1.State of art of the collaborative project
Modern theory of dynamical systems originates from Newtonian mechanics, an active area of mathematics and theoretical physics used to describe the long term quantitative behaviour of complex dynamics, usually by employing the iterations of maps (discrete time) or differential equations (continuous time). In reality, such long term quantitative behaviour is often primarily mechanical or otherwise physical in nature.

The project is meant to lie in the extending geometric and topological aspects of dynamical system studies, in particular on topics in stability, which is one subject in dynamical systems with the longest history. Despite the apparent differences in the type of questions, there is a natural and global coherence. We will also aim to apply these theoretic results in realistic models arising from biomedical and power grid systems.

2. Preliminary works:
Theoretic part:
General goal: Roughly speaking, stability theory studies the stability of solutions of differential equations and dynamical systems. Contexts in stability include (local) stability discussions on equilibrium points (e.g. Lyapunov stability, linearization et.al) or existences of (quasi)periodic solution or limit cycles, or the globally structural stability.

Famous mathematical theories such as Hartman-Grobman theorem, central manifold theory as well as normal form theory have already given solutions for 2D and even abundant 3D dynamical systems systematically. However, once the system is 4D and above, or it is highly nonlinear, or coupled, or with hysteresis, classical methods (mentioned above) will have shortages here and there. Therefore, old theories need to be developed, and new approaches need to be created.

1.1 Local qualitative analysis of essentially non-linear systems of differential equations, and stability analysis for hysteresis systems.
Within the framework of the project it is planned by Prof. Ilyin Yury A. to continue the study of essentially nonlinear systems of differential equations in the vicinity of the rest point. The main goal is to find the conditions under which statements similar to key statements for quasilinear systems hold for essentially nonlinear systems, for example, the Poincaré - Lyapunov - Hadamard - Perron theorem, the Grobman - Hartman theorem, theorems on a neutral manifold, theorems on the existence of an invariant foliation, various stability theorems for the rest point. In the course of such studies, we find paradoxical facts and phenomena that are impossible in the classical quasilinear theory. Such studies are extremely important and interesting for locally qualitative analysis of complex rest points. They also enrich the mathematical apparatus of nonlinear analysis. The study of stability can be useful in the analysis of complex resting points in applied models. Relying on the existence of an invariant bundle in a neighborhood of a neutral manifold, it is planned to prove a reduction principle similar to the well-known V.A. Pliss reduction principle for quasilinear systems. The main attention will be paid to find effectively verifiable coefficient conditions on the right-hand sides of the system, which is important for practical applications of the results obtained. The main results of prof. Ilyin Yury A. on this topic were published in [1,2]

Within the framework of the project it is planned by Prof. Zvyagintseva T.E. to continue researching second-order automatic control discrete-time systems with various nonlinearities. In particular, with nonlinearity that satisfies the generalized Routh-Hurwitz condition. Systems of this type are mathematical models that describe the dynamics of the behavior of solutions in many applied systems that arise in modern engineering problems, motion control theory and problems of natural science. A discrete representation of a mathematical model is necessary in decision theory, logical control, digital automata.

A large number of works in recent decades have been devoted to the study of such systems, the formulation of conditions for absolute stability, and methods of searching for periodic oscillations in systems with discrete time. In the papers published by W. Heath, J. Carrasco, M. de la Seine (2015) [3-5], two examples of two-dimensional discrete systems with nonlinearities satisfying the Routh-Hurwitz conditions, which have periodic trajectories, were constructed. In the papers of V.P. Heath and J. Carrasco (2015) the global asymptotic stability of discrete systems is investigated. In the papers (2020, 2021) [6, 7] a second-order discrete-time automatic control system is studied. It is assumed that the nonlinearity of this system is 2- or 3-periodic and satisfies the generalized Routh–Hurwitz condition. This system is analyzed for all acceptable values of the parameters. Conditions on the parameters under which it is possible to construct the nonlinearity in the way that the system has a family of cycles of period three, four or period six are given.

Scientific questions: the existence of integral manifolds in the vicinity of the rest points of such systems, topological equivalence, Lyapunov stability and asymptotic behaviour of solutions. It is planned to obtain new criteria for absolute stability for specific discrete systems with a given nonlinearity and criteria for the existence of periodic motions and limit cycles in such systems.

1.2 Chaos analysis in dynamical systems.
In contrast to stability, chaos theory is the study of deterministic difference (differential) equations that display sensitive dependence upon initial conditions in such a way as to generate time paths that look random.

For hyperbolic systems, various mechanisms such as homoclinic points and heteroclinic contours with tangency, or positive Lyapunov exponent will indicate the existence of Chaos.

Within the framework of the project it is planned by prof. Vasil’eva E.V. to continue the study of the properties of parametric sets of dynamical systems under the assumption that elements of these sets can be dynamical systems with non-transversal heteroclinic contours or homoclinic points, and the points of non-transversal intersection of stable and unstable manifolds are not finite order tangent points. It is supposed to study invariant sets of points whose trajectories do not leave a small neighborhood of the trajectory of non-transversal homoclinic point. In most of the previously published papers, for instance [8, 9], it was assumed that the points of non-transversal intersection of such manifolds are tangency points of finite order. The main results of prof. Vasil’eva E.V. on this topic were published in [10, 11]

Within the framework of the project N.A. Begun plans to continue the research in the
following directions.

First, it is proposed to continue the study of weakly hyperbolic invariant sets. In a series of papers published by N. A. Begun (together with V. A. Pliss and G. Sell) [12, 13] the stability of such structures was proved even in the case when the stable, unstable, and neutral linear spaces of the corresponding linearized system do not satisfy the Lipschitz condition. During the proof, they found continuous map such that the image of the weakly hyperbolic invariant set of the unperturbed system under the action of this map is equal to the weakly hyperbolic invariant set of the perturbed system. However, the question of whether this map is homeomorphic remains open. As it turned out, this issue is closely related to the so-called plague expansivity conjecture, which has been studied in many works over the past 70 years.

Second, and the main, it is proposed to continue the study of chaotic systems with a hysteresis stop operator. An important property of the stop operator (as, indeed, of any other hysteresis operator) is the fact that its value depends not only on the argument, but also on the pre-history of the system. The stop operator finds applications in a wide variety of fields, from medicine and biology to economics and finance. It was made an attempt to use the stop operator to build a macroeconomic model of inflation. The stop operator was interpreted as the risk assumed by the trading agent (we also note that for the first time the stop operator was adapted for discrete time). The main results of this topic were published in [14, 15]. It is supposed to find the areas of parameters where chaotic scenarios arise. In these areas describe as detailed as possible the behavior of the system. In other areas totally describe observed dynamics.

Scientific questions: Investigating invariant sets of general diffeomorphisms (beyond Anosov), non-transversal homoclinic points and heteroclinic contours with tangency that is not a tangency of finite order. Investigating stability of weakly hyperbolic invariant sets, systems with a hysteretic stop operator, and shadowing properties.

1.3 Global bifurcation.
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves. Global bifurcations occur when 'larger' invariant sets, such as periodic orbits, collide with equilibrium. This causes changes in the topology of the trajectories in the phase space which cannot be confined to a small neighbourhood, as is the case with local bifurcations. In fact, the changes in topology extend out to an arbitrarily large distance.
As part of the project, prof. Basov V.V. will continue the study of bifurcations of invariant tori in systems with a small parameter with periodic perturbations and a Hamiltonian unperturbed part for various Hamiltonians. In works [16-18], various Hamiltonians were considered using the developed by prof. Basov V.V. method. The universal method, meaning it can be applied to any harmiltonian, called the method of the generating tori splitting (GTS method), is developed and applied to the research of such systems. For arbitrary system of any class, this method allows to find the sets of the initial values for the solutions of the corresponding unperturbed system, and for each such set, to provide the explicit conditions on the system perturbations independent of the parameter. Any such set, that satisfies the aforementioned conditions, determines the solution of the unperturbed system, which parametrizes the generating cycle. The obtained cycle is a generatrix of an invariant cylindrical surface. It is proven that the system has two-periodic invariant surface, homeomorphic to torus, if time is factored with the respect to the period, in the small with the respect to a small parameter neighborhood of this surface. The formula and the asymptotic extension are provided for this surface, and the number of properties is discovered.
An example of the set of systems with eleven invariant tori and a perturbation, which average value is a three-term polynomial of the third degree, is constructed in [16] as a demonstration of the practical use of the GTS method.
The GTS method is a universal alternative to the so-called method of detection functions and the Melnikov function method, which are used in studies concerning the weakened XVI Hilbert's problem on the evaluation of a number of limit cycles of autonomous systems with the hamiltonian unperturbed part.
The GTS method allows not only to evaluate the lower bound of the analogue of the Hilbert's cyclicity value, which determines the amount of the invariant tori in the periodic systems with "slow" time, but also to solve the same problem for the periodic systems of any even degree with the common factor a small parameter in its right-hand side. The results can be put into practice, while researching the systems of the ordinary differential equations of the second degree, which describe the oscillations of the weakly-coupled oscillators.
Other possible applications of the VPT method and the Hamiltonian for further research can be found, for example, in [14,19].

Application part:
1. Biomedical systems such as the neuron network and its associated neuron disorder is also a complicated network. Stability and bifurcation analysis of systems is a key element in studying biomedical systems about maintenance of health conditions. In particular, neuron disorder such as Parkinson disease and epilepsy has received wide interests in both biomedical research and mathematical modelling [20-21]. Early warming for these diseases would have a high potential to enhance the living conditions of patients [21-22]. We propose to use the recent progress on chaos and stability to study the neuron network stability and study key mechanisms in neuron disorder.

Scientific Questions: how to build a realistic model for neuron disorder based on real data and how to understand the mechanism underlying the disorder by stability and bifurcation analysis?

2. Power grid system is an interconnected network for electricity delivery from producers to consumers. Renewable energy power generation, direct current transmission and efficient utilization of electric energy have promoted the historic transformation of power equipment from electro magnetization to power electronics, leading to major changes in the dynamic mechanism of the new generation of power systems [23-24]. The safe and stable operation of power systems is facing major challenges. The basic theory of power system dynamics and the innovation of key technologies put forward an urgent need. Using the theory of complex large-scale systems to study the mathematical and physical nature of dynamic problems in power electronic power systems is the core topic to deal with this major challenge. Stability analysis of such large-scale power systems and its associated reduced system is of particular importance. We aim to use recent progress on stability and bifurcation theory to the application of the power system and propose methods to quantify how the reduced system captures the realistic system, see [25].

Scientific Questions: how to reduce the large-scale power system while maintaining the stability?

References
[1] Il’in Y A. On the existence of a local-integral manifold of neural type for an essentially nonlinear system of differential equations. Vestnik St. Petersburg University: Mathematics, 2007, 40(1): 36-45.
[2] Il’in Y. Existence of an Invariant Foliation Near a Locally Integral Surface of Neutral Type. Lobachevskii Journal of Mathematics, 2021, 42(14): 3438-3450.
[3] Heath W P, Carrasco J. Global asymptotic stability for a class of discrete-time systems//2015 European Control Conference (ECC). IEEE, 2015: 969-974.
[4] Heath W P, Carrasco J, de la Sen M. Second-order counterexamples to the discrete-time Kalman conjecture. Automatica, 2015, 60: 140-144.
[5] Carrasco J, Heath W P, de la Sen M. Second-order counterexample to the discrete-time Kalman conjecture//2015 European control conference (ECC). IEEE, 2015: 981-985.
[6] Zvyagintseva T E. On the Aizerman problem: Coefficient conditions for the existence of a four-period cycle in a second-order discrete-time system. Vestnik St. Petersburg University, Mathematics, 2020, 53: 37-44.
[7] Zvyagintseva T E. On the Conditions for the Existence of Cycles in a Second-Order Discrete-Time System with a Sector Nonlinearity. Vestnik St. Petersburg University, Mathematics, 2021, 54: 50-57.
[8] Gonchenko S V, Sten'kin O V. Homoclinic Ω-explosion: Hyperbolicity intervals and their boundaries. Russian Journal of Nonlinear Dynamics, 2011, 7(1): 3-24.
[9] Gonchenko S V, Gonchenko A S, Kazakov A O. Three types of attractors and mixed dynamics of nonholonomic models of rigid body motion. Proceedings of the Steklov Institute of Mathematics, 2020, 308: 125-140.
[10] Vasil’eva E V. Stable and Completely Unstable Periodic Points of Diffeomorphism of a Plane with a Heteroclinic Contour. Vestnik St. Petersburg University, Mathematics, 2020, 53: 261-269.
[11] Vasil’eva E V. One-Parameter Set of Diffeormorphisms of the Plane with Stable Periodic Points. Lobachevskii Journal of Mathematics, 2021, 42(14): 3543-3549.
[12] Begun N A, Pliss V A, Sell G R. On the stability of weakly hyperbolic invariant sets. Journal of Differential Equations, 2017, 262(4): 3194-3213.
[13] Begun N A, Pliss V A, Sell J R. On the stability of hyperbolic attractors of systems of differential equations. Differential Equations, 2016, 52: 139-148.
[14] Arnold M, Begun N, Gurevich P, et al. Dynamics of discrete time systems with a hysteresis stop operator. SIAM Journal on Applied Dynamical Systems, 2017, 16(1): 91-119.
[15] Cross R, McNamara H, Pokrovskii A, et al. A new paradigm for modelling hysteresis in macroeconomic flows. Physica B: Condensed Matter, 2008, 403(2-3): 231-236.
[16] Basov V V, Zhukov A S. Weakened XVI Hilbert's Problem: Invariant Tori of the Periodic Systems with the Nine Equilibrium Points in Hamiltonian Unperturbed Part. arXiv preprint arXiv:2302.02397, 2023.
[17] Basov V V. Invariant surfaces of standard two-dimensional systems with conservative first approximation of the third order. Differential Equations, 2008, 44: 1-18.
[18] Basov V V, Zhukov A S. Invariant Surfaces of Periodic Systems with Conservative Cubic First Approximation. Vestnik St. Petersburg University, Mathematics, 2019, 52(3): 244-258.
[19] Li Q, Huang Q. Bifurcations of limit cycles forming compound eyes in the cubic system. Chinese Annals of Mathematics, 1987, 4.
[20] Parakkal Unni M, Menon P P, Wilson M R, et al. Ankle push-off based mathematical model for freezing of gait in parkinson's disease. Frontiers in Bioengineering and Biotechnology, 2020, 8: 552635.
[21] Arbab M, Baars S, Geijsen N. Modeling motor neuron disease: the matter of time. Trends in neurosciences, 2014, 37(11): 642-652.
[22] Creaser J, Lin C, Ridler T, et al. Domino-like transient dynamics at seizure onset in epilepsy. PLoS computational biology, 2020, 16(9): e1008206.
[23] Menck P J, Heitzig J, Kurths J, et al. How dead ends undermine power grid stability. Nature communications, 2014, 5(1): 3969.
[24] Zhang Y, Zhang C, Cai X. Large-signal grid-synchronization stability analysis of pll-based vscs using lyapunov's direct method. IEEE Transactions on Power Systems, 2021, 37(1): 788-791.
[25] Ma R, Zhang Y, Yang Z, Kurths J, Zhan M, Lin C. Synchronization Stability of Power-grid-tied Converters, accepted in Chaos, 2023.

Planned activities
Year 1: propose to hold online seminars, from both interior and exterior. The interior seminar will be participants in this group and will make presentations on their scientific interests, exterior seminar will invite experts worldwide, to introduce the latest updates from the topics proposed in this proposal.

Year 2: Project member visits between SPBU and HUST to establish output from the project. We will also suggest reports in the Joint Seminar at Sino-Russia Mathematical center in Peking University.
Please detail the specific outputs that will arise from this collaborative project. Refer to the funding call guidance for more detail.

We aim to output:
1.collaborative works (aiming for 3 or more papers) published in top international peer reviewing journals;
2.foster students together, promote mobility of students, and collaborate to apply for further international collaboration project;
3. create new collaboration opportunity to provide interdisciplinary studies using the knowledge of mathematics(especially theoretic and applied dynamical theory ) between China and Russia.
4. enhance the impact on joint mathematical communications between China and Russian especially in Center China, and aim to establish a Wuhan base for Sino-Russia Mathematical center.



Sustainability and Year 2 Plan
Please outline a year 2 plan and explain how this funding may lead to future research or external funding that would sustain the collaboration.

Ideas of the research project are formulated for particular cases of dynamical systems. This is why, the majority of them admit various generalizations and the research area is quite strong regarding the output. This is why a long research in the suggested area is possible.

Academically,
Based on our previous research basis, we will sustain
1)Investigate the comparison of theoretical results and numerical studies will be given.

2) investigate biomedical models with hysteresis effect.

3) investigate the method of small parameter for the biomedical and renewable energy models.


In year 2, we are aiming to apply for international collaboration funding from NSFC-RSF (e.g. RSF-NSFC Cooperation: Possibility for Joint Russian-Chinese Project) and also research grants opportunities from Sino-Russia Mathematical center. We are also aiming to suggest reports in the Joint Seminar at Sino-Russia Mathematical center, and aiming to establish a Wuhan base for Sino-Russia Mathematical center, as Sichuan University did for the Chengdu base for Sino-Russia Mathematical center.


Moreover, we will promote mobility of PhDs, submit a significant external funding application, and establish basis for future strategic partnership strategic cooperative partnership.