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Abstract <p> The article discusses the issues of studying stability and bifurcations in thereaction–diffusion system in a bounded domain with homogeneous Neumann boundary conditions.The main results of the article concern the study of problems of local bifurcations in the vicinityof spatially homogeneous equilibrium positions. A general scheme is proposed that allows us toobtain new formulas for studying the main characteristics of multiple equilibrium bifurcation andbifurcation Andronov–Hopf: sufficient signs of bifurcations, their type, approximate constructionof solutions, stability analysis. The proposed approaches do not require complex and cumbersometransformations, the results obtained are brought to calculation formulas and algorithms. Someapplications in problems of diffusion instability and corresponding bifurcations inreaction–diffusion systems are also discussed. The distributed model of the “brusselator” isconsidered as the main illustrative example.</p>

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Operator Methods for Investigating Stability and Bifurcation Problems in a Reaction–Diffusion System and Applications

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • M. G. Yumagulov,
  • N. A. Vasenina,
  • R. I. Gabdrahmanov

摘要

Abstract

The article discusses the issues of studying stability and bifurcations in thereaction–diffusion system in a bounded domain with homogeneous Neumann boundary conditions.The main results of the article concern the study of problems of local bifurcations in the vicinityof spatially homogeneous equilibrium positions. A general scheme is proposed that allows us toobtain new formulas for studying the main characteristics of multiple equilibrium bifurcation andbifurcation Andronov–Hopf: sufficient signs of bifurcations, their type, approximate constructionof solutions, stability analysis. The proposed approaches do not require complex and cumbersometransformations, the results obtained are brought to calculation formulas and algorithms. Someapplications in problems of diffusion instability and corresponding bifurcations inreaction–diffusion systems are also discussed. The distributed model of the “brusselator” isconsidered as the main illustrative example.