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Campus inMoscow
• HSE University
• Research projects
• Construction of mathematical models designed for the analysis of data series, prediction of big systems’ evolution, and the control of the dynamics of these systems

# Construction of mathematical models designed for the analysis of data series, prediction of big systems’ evolution, and the control of the dynamics of these systems

Priority areas of development: IT and mathematics
2019
The project has been carried out as part of the HSE Program of Fundamental Studies.

#### Goal of research

The design of imitative models and numerical methods intended to reconstruct the parameters of complex systems among which are the van der Pol and Kuramoto oscillators with a moderate coupling, self-organized critical model on the self-similar lattice, and the network of the market agents which are endowed by incomplete information about consumer demand.

#### Methodology

Depending on the system in question, we use the following approaches.

A. We investigate two van der Pol equations with a moderate coupling between them. The algorithm, which finds the limit cycle of the equations numerically, is constructed. For any initial condition a quasi-cycle of the solution is defined. It induces the determination of the quasi-period of the solutions. Then we compute the correlation of the stabilized oscillators within the quasi-periods. This correlation underlies the estimate of the coupling between the oscillators. The estimate is compared with that found for the Kuramoto model. We find the conditions on the correlation between the solutions that provide similar reconstruction of the coupling in the two models.  Further, three Kuramoto equations are investigated. An explicit relationship between the edge oscillators is not assumed. Solving the direct problem, we find the phase of the middle oscillator when the natural frequencies and the coupling do not depend on time. We also determine sufficient conditions that lead to the stability of the found phases. Then we turn to the inverse problem reconstructing the phases, given the coupling between the oscillators. The stability of the reconstruction is justified by standard methods of the differential equations. They reveal singular values of the parameters that require specific consideration. We found those that lead to the non-existence of the reconstruction or its non-uniqueness.

B. Complex magnetoplasma structures were simulated in the magnetospheres of the planets of the solar system. It is shown that high relative concentrations of oxygen ions, as well as their relatively high temperatures and drift velocities lead to a significant thickening of such structures and the formation of an additional nesting scale.

C. We construct a numerical algorithm that constructs the Bak-Tang-Wiesenfeld mechanism on the self-similar lattice. This mechanism is characterized by slow constant loading and rare immediate stress-release. As a result, the constructed system exhibits oscillations around the critical state. The research deals with the level of the load associated with the critical state. We establish that the system oscillating around the critical level follows a power-law size-frequency distribution of the events on the main interval of the sizes. An abrupt downward bend changes the power-law segment at the right part of the distribution. We investigate the predictability of the events that are located on the downward bend. The efficiency of the prediction algorithms are estimated with modified statistical errors of type I and II. They are the rate of the unpredicted events and the alarm rate. The simplest prediction algorithm deals with the inter-event time distribution. The reference point of the unpredictability then is the exponential distribution. We analyze the proximity of the observed and theoretical distributions numerically and hypothesize that the scenario of the extreme events involves the transition of the system into the super-critical state.

D. We define a structural model of general equilibrium with separable consumer preferences and demand uncertainty observed by producers. In mathematical terms, the model consists of the consumer optimization problem, firm optimization problem, and balances which close the underlying system of algebraic equations. The existence and uniqueness of the equilibrium is established analytically. The equilibrium is described through the system of the equations that relate the variables of interest to the parameters of the model. The standard deviation of the random variable, which defines the uncertainty, is among those parameters. Evaluating the derivative of the equilibrium variables with respect to the standard deviation we end up with the influence of uncertainty on these variables.

E. Testing is performed with model with structured data of general type. We define classes of the functions used in the model. Then testing is described with the corresponding graphs. We formulate and solve numerically the problem regarding non-availability of several vertices. Then we determine the number of random tests required for an efficient adjustment of the constructed algorithm.

#### Empirical base of research

The databases of index ISSN of the sunspots (http://www.sidc.be/silso/datafiles), sunspot groups RGO (https://solarscience.msfc.nasa.gov/greenwch.shtml), and solar faculae (https://dataverse.harvard.edu/dataverse/solardynamo).

#### Results of research

A. We introduce a numerical procedure that determine the amplitude and phase of the oscillators given by two van der Pol equations with the moderate coupling between them. It is established that if the correlation between the solutions of the corresponding equations is positive then the inverse problem on the reconstruction of the coupling is well posed. We find the sufficient conditions providing that the reconstructions of the coupling in the van der Pol and Kuramoto equations lead to similar values. The reconstruction of the profile of the meridional flow in the Sun gives an application of the developed methodology. We perform a theoretical analysis of the Kuramoto model with three oscillators, where the coupling between the edge oscillators is designed through the middle one and the relationship within any pair of the oscillators is symmetrical. The correctness of the direct and inverse problems for the synchronized oscillators is established rigorously. We prove the existence and stability of the reconstruction of the coupling for almost all values of the phase difference. The case of singular values of the parameters is fully investigated.

B1. A hybrid model of a complex magnetoplasma system – a thin current sheet in a plasma consisting of three components: protons, electrons, and oxygen ions – has been developed. It is shown that the corresponding profiles of current density, magnetic field, and plasma density have a multiscale embedded character.

B2. In the presence of oxygen ions, the thickness of the current sheet can increase significantly compared with the proton-electron plasma. At a relative concentration of heavy ions above 5%, their high thermal and directional velocities, kinks (jumps in derivatives) on the current density and magnetic field profiles become noticeable, indicating a change in the regions of dominance of different types of particles.

B3. The results of this work are consistent with observational data at a qualitative and quantitative level.

C. We gave a complete description of the extreme events in a model of self-organized criticality on the self-similar lattice. The power-law exponent is found. The proximity between observed and theoretical inter-event distributions is estimated in terms of elementary prediction algorithms and type I and II statistical errors. We claim that the deviation of the observed inter-event distribution from the exponential one is insufficient to predict the extreme events efficiently based only on the information regarding their occurrence.

D. We assess the impact of the uncertainty in demand on the welfare of economic agents if the consumer preferences are given by separable utilities of a general type. We propose a mechanism staying beyond a positive influence of the uncertainty on the economy. This mechanism relates the uncertainty to a growth of abilities. In the model, the equilibrium is described analytically and the existence of the above mechanism is derived under quite general conditions.

E. We develop a prototype for the interactive event clusters in the list of the distributed trading systems. The efficiency of the preliminary tests is confirmed with standard inputs.

#### Publications:

E. E. Grigorenko, Zelenyi L., Dibraccio G., Ermakov V., Shuvalov S., Malova H., V. Y. Popov, Halekas J. S., Mitchell D. L., Dubinin E. Thin Current Sheets of Sub‐ion Scales observed by MAVEN in the Martian Magnetotail // Geophysical Research Letters. 2019. Vol. 46. P. 6214-6222. doi
Zelenyi L., Malova H., Grigorenko E., Victor Popov, Delcourt D. Current sheets in planetary magnetospheres // Plasma Physics and Controlled Fusion. 2019. Vol. 61. P. 1-12. doi
Parkhomenko E. I., Malova H. V., Grigorenko E. E., Popov V., Petrukovich A. A., Delcourt D. C., Kronberg E. A., Daly P. W., Zelenyi L. M. Acceleration of plasma in current sheet during substorm dipolarizations in the Earth’s magnetotail: Comparison of different mechanisms // Physics of Plasmas. 2019. Vol. 26. No. 4. P. 1-9. doi
Savostyanov A., Shapoval S., Shnirman M. G. The inverse problem for the Kuramoto model of two nonlinear coupled oscillators driven by applications to solar activity // Physica D: Nonlinear Phenomena. 2020. Vol. 401. P. 132160. doi
Домрин В. И., Малова Х. В., Попов В. Ю., Григоренко Е. Е., Петрукович А. А. ВЛИЯНИЕ ИОНОВ КИСЛОРОДА НА СТРУКТУРУ ТОНКОГО ТОКОВОГО СЛОЯ В ХВОСТЕ ЗЕМНОЙ МАГНИТОСФЕРЫ // Геомагнетизм и аэрономия. 2020. Т. 60. № 2. С. 173-186. doi
Savostianov A., Shapoval S., Shnirman M. Reconstruction of the coupling between solar proxies: When approaches based on Kuramoto and Van der Pol models agree with each other // Communications in Nonlinear Science and Numerical Simulation. 2020. Vol. 83. P. 105149. doi
E. V. Maiewski, Kislov R. A., Khabarova O. V., Malova H. V., V. Yu. Popov, Petrukovich A. A., Zelenyi L. M. Magnetohydrodynamic Modeling of the Solar Wind Key Parameters and Current Sheets in the Heliosphere: Radial and Solar Cycle Evolution // Astrophysical Journal. 2020. Vol. 892. No. 1. P. 1-17. doi
Dayana Mukhametshina, Shapoval S., Shnirman M. G. Self-Organized Criticality on Self-Similar Lattice: Exponential Time Distribution between Extremes, in: Proceedings of the MACSPro Workshop 2019 / Ed. by Irina Lomazova, Anna Kalenkova, Р. Яворский. Vol. 2478: CEUR Workshop Proceedings. CEUR-WS.org, 2019. P. 127-138.
Fomenko M., Alexander Shapoval. Modeling of Necessity Entrepreneurship via General Equilibrium Approach, in: Proceedings of the MACSPro Workshop 2019 / Ed. by Irina Lomazova, Anna Kalenkova, Р. Яворский. Vol. 2478: CEUR Workshop Proceedings. CEUR-WS.org, 2019. P. 185-190.