Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 41. no. 4. pp. 47–65. ISSN 2079-6641

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MSC 26A33, 85-08

Research Article

Mathematical modeling in MATLAB of solar activity cycles according to the growth-decline of the Wolf number

D. A. Tverdyi, R. I. Parovik

Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 684034, p. Paratunka, Mirnaya st., 7, Russia


In this article, mathematical modeling of the dynamics of solar activity is carried out. Observational data on the average monthly number of sunspots, called the Wolf number, for the period of 24.5 years from May 1996 to October 2022 are studied. Based on the results of a similar study of data on this process, using the Riccati equation of a fractional constant order, that the rise and fall of the Wolf number over time occurs along a curve very close to the generalized logistic curve, this article also proposes a mathematical model based on the Riccati equation. Since the Riccati equation describes well the processes that obey the logistic law. However, the equation is generalized to the integro-differential Riccati equation by introducing a fractional derivative of the Gerasimov-Caputo type of variable order, and a fractional derivative with a variable order, allows you to get a more precise mathematical model of Wolf number cycles with saturation, and allows you to take into account the effect of variable memory. All model calculations, data processing and visualization are carried out in the FDRE 3.0 program developed in the MATLAB package. Modeling parameters are refined by approximation of known data under study, using regression analysis. As a result, the model curves and graphs of the observed data known for 24.5 years show good agreement with each other. With the help of a refined mathematical model, a forecast is made for the next 9 years, which visually agrees well with the known model results of solar activity.

Key words: solar activity, Wolf number, mathematical modeling, saturation effect, hereditarity, Riccati equation, Gerasimov-Caputo type derivative.

DOI: 10.26117/2079-6641-2022-41-4-47-65

Original article submitted: 19.11.2022

Revision submitted: 22.11.2022

For citation. Tverdyi D. A., Parovik R. I. Mathematical modeling in MATLAB of solar activity cycles according to the growth-decline of the Wolf number. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 41: 4, 47-65. DOI: 10.26117/2079-6641-2022-41-4-47-65

Competing interests. The authors declare that there are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

The content is published under the terms of the Creative Commons Attribution 4.0 International License (

© Tverdyi D. A., Parovik R. I., 2022

Funding. The research was carried out within the framework of the state task of IKIR FEB RAS on the topic AAAA-A21-121011290003-0.


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Tverdyi Dmitrii Alexsandrovich – Ph. D. (Phys. & Math.), Lead programmer laboratory of electromagnetic propogation Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0001-6983-5258.

Parovik Roman Ivanovich – D. Sci. (Phys. & Math.), Associate Professor, Leading researcher laboratory of modeling physical processes Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0002-1576-1860.