Annotation of the results obtained in 2021
The question of the local (in a neighborhood of the equilibrium state) dynamics of chains of Van der Pol equations connected in a ring and chains of systems of Van der Pol equations has been considered. Critical cases in the stability problem are singled out, and it is shown that they have infinite dimension. In each of these cases, special nonlinear equations are constructed that play the role of normal forms. It is shown that the considered critical cases in the problem of the stability of a distributed chain of logistic equations with delay have infinite dimension. This leads to the fact that the description of their local dynamics is reduced to the study of the nonlocal behavior of solutions of boundary value problems of the Ginzburg – Landau type. It is shown that in a number of cases the solutions contain components that oscillate rapidly and slowly with respect to the spatial variable. The dynamic properties of the original system are determined by a quasinormal form, which includes a parameter. For different values ​​of this parameter, the dynamics of the original boundary value problem can change. Hence it follows that an endless process of forward and backward bifurcations can occur. It is shown that the quasinormal forms that determine the dynamics of the original boundary value problem are the Ginzburg – Landau equations. In particular, it has been established that the properties of the simplest solutions of these equations and their stability are largely determined by the imaginary components of the diffusion coefficients and the Lyapunov value. Numerical analysis of the corresponding criterion made it possible to formulate a conclusion about the instability of the simplest solutions. Thus, in the considered chains, solution synchronization is a rather rare phenomenon. In addition, the orders (with respect to the parameter) of the diffusion coefficients in the original boundary value problem are revealed, which make the contribution of the diffusion term comparable with the term providing the connection between the elements of the chain. The questions of the local dynamics of chains of coupled systems spatially distributed in a two-dimensional region in critical cases have been considered. These critical cases are of infinite dimension. A consequence of this circumstance is the fact that the constructed quasinormal forms are partial differential equations. They contain four spatial variables with boundary conditions for each of them. Their nonlocal dynamics makes it possible to determine the asymptotics of the leading terms of solutions to the original problem. The results obtained are presented in the works Kashchenko S.A., "Dynamics of chains of coupled systems of equations spatially distributed in a two-dimensional domain", Matem. zametki, 110:5 (2021); Math. Notes, 110:5 (2021). https://doi.org/10.4213/mzm13031 Kashchenko S.A., Tolbey A.O., "New Irregular Solutions in the Spatially Distributed Fermi–Pasta–Ulam Problem" // Mathematics. 2021. Vol. 9, no. 22. P. 2872. https://doi.org/10.3390/math9222872 A new definition of an infinite-dimensional torus is proposed and substantiated, the main advantage of which is that within its framework a torus is an analytic Banach manifold with a Finsler metric. This allows for an arbitrary diffeomorphism to define the concept of hyperbolicity in the usual way. A new class of torus diffeomorphisms is introduced, and a hyperbolicity criterion is developed for mappings from this class, which is new not only in the infinite-dimensional but also in the finite-dimensional case. The following paper has been published Glyzin S.D., Kolesov A.Yu., "On some modifications of Arnold's cat map", Doklady Mathematics (2021). https://doi.org/10.31857/S2686954321050064 One of the directions of the project is to study the phenomenon of supratransmission in physical systems modeled by one-dimensional and two-dimensional Hamiltonians, consisting of a large number of particles with local potentials and global (long-range) interactions. We have shown that: (a) For various lattice models with analytic particle interaction potentials, supratransmission appears at higher and higher critical amplitudes of the periodic forcing as the range of interaction becomes longer and longer (i.e. alfa → 0). In the absence of localized on-site potentials in the lattice, this continues down to the longest possible range of interactions alfa=0. (b) When localized potentials are present at every site, we discovered that this monotonic growth of the critical amplitudes is reversed over 0<alfa<2, where the critical amplitudes monotonically decrease towards zero. Following these results, we began to study with several colleagues (mathematicians, physicists, and engineers) energy transport in 1-dimensional Hamiltonian lattices with non-analytic potentials, as well as hysteretic (rather than viscous) damping. In particular, with regard to non-analytic potentials, we studied separately graphene and also Hollomon systems of oscillators and found significant differences from what one finds with analytic potentials. Our studies extended from the stability of fundamental periodic solutions and the discovery of supratransmission effects, to wave packet propagation, where a middle section of the lattice is excited and a “wave” of energy proceeds towards the right and left ends. Our models are very important in engineering and were investigated both for local as well as non local hysteretic damping. We found that many phenomena such as supratransmission and “breather” (localized oscillations) in our hysteretic models occur upon periodic forcing (in the former case) and impulsive driving (in the latter), much like the viscous damping cases. However, in the case of “breather” transmission along the lattice, which had been observed by other authors for models with viscous driving, we found that our models with non local hysteretic damping are close to describing accurately engineering experiments. The following paper has been published T. Bountis, K. Kaloudis, J. Shena, Ch. Skokos, and Chr. Spitas, "Energy Transport in 1-Dimensional Oscillator Arrays With Hysteretic Damping", European Physical Journal. Special Topics (2022). https://arxiv.org/abs/2111.10816 We also studied the topic of integrable and non-integrable perturbations of Lotka-Volterra Hamiltonian (LVH) equations of competing species. We extended the results we had obtained in an earlier paper on LVH systems without linear terms, by generalizing to a system that includes an arbitrary set of linear terms that preserve the Hamiltonian integral. We thus discovered a wide class of LVH systems which are integrable, in the sense that they possess the Painlevé property. Next, we focused on the case n=3, varied some parameters and included additional nonlinearities, which made the system non-integrable. Our results generally yielded simple dynamics far removed from the type of complexity one expects from non-integrable 3-dimensional nonlinear systems. The following work has been published T. Bountis, Z. Zhunussova, K. Dosmagulova, G. Kanellopoulos. "Integrable and non-integrable Lotka-Volterra systems", Physics Letters A (2021). https://doi.org/10.1016/j.physleta.2021.127360 The problem of the existence of invariants of the Koopman operator has been considered, in particular, sufficient conditions for the existence of a complete involutive set of "first integrals" have been obtained. More precisely, the Koopman operator generated by an invertible transformation of a space with a finite countably additive measure has been considered. If the square of this transformation is ergodic, then the orthogonal Koopman operator is a symplectic transformation on the real Hilbert space of square-summable functions with zero mean. An infinite set of quadratic invariants of the Koopman operator is indicated, which are pairwise in involution with respect to the corresponding symplectic structure. For transformations with a discrete spectrum, these quadratic invariants are functionally independent and form a complete involutive set. The following paper has been published Kozlov V.V., "The symplectic geometry of the Koopman operator", Doklady Mathematics (2021). https://doi.org/10.31857/S268695432104010X The problem of the existence of integrals quadratic in momenta with circulating forces has been considered. Conditions are given under which a system with nonpotential forces can be represented in Hamiltonian form with a certain symplectic structure, where the role of the Hamiltonian is played by a quadratic integral, or in a conformally Hamiltonian form. The existence of an integral quadratic in momenta allows one to draw conclusions about the stability of the equilibria of circulation systems. The following work has been published Kozlov V.V. "Integrals of Circulatory Systems Which are Quadratic in Momenta", Regular and Chaotic Dynamics (2021). https://doi.org/10.1134/S1560354721060046 An analysis of the equations of dynamics and a study of the Liouville integrability of the equations of motion of vortex structures in a Bose-Einstein condensate placed in an axially symmetric field has been carried out. Equations of motion are obtained for several vortex filaments in a Bose-Einstein condensate. The equations of motion are presented in Hamiltonian form. The symplectic structure and phase space of this system are obtained. An additional first integral is indicated, and it is shown that the indicated system is completely Liouville integrable. The results of this research can be used as independent results for application in problems of vortex dynamics in Bose-Einstein condensate, in the analysis of the dynamics of vortex structures in a classical ideal fluid, in problems of the dynamics of magnetic vortices in superconductors of reduced dimension, etc. An article has been prepared for publication in the journal "Russian Journal of Mathematical Physics". For a nonlinear system of equations for shallow water in a basin with gentle banks, an asymptotic solution to the Cauchy problem with initial data of small amplitude has been constructed, taking into account the dependence of the domain of definition of the solution on the solution itself and allowing, in the first approximation, to calculate the amplitude of the wave on the bank and the magnitude of the splash by solving the linearized system. The construction of the asymptotic solution relies on a change of variables (such as the simplified Carrier – Greenspan transform) that depends on the unknown solution itself and transforms the domain in which the latter is defined into an unperturbed domain that does not depend on the solution, and leads to the proof of the existence and uniqueness of the asymptotic solution. The successive terms of the asymptotic expansion are the solutions of the Cauchy problem for a linearized system of shallow water equations - a hyperbolic system with degeneration at the boundary of the domain and with right-hand sides depending on the previous expansion terms. It is shown by the uniformization method that this system has a unique solution for smooth initial data and right-hand sides. Based on the results, an article “On asymptotic solutions of the Cauchy problem for a nonlinear system of shallow water equations in a basin with gentle banks” was prepared and accepted for publication in the journal “Russian Journal of Mathematical Physics”. In October 2021, a school-conference for young scientists was held in Yaroslavl with the invitation of leading scientists to give lectures on the project's topics. More than 100 scientists, postgraduates, and students from Russia, UK, Greece, Cyprus, and other countries took part in the school. The main topics of the lectures were the theory of integrable systems and nonlinear dynamics. Lectures at the school prepared young scientists, postgraduates and students to participate in the international scientific conference on integrable systems and nonlinear dynamics (ISND 2021), which was held simultaneously with the school. In addition, some young school participants (including students and postgraduates) had the opportunity to present their scientific results at the conference. Information about the school and conference is available on the website https://lomonosov-msu.ru/eng/event/6844/

 

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Annotation of the results obtained in 2022
Bifurcation dynamical effects of a wide class of chains are studied under the condition that the number of elements in the chain is sufficiently large. It is shown that critical cases in stability problems have infinite dimension. The so-called quasi-normal forms (CNF) are constructed for fully connected systems, for systems with unidirectional and bidirectional constraints and with diffusion type constraints. These CNFs determine primarily the dynamical properties and asymptotics of the solutions of the original systems. In 2022, we have extended our previous results on supertransmission and verified that in the one-dimensional Hamiltonian lattices we studied, as localized potential terms are gradually removed, the supertransmission thresholds increase significantly. We have also studied a number of classes of nonlinear perturbations of Lotka-Volterra integrable systems and determined that significant nonlinear terms must be introduced before important bifurcation phenomena occur. We are currently studying concrete examples, in order to determine exactly what types of new nonlinearities are needed to create models that can describe the realistic behavior expected in complex biological phenomena. We have continued to construct solutions and study their properties for the Zamolodchikov tetrahedron equation, which has fundamental applications in many areas of physics and mathematics, including statistical mechanics, quantum field theories, algebraic topology, and the theory of integrable systems. We have constructed new solutions for this equation related to matrix refactorization problems and associative rings, and showed that some of the obtained solutions are Liouville integrable maps. The problem of exact integrability of the equations of motion of circulation systems with two degrees of freedom is studied. Topological obstacles to integrability have been discovered: if the genus of the configuration space is greater than 1, then in the analytic case the equations of motion do not admit any single-valued first integrals that are polynomial in velocities. A similar result is also valid for systems on a two-dimensional torus under a certain additional condition on the conformal factor of the Riemannian metric on the torus, given by the kinetic energy of the system. A new general theorem on the instability of equilibrium states of dynamical systems with an integral invariant is proved. The conditions imposed on the Lyapunov function are much weaker than in the classical theorems of Lyapunov, Chetaev, and Krasovskii. Asymptotic time-periodic solutions of nonlinear shallow water equations in a bounded basin are constructed, which are related to integrable billiards with semirigid walls and describe coastal waves. For a ring of coupled optical-electronic oscillators, it is shown that there is a one-parameter family of solutions in the form of continuous waves. The asymptotics of the solutions and sufficient conditions for the stability and instability of these solutions are found. It is shown that as a small parameter tends to zero in the system under study, the number of coexisting stable solutions of the type under consideration increases indefinitely, i.e. the phenomenon of hypermultistability is observed. The global dynamics of the Ziegler pendulum, a flat two-link pendulum in the field of a following force, which is always directed along one of the links of the pendulum, is studied. In the classical Ziegler pendulum, it is assumed that there are springs of linear stiffness in both nodes of the pendulum. We assumed that there is no spring at the pendulum suspension point. This allows us to reduce the system and proceed to the study of the system in three-dimensional phase space. It is shown that some open region of this phase space is stratified into periodic solutions, i.e., in particular, the system is integrable by the classical ODE integrability theorem, since it has two independent first integrals. However, apparently, the system is not integrable everywhere, as the results of numerical analysis show. A system of three ring-coupled generators with asymmetric nonlinearity is considered. Asymptotic methods are used to study the problem of oscillatory regimes branching from equilibrium states. In our work, a numerical experiment has been carried out, which allows one to establish the boundaries of asymmetry, at which the system generates periodic and chaotic oscillations. The dependence of the system dynamics on the degree of asymmetry of the cubic nonlinearity describing the characteristic of the nonlinear element is studied. A mathematical model has been developed that describes in detail the nonlinear-dynamic, self-oscillatory properties of the circuits of the autonomic regulation of the heart rhythm, mean arterial pressure, chemoreceptor regulation of the frequency and depth of breathing, taking into account the bidirectional interaction between the circulatory and respiratory systems. The proposed mathematical model made it possible to carry out a numerical experiment aimed at studying the causes of heart rhythm irregularity. It has been found that estimates of a number of important physiological parameters, including heart rate, systolic blood pressure, diastolic blood pressure, breathing depth, minute volume of inhaled air, LF-index and HF-index, demonstrate significant changes after the blockade of the regulation of the frequency and depth of breathing. The results indicate that the transition of the dynamics of the respiratory system from irregular to periodic does not have a significant effect on the values of the senior Lyapunov exponent and the correlation dimension, estimated from the signals of RR intervals. In 2022, in the framework of the project we have continued the study of the phase topology of completely Liouville-integrable problems of vortex dynamics and rigid body dynamics in an ideal fluid in the presence of vortex filaments. In particular, the problem of the motion of a cylindrical rigid body and two vortex filaments parallel to the generatrix of the cylinder is studied. A method is proposed for obtaining asymptotic representations for solutions of discrete systems containing oscillatory decreasing coefficients. The method is illustrated by the example of the problem of constructing asymptotic formulas for solutions of a higher order scalar difference equation. A method for constructing asymptotic formulas for solutions of a certain class of differential equations in a Banach space is proposed. This method is used for asymptotic integration of some systems of differential equations with distributed parameters. In particular, asymptotics are obtained for solutions of a perturbed heat equation. In the period from 27.06.2022 to 01.07.2022, the Scientific School "Nonlinear Days" was held by the P.G. Demidov Yaroslavl State University on the basis of a partner organization -the Information Technology Company "Tensor". More than 100 students and scientists from Russia, as well as from more than 10 countries, including the UK, Italy, Greece, USA, South Africa, Israel, Kazakhstan, Benin, the Republic of Korea, and Slovakia, took part in the scientific school. Participants of the RSF project 21-71-30011 made interesting presentations that cover a wide range of topics related to the RSF project 21-71-30011, including applied problems, such as the study of the stability of suspension pedestrian bridges, as well as more abstract problems, such as the use of algebraic-geometric constructions in solving the Zamolodchikov tetrahedron equation. In addition, some young participants of the school (including undergraduate and graduate students) had the opportunity to present their scientific results at the conference. Information about the school is available on the website https://lomonosov-msu.ru/eng/event/6251/

 

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Annotation of the results obtained in 2023
A model of a neuron generating short-term, high-amplitude impulses, based on an equation with a delay, has been developed and studied. It is shown that under certain and very natural assumptions, the specific form of the functions included in the equation is not significant. A model of interaction between neurons, and thus a neural network, is proposed. Mathematical methods for studying systems of equations for a neural population have been developed. It has been proven that adaptation of the strength of interaction between neurons in the model forms a population that generates a given periodic sequence of impulses (the ability to record information). A model of synchronization of neural structures has been constructed, which makes it possible to establish the identity of pulse sequences. We proposed a retarded integro-differential model to describe the dynamics of a ring of coupled lasers if the number of elements is large enough. Such a distributed model allows one to analyze the critical conditions under which the stationary state becomes unstable. For a wide class of diffeomorphisms on an infinite-dimensional torus, the following statements are substantiated: the theorem on the topological conjugacy of a hyperbolic diffeomorphism with a linear hyperbolic automorphism; topological mixing theorem; theorem on structural stability. For the infinite-dimensional torus introduced by the authors, the classical question in the theory of dynamical systems about the minimality of the shift mapping on it is studied. The problem of finding sufficient conditions guaranteeing the absence of the minimality property is solved. Several new results have been established, some of which have no analogues in the finite-dimensional case. The classical hypothesis of instability of isolated fixed points of a mechanical system in a divergence-free vector field is studied. The instability results are proved in a general case and in a typical degenerate case assuming that the vector field in the right-hand side is smooth enough. Moreover, it is proved that, for the considered degenerate situation, there exists a trajectory that leaves the equilibrium. A topological-analytical method for proving some results of the N.N. Bogolyubov method of averaging, for the case in which the time interval is infinite, has been proposed. The considered approach allows us to abandon the non-degeneracy condition on the Jacobi matrix from the classical theorems of the averaging method. Asymptotic solutions of a system of nonlinear equations of shallow water in a pool with gentle shores, describing coastal waves and associated with the so-called billiards with “semi-rigid” walls, are obtained. A variational method has been developed for solving the problem of reflecting the ray characteristics of long ocean waves from the coastline with given positions of the source and the wave registration point. It is shown that the original boundary value problem can be reduced to calculating stationary points of the functional of the time of wave propagation along the ray. A feature of the proposed approach is the optimization of the beam reflection point along a given coastline. Calculations were carried out for several model situations and good agreement with the trajectories was obtained in the case of exactly solvable models. A new approach has been developed for averaging linearized shallow water equations in the problem of wave propagation over rapidly changing areas of the bottom. Effective formulas defining the averaged equations are derived. The difference from standard approaches is that the approach is applied to “not very long” waves, which leads to the appearance of dispersion effects and isotropy effects during propagation. An algorithm has been compiled based on a combination of the above approaches (variational and averaging), calculation of trajectories and fronts over oscillating inclined sections of the bottom near the coastline. The algorithm is tested on several examples. At the heuristic level, an asymptotic formula is derived for the exponentially small splitting of weakly excited lower energy levels of the multidimensional Schrödinger operator with a symmetric potential of the symmetric double well type. Ferromagnetic dissipative systems, described by the isotropic Landau–Lifshitz–Hilbert equation, were studied from the point of view of their spatially localized dynamic excitations. In particular, dissipative soliton solutions of the nonlocal nonlinear Schrödinger equation, into which the Landau Lifshitz–Hilbert equation is transformed, were constructed. To prove the existence of these solutions at sufficiently small scattering, Melnikov's theory was used. To verify the reliability of the obtained analytical results, pseudospectral numerical methods and physically informed neural networks were used in a machine learning scheme. Such localized structures were discovered experimentally in magnetic systems and observed in nanooscillators, and magnetic drops described by dissipative solitons were studied theoretically and observed experimentally. We extended earlier results using a 4-dimensional extension of the 2D MacMillan map and showed that the symplectic model of two coupled MacMillan maps also exhibits stickiness phenomena in limited regions of phase space. We also investigated a class of one-dimensional (1D) Hamiltonian lattices of N particles whose binary interactions are quadratic and/or fourth-order in potential. Using a sinusoidal perturbation at one end of the grating and an absorbing boundary at the other, we investigated the phenomenon of overtransmission and its dependence on two ranges of interactions as the influence of local potential terms of the Hamiltonian changes. Methods have been developed for constructing mappings of 3-simpex and 4-simplex using matrix refactorisation problems of local Yang-Baxter and Zamolodchikov type equations. Using the developed methods, we constructed new 3-simplex and 4-simplex maps, including birational non-involutive mappings on groups and division rings (associative rings with division). Matrix differential-difference Lax pairs play one of the key roles in the theory of nonlinear integrable differential-difference equations. We obtained sufficient conditions that a given matrix Lax differential-difference pair can be simplified using gauge transformations, and developed a procedure for such simplification. We have demonstrated how our procedure works to simplify Lax pairs of scalar differential-difference equations, including equations of the Narita-Ito-Bogoyavlensky type. We also showed that from some Lax pairs simplified by our method, new nonlinear integrable equations can be obtained. Value ranges have been selected for the parameters of the traffic flow model. The software package has been improved - new scenarios for modeling transport situations have been added. For the developed prototype of an aquatic robot driven by two internal moving masses, a theoretical model of motion in liquid is proposed. The equations of motion are written in the form of the Kirchhoff equations to describe the motion of a rigid body in an ideal fluid, which are supplemented with terms of viscous resistance. Also, hydrodynamic forces and moment of forces that act on the tail fin mounted on the robot body were added to the equations. This robot is controlled by changing the angular velocities of the moving masses. The theoretical model showed that when two masses rotate at a constant speed, but in different directions, the robot moves along a straight line, and when rotated in the same direction, the robot turns. In this way, movement can be carried out along any trajectories. For the limiting system of Mackey-Glass equations, sufficient conditions for the parameters were found under which a solution to the system exists in the form of a discrete traveling wave.

 

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