# Roman Ivanovich ParovikRussian Academy of Sciences | RAS · Laboratory of Physical Process Modeling

Roman Ivanovich Parovik

DSc

Leading Researcher, Laboratory for Modeling Physical Processes, IKIR FEB RAS

## About

106

Publications

9,747

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525

Citations

Citations since 2016

Introduction

Qualitative analysis of fractional oscillators

Additional affiliations

December 2014 - present

**Kamchatka State University**

Position

- Head of Faculty

Education

September 2001 - July 2006

**Kamchatka State University**

Field of study

- Applied Mathematics and Computer Science

## Publications

Publications (106)

A non-linear fractional Selkov dynamic system for mathematical modeling of microseismic phenomena is proposed. This system is a generalization of the previously known Selkov system, which has self-oscillatory modes and is used in biology to describe glycolytic vibrations of the substrate and product. The Selkov fractional dynamical system takes int...

During the eruption of the Kizimen volcano in 2010-2013 there was a uniform squeezing of the viscous lava flow. Simultaneously with its movement, earthquakes with an unusual quasi-periodicity were recorded, the "drumbeats" mode. In this work, we show that these earthquakes were generated by the movement of the flow front, which was observed for the...

The article proposes a computer program for calculating economic crises according to the generalized mathematical model of S.V. Dubovsky. This model is represented by a system of ordinary nonlinear differential equations with fractional derivatives in the sense of Gerasimov-Caputo with initial conditions. Furthermore, according to a numerical algor...

The paper is devoted to derivation of the optimal interpolation formula in W2(0,2)(0,1) Hilbert space by Sobolev’s method. Here the interpolation formula consists of a linear combination ΣNβ=0Cβφ(xβ) of the given values of a function φ from the space W2(0,2)(0,1). The difference between functions and the interpolation formula is considered as a lin...

In this paper, a numerical analysis of the oscillation equation with a derivative of a fractional variable Riemann–Liouville order in the dissipative term, which is responsible for viscous friction, is carried out. Using the theory of finite-difference schemes, an explicit finite-difference scheme (Euler’s method) was constructed on a uniform compu...

In this study, the model Riccati equation with variable coefficients as functions, as well as a derivative of a fractional variable order (VO) of the Gerasimov-Caputo type, is used to approximate the data for some physical processes with saturation. In particular, the proposed model is applied to the description of solar activity (SA), namely the n...

The monograph is devoted to some aspects of the study of chaotic and regular regimes of fractional oscillators. Using numerical methods, an algorithm for calculating the spectra of maximum Lyapunov exponents for determining the chaotic dynamics of nonlinear fractional oscillators has been developed. A numerical algorithm has been developed for calc...

Complex methods and results of studies of physical processes and their interactions in the system of near space and geospheres under conditions of increased variability of solar, cyclonic and seismic activity are presented. The results of long-term observations are summed up. Methods of system analysis have been developed. Models of nonlinear and r...

The monograph is devoted to the study of dynamic processes with saturation and memory using the fractional Riccati equation. Numerical methods for solving using non-local explicit and implicit finite-difference schemes are proposed, questions of stability and convergence of methods are investigated, test calculations are carried out, and computer p...

The results of modeling the conversion factor from rainfall-deposited unit activity of gamma-emitting radon and thoron daughter decay products to their created gamma-radiation dose rate as a function of height above the Earth’s surface using the Geant4 toolkit are presented in this paper. Thin layers of water, soil, and air, with the height of 0.1–...

The article discusses different schemes for the numerical solution of the fractional Riccati equation with variable coefficients and variable memory, where the fractional derivative is understood in the sense of Gerasimov-Caputo. For a nonlinear fractional equation, in the general case, theorems of approximation, stability, and convergence of a non...

The paper investigates the dynamic modes of the Sel’kov fractional self-oscillating system in order to simulate the interaction of cracks. The spectra of the maximum Lyapunov exponents, constructed depending on the parameters of the dynamic system, are used as a research tool. The maximum Lyapunov exponents were constructed according to the Benetti...

The article proposes a mathematical model based on the fractional Riccati equation to describe the dynamics of COVID-19 coronavirus infection in the Republic of Uzbekistan and the Russian Federation. The model fractional Riccati equation is an equation with variable coefficients and a derivative of a fractional variable order of the Gerasimov-Caput...

The features of the atmospheric γ-background reaction to liquid atmospheric precipitation in the form of bursts is investigated, and various forms of them are analyzed. A method is described for interpreting forms of the measured γ-background response with the determination of the beginning and ending time of precipitation, the distinctive features...

The article proposes a nonlocal explicit finite-difference scheme for the numerical solution of a nonlinear, ordinary differential equation with a derivative of a fractional variable order of the Gerasimov–Caputo type. The questions of approximation, convergence, and stability of this scheme are studied. It is shown that the nonlocal finite-differe...

In this paper, we study the Cauchy problem for the Riccati differential equation with constant coefficients and a modified Gerasimov-Caputo type fractional differential operator of variable order. Using Newton’s numerical algorithm, calculation curves are constructed taking into account different values of the Cauchy problem parameters. The calcula...

In this paper, we study the conditions of global solvability and unsolvability in time of solutions to the nonlinear diffusion problem based on self-similar analysis. We constructed various self-similar solutions of the nonlinear diffusion problem in the slow diffusion case. We established critical exponents of the Fujita type and critical exponent...

The paper presents a numerical analysis of the class of mathematical models of linear fractional oscillators, which is the Cauchy problem for a differential equation with derivatives of fractional orders in the sense of Gerasimov-Caputo. A method based on an explicit nonlocal finite-difference scheme (ENFDS) and the Adams-Bashfort-Moulton (ABM) met...

The paper proposes a new mathematical model of economic cycles and crises, which generalizes the well-known model of Dubovsky S.V. The novelty of the proposed model lies in taking into account the effect of heredity (memory), as well as the introduction of harmonic functions responsible for the arrival of investments in fixed assets and new managem...

With regard to reconstructing the gamma background dose rate, existing models are either empirical with limited applicability or have many unknown input parameters, which complicates their application in practice. Due to this, there is a need to search for a new approach and build a convenient, easily applicable and universal model. The paper propo...

During the eruption of the Kizimen volcano in 2010-2013. There was a uniform squeezing of the viscous lava flow. Simultaneously with its movement, earthquakes with an unusual quasi-periodicity were recorded, the "drumbeats" mode. In this work, we show that these earthquakes were generated by the movement of the flow front, which was observed for th...

In this paper, we consider some aspects of the numerical analysis of the mathematical model of fractional Duffing with a derivative of variable fractional order of the Riemann-Liouville type. Using numerical methods: an explicit finite-difference scheme based on the Grunwald-Letnikov and Adams-Bashford-Moulton approximations (predictor-corrector),...

Results and prospects of work of the integrative laboratory "Natural disasters of Kamchatka - earthquakes and volcanic eruptions".

The eruption of the Kizimen volcano in 2011-2012 characterized by stable, almost uniform
squeezing of a viscous lava flow with a volume of 0.3 km3. The formation of the lava
flow was accompanied by the occurrence of quasiperiodic earthquakes of the “drumbeats” mode with energy classes Ks < 7, recorded at long time intervals. Shown that earthquakes...

The article proposes a computer program for calculating economic crises according to the generalized mathematical model of S.V. Dubovsky. This model is represented by a system of ordinary nonlinear differential equations with fractional derivatives in the sense of Gerasimov-Caputo with initial conditions. Furthermore, according to a numerical algor...

The paper proposes a new mathematical model of economic cycles and crises, which generalizes the well-known model of Dubovsky S.V. The novelty of the proposed model lies in taking into account the effect of heredity (memory), as well as the introduction of harmonic functions responsible for the arrival of investments in fixed assets and new managem...

In this paper, we study the Cauchy problem for the Riccati differential equation with constant coefficients and a modified Gerasimov-Caputo type fractional differential operator of variable order. Using Newton's numerical algorithm, calculation curves are constructed taking into account different values of the Cauchy problem parameters. The calcula...

The paper presents a numerical analysis of the class of mathematical models of linear fractional oscillators, which is the Cauchy problem for a differential equation with derivatives of fractional orders in the sense of Gerasimov-Caputo. A method based on an explicit nonlocal finite-difference scheme (ENFDS) and the Adams-Bashfort-Moulton (ABM) met...

A new fast method for pupil detection and eyetracking real time is being developed based on the study of a boundary-step model of a grayscale image by the Laplacian-Gaussian operator and finding a new proposed descriptor of accumulated differences (point identifier), which displays a measure of the equidistance of each point from the boundaries of...

In this paper we suggest an explicit finite-difference scheme for numerical simulation of the Cauchy problem with an integro-differential nonlinear equation that describes an oscillatory process with friction and memory (hereditarity), and with the corresponding local initial conditions. The problems of approximation, stability, and convergence of...

The article proposes a mathematical model of radon accumulation in a chamber, which takes into account the hereditary properties of the environment in which radon migrates. The model equation is the fractional Riccati equation with a derivative of a fractional variable order of the Gerasimov-Caputo type, taking into account heredity, as well as tak...

The paper presents a mathematical model of radon accumulation in a chamber, which takes into account the hereditary properties of the environment in which radon migrates, and also uses a nonlinear function that is responsible for the mechanisms of radon entering the chamber. The simulation of accumulation is performed in comparison with real data....

Microseismic phenomena are studied by a Sel'kov generalized nonlinear dynamic system. This system is mainly applied in biology to describe substrate and product glycolytic oscillations. Thus, Sel'kov dynamic system can also describe interaction of two types of fractures in an elastic-friable medium. The first type includes seed fractures with lower...

The article investigates a mathematical model of the Duffing oscillator with a variable fractional order derivative of the Riemann-Liouville type. The study of the model is carried out using a numerical scheme based on the approximation of the fractional derivative of the Riemann-Liouville type by a discrete analog-the fractional derivative of Grun...

In the present paper the problem of construction of optimal quadrature formulas in the sense of Sard in the space L2(m)(0,1) is considered. Here the quadrature sum consists of values of the integrand at nodes and values of the first and the third derivatives of the integrand at the end points of the integration interval. The coefficients of optimal...

In this work, based on Newton's second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov-Caputo fractional derivatives is carried out. For the resulting model equation, local initial conditions are set (...

A non-linear fractional oscillator is a generalization of a classical non-linear oscillator in consideration of the hereditary or the memory effect. The memory effect is a property of a dynamic system in which its current state depends on a finite number of its previous states. Therefore, a non-linear fractional oscillator can be mathematically des...

Analytical formulas for the amplitude–frequency and phase–frequency characteristics, as well as forced oscillations, of a fractional oscillator have been derived using the harmonic balance method. It has been found that these characteristics depend on the dissipative properties of a medium—memory effects, which are described by fractional-order der...

A non-linear fractal oscillator is a generalization of a classical non-linear oscillator in consideration of the hereditary or the memory effect. The memory effect is a property of a dynamic system in which its current state depends on a finite number of its previous states. Therefore, a non-linear fractal oscillator can be mathematically described...

Using the harmonic balance method, analytical formulas are obtained for calculating the amplitude-frequency and phase-frequency characteristics, as well as the quality factor of the forced oscillations of a linear fractional oscillator. It was established that the characteristics under study depend on the dissipative properties of the medium - memo...

A mathematical model of a Duffing type oscillator with a variable fractional derivative of the Riemann-Liouville derivative is studied. Using the harmonic balance method, algorithms for constructing amplitude-phase characteristics were found. The amplitude-frequency and phase-frequency characteristics were built. The inverse dependence of the Q fac...

The Cauchy problem for a linear fractional oscillator is investigated, the Green function is found using the Laplace integral transform, through which the solution of the original problem is represented.

The relationship is substantiated between the amplitude-frequency and phase-frequency performances of forced oscillations of a nonlinear fractional oscillator and the orders of fractional derivatives in its model equation. Using computer simulation, it is shown that the orders of fractional derivatives are related with the quality factor of an osci...

Using numerical modeling, oscillograms and phase trajectories were constructed to study the limit cycles of a van der Pol Duffing nonlinear oscillatory system with a power memory. The simulation results showed that in the absence of a power memory (α = 2, β = 1) or the classical van der Pol Duffing dynamical system, there is a single stable limit c...

Recently, hereditary oscillatory systems (oscillators) are of more interest.
Hereditary oscillators have a memory effect, a property of the environment in which
the current state of a dynamic system depends on a nite number of previous states.
Memory effects are characteristic of viscoelastic media or media with fractal properties.

С помощью метода гармонического баланса получены аналитические формулы для расчета амплитудночастотной и фазово-частотной характеристик, а также добротности вынужденных колебаний дробного
линейного осциллятора. Установлено, что исследуемые характеристики зависят от диссипативных свойств
среды — эффектов памяти, которые описываются производными дроб...

В работе проведено исследование хаотических режимов дробного аналога осциллятора типа Дуффинга. Для этого по алгоритму Вольфа с ортогонализацией Грама-Шмидта были рассчитаны спектры максимальных показателей Ляпунова в зависимости от значений управляющих параметров, на основе которых были построены бифуркационные диаграммы. Бифуркационные диаграммы...

Into this paper, the amplitude-frequency and phase-frequency characteristics of the Van der Polar fractional oscillator are studied in order to establish their relationship with the orders of fractional derivatives included in the model equation. Using the harmonic balance method, analytical formulas were obtained for the amplitude-frequency, phase...

Abstract. In this paper, oscillograms and phase trajectories are constructed using numerical
simulation to study the limiting cycles of a nonlinear FitzHugh-Nagumo oscillatory system with
power memory. The simulation results showed that in the absence of power memory (α=2, β=1) or the
classical dynamic FitzHugh-Nagumo system, there is a single s...

In the training course of the theory of differential equations, there exists a section on the investigation of the stability of systems of differential equations. If the system of differential equations consists of differential equations of integer order, then the stability theory of Lyapunov is usually used to research the stability of their rest...

In the paper, a fractal nonlinear oscillator was investigated with the aim of identifying its chaotic oscillatory regimes. The measure of chaos for a dynamic system is the maximum Lyapunov
exponents. They are consid- ered as a measure of the dispersal of several phase trajectories constructed under different initial conditions. To determine the ma...

A mathematical model is proposed for describing a wide class of radiating or memory oscillators. As a basic equation in this model is an integro-differential equation of Voltaire type with difference kernels - memory functions, which were chosen by power functions. This choice is due, on the one hand, to broad applications of power law and fractal...

In this paper we study the conditions for the existence of chaotic and regular oscillatory regimes of the hereditary oscillator FitzHugh-Nagumo (FHN), a mathematical model for the propagation of a nerve impulse in a membrane. To achieve this goal, using the non-local explicit finite-difference scheme and Wolf’s algorithm with the Gram-Schmidt ortho...

In study with the help of the spectrum of maximal Lyapunov exponents, dynamic regimes of the stick-slip effect were studied with allowance effect of hereditarity. Spectrum of the Lyapunov exponents were constructed using the Wolff algorithm with Gram-Schmidt orthogonalization depending on the values of the control parametersfriction and adhesion co...

Монография посвящена вопросам математического моделирования эредитарных нелинейных колебательных систем, которые учитывают эффекты «памяти». Разработаны математические модели эредитарных нелинейных осцилляторов на основе формализма дробного интегро-дифференцирования, построены численные алгоритмы их решения с помощью конечно-разностных схем. На осн...

For the hereditary dynamical system, the Cauchy problem is posed: ∂ β 0t x (η) + λ∂ γ 0t x (η) = f (x (t) , t) , x (0) = α1, ˙ x (0) = α2, 1 < β < 2, 0 < γ < 1, (1) where ∂ β 0t x (η) = 1 Γ(2−β) t 0 ¨ x(η)dη (t−η) β−1 , ∂ γ 0t x (η) = 1 Γ(1−γ) t 0 ˙ x(η)dη (t−η) γ-operators of Gerasimov-Kaputo fractional orders β and γ, α 1 , α 2-constants, t ∈ [0,...

The mathematical model of the hereditary dynamic system is proposed, which describes free oscillations with allowance for the variable memory. Variable memory is reflected in the model in the form of derivatives of fractional order variables. Put the corresponding Cauchy problem, which was solved by the theory of finite-difference schemes. The cons...

Work is devoted to mathematical modelling hereditarity oscillatory systems with the help of the mathematical apparatus of fractional calculus on the example of an Airy oscillator with friction. Model Airy equation was written in terms of Gerasimov - Caputo fractional derivatives. Next a finite-difference scheme to this generalized equation for nume...

The model of radon transfer is considered in a porous ground layer of finite power. With the help of the Laplace integral transformation, a numerical solution of this model is obtained which is based on the construction of a generalized quadrature formula of the highest degree of accuracy for the transition to the original - the function of solving...

В работе предложена обобщенная математическая модель осциллятора Дуффинга с трением, которая учитывает эффект «памяти» или эредитарность в колебательной системе. Описание этого эффекта дается формальной заменой в модельном уравнении целочисленные производные на производные дробных порядков в смысле Римана-Лиувилля. Была построена явная конечно разн...

The paper deals with the model of variable-order nonlinear hereditary oscillator based on a numerical finite-difference scheme. Numerical experiments have been carried out to evaluate the stability and convergence of the difference scheme. It is argued that the approximation, stability and convergence are of the first order, while the scheme is sta...

В статье рассматривается обобщение нелинейного осциллятора на случай учета свойства эредитарности или памяти. В этом случае модельное уравнение осциллятора приводится к уравнению с производными дробных порядков. Далее ставится задача Коши, которая решается с помощью явной конечно-разностной схемы. Исследованы вопросы аппроксимации, устойчивости и с...

The work was a mathematical model that describes the effect of the sliding attachment (stick-slip), taking into account hereditarity. explicit finite-difference scheme for the corresponding. Cauchy problem was constructed. Built on the basis of its waveform and phase trajectories

This paper presents a mathematical model that generalizes the famous Kondratiev cycles model (Dubovskiy model) used to predict economic crises. This generalization is to integrate the memory effect, which occurs frequently in the economic system. With the help of numerical methods, a generalized model was received, according to which the phase path...

The study proposes algorithms for the mittag-leffler function to be correctly calculated with a fractional parameter 1<α<2 in the MAPLE mathematical package.

Монография посвящена вопросам математического моделирования эредитарных линейных колебательных систем, которые учитывают эффекты «памяти». Разработаны математические модели эредитарных линейных осцилляторов на основе формализма дробного интегро-дифференцирования, построены численные алгоритмы их решения с помощью конечно-разностных схем. На основе...

The paper deals with the explicit finite difference schemes for the fractional oscillator. The questions of approximation, stability and convergence of these schemes.

We consider solutions of Mathematical Olympiad «Vitus Bering – 2015» for high school students. It was held at Kamchatka State University in November 2015.

The paper considers hereditarity oscillator which is characterized by oscillation equation with derivatives of fractional order $\beta$ and $\gamma$, which are defined in terms of Gerasimova-Caputo. Using Laplace transform were obtained analytical solutions and the Greens function, which are determined through special functions of Mittag-Leffler an...

В работе предложена новая математическая модель изменения заряда облачных
капель в грозовых облаках. Модель учитывает фрактальные свойства грозовых облаков, а ее решение было получено с помощью аппарата дробного исчисления.

В работе рассматривается нелинейная фрактальная колебательная система Дуффинга
с трением. Проведен численный анализ этой системы с помощью конечно-разностной
схемы. Построены решения системы в зависимости от дробных параметров, а также
фазовые портреты.

The paper proposes a new mathematical model of cloud droplet charge change in storm clouds. The model takes into account the fractal properties of storm clouds, and the solution was obtained using the apparatus of fractional calculus.