ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Gromov V. A., Tomashchuk K. K., Beschastnov Y. N., Sidorenko A. A., Kakurin V. V. A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem. Izvestiya VUZ. Applied Nonlinear Dynamics, 2025, vol. 33, iss. 4, pp. 435-465. DOI: 10.18500/0869-6632-003160, EDN: NXMRAP

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Russian
Heading: 
Bifurcation in Dynamical Systems. Deterministic Chaos. Quantum Chaos.
Article type: 
Article
UDC: 
530.182
DOI: 
10.18500/0869-6632-003160
EDN: 
NXMRAP

A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem

Autors: 
Gromov Vasily Alexandrovich, National Research University "Higher School of Economics"
Tomashchuk Korney Kirillovich, National Research University "Higher School of Economics"
Beschastnov Yury Nikolaevich, National Research University "Higher School of Economics"
Sidorenko Artem Aleksandrovich, National Research University "Higher School of Economics"
Kakurin Vasily Vladimirovich, National Research University "Higher School of Economics"
Abstract: 

The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem.

Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations.

Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions.

Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.
 

Key words: 
bifurcation analysis
nonlinear partial differential equations
boundary value problems
KolmogorovArnold theorem
Acknowledgments: 
This work/article is an output of a research project implemented as part of the Basic Research Program at the HSE University and Strategic Project “’Human Brain Resilience: Neurocognitive Technologies for Adaptation, Learning, Development and Rehabilitation in a Changing Environment”’, which is part of Higher School of Economics’ development program under the “Priority 2030” academic leadership initiative. The “Priority 2030” initiative is run by Russia’s Ministry of Science and Higher Education as part of National Project “Science and Universities”.
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Received: 
12.09.2024
Accepted: 
24.01.2025
Available online: 
29.01.2025
Published: 
31.07.2025
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