Application of the computer analogy method for solving complex nonlinear systems of differential equations

Aristov V.V., Muzyka A.A., Stroganov A.V.
 pdf (670K)

This study develops a previously proposed Method of Computer Analogy (MCA) based on formalization of digital computer operations. The paper discusses the position of the proposed approach among other well-known methods. It is emphasized that the primary objective is to derive analytical solutions, although in some cases they have to resort to semianalytical approximations. The paper focuses on constructing solutions for systems which, for certain parameter values, demonstrate the deterministic chaos behavior, namely Lorenz, Marioka – Shimitsu and R¨ossler systems. The paper also considers obtaining solution for Van der Pol equation (reduced to a nonlinear system). The aim of the study is to construct semi-analytical solutions represented as a segment of a power series in a step size of approximating difference scheme. To prevent overflow, authors formalize rank transfer operation. The authors apply a convergent difference scheme, referred to as the “guiding” scheme, to advance to the next step of the independent variable. The resulting approximation by a sum with only a few terms provides an approximation to the solution with any accuracy in accordance with the accuracy of the governing difference scheme. The senior digits in the resulting approximation exhibit probabilistic properties that can be modeled by known distributions, thereby enabling the derivation of analytical and semi-analytical approximations. The paper presents linear approximations that are the base for a complete approximations of solutions and provide important qualitative as well as some quantitative properties of solutions of considered systems. This work describes approximations of various orders, including those that do not guarantee convergence to the exact solution, but simplify the analysis of certain properties of nonlinear equations and systems. In particular, for the Van der Pol equation, authors demonstrate that its corresponding system has a cyclic solution and provide an estimate of its scale. A modification of the MCA that has features of the Monte Carlo method makes it possible to remove recurrent sequences and construct complete solutions in simple situations. The authors mention a promising approach for representing the solution using branched continued fractions.

Keywords: computer analogy method, solution of differential equations, Cauchy problem, solution of systems of differential equations, Van der Pol equation, Lorenz system, Marioka – Shimitsu system, R¨ossler system
Citation in English: Aristov V.V., Muzyka A.A., Stroganov A.V. Application of the computer analogy method for solving complex nonlinear systems of differential equations // Computer Research and Modeling, 2025, vol. 17, no. 6, pp. 1083-1104
Citation in English: Aristov V.V., Muzyka A.A., Stroganov A.V. Application of the computer analogy method for solving complex nonlinear systems of differential equations // Computer Research and Modeling, 2025, vol. 17, no. 6, pp. 1083-1104
DOI: 10.20537/2076-7633-2025-17-6-1083-1104
URL: http://crm-en.ics.org.ru/journal/article/3668/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2025 Aristov V.V., Muzyka A.A., Stroganov A.V.

Indexed in Scopus

Full-text version of the journal is also available on the web site of the scientific electronic library eLIBRARY.RU

The journal is included in the Russian Science Citation Index

The journal is included in the RSCI

International Interdisciplinary Conference "Mathematics. Computing. Education"

Website Copyright © 2009–2025 Institute of Computer Science