. 13 P. 269–299.
This is an outline of a theory to be created, as it was seen in April 2015. An addendnum to the proofs at the end of the chapter describes the recent developments. ...
Added: December 15, 2016