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Hexagonal patterns and resonant Turing–Turing bifurcation in the Brusselator model with cross-diffusion on a 2D domain.

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Title: Hexagonal patterns and resonant Turing–Turing bifurcation in the Brusselator model with cross-diffusion on a 2D domain.
Authors: Cao, Xun1 (AUTHOR) caoxun@hit.edu.cn, Jiang, Weihua1 (AUTHOR) jiangwh@hit.edu.cn
Source: Mathematics & Computers in Simulation. Oct2026, Vol. 248, p42-69. 28p.
Subjects: Pattern formation (Physical sciences), Reaction-diffusion equations, Hopf bifurcations, Diffusion kinetics, Stability of linear systems, Bifurcation theory
Abstract: Besides self-diffusion, cross-diffusion also plays an important role in biochemical reactions. This paper focuses on the 2D Brusselator model with cross-diffusion, and investigates the effects of cross-diffusion on spatial pattern formation (e.g., hexagonal patterns arising from resonant Turing–Turing bifurcation). Firstly, with the aid of linear stability analysis, we establish the conditions for the occurrence of (equivariant) Turing/Hopf bifurcations and their interactions (e.g., (equivariant) Turing-(equivariant) Turing bifurcation, (equivariant) Turing–Hopf bifurcation), and finally determine the first (Turing/Hopf) bifurcation curves and stability regions of the unique coexistence equilibrium. Roughly speaking, the emergence of cross-diffusion renders the coexistence equilibrium more prone to losing stability through (equivariant) Turing bifurcation, but exerts no influence on the occurrence of Hopf bifurcation. Then, with the aid of the normal form method and center manifold theory, we establish the third-order normal form of resonant Turing–Turing bifurcation for the 2D Brusselator model with cross-diffusion, and then investigate spatial hexagonal pattern formation arising from the interacting modes (0 , 2) / (2 , 1) , and finally theoretically predict and numerically display bistable spatial hexagonal patterns , their coexistence with uniform/stripe patterns , as well as transient flipped hexagonal patterns. [ABSTRACT FROM AUTHOR]
Copyright of Mathematics & Computers in Simulation is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Hexagonal patterns and resonant Turing–Turing bifurcation in the Brusselator model with cross-diffusion on a 2D domain.
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  Data: <searchLink fieldCode="AR" term="%22Cao%2C+Xun%22">Cao, Xun</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> caoxun@hit.edu.cn</i><br /><searchLink fieldCode="AR" term="%22Jiang%2C+Weihua%22">Jiang, Weihua</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> jiangwh@hit.edu.cn</i>
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  Data: <searchLink fieldCode="JN" term="%22Mathematics+%26+Computers+in+Simulation%22">Mathematics & Computers in Simulation</searchLink>. Oct2026, Vol. 248, p42-69. 28p.
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  Data: <searchLink fieldCode="DE" term="%22Pattern+formation+%28Physical+sciences%29%22">Pattern formation (Physical sciences)</searchLink><br /><searchLink fieldCode="DE" term="%22Reaction-diffusion+equations%22">Reaction-diffusion equations</searchLink><br /><searchLink fieldCode="DE" term="%22Hopf+bifurcations%22">Hopf bifurcations</searchLink><br /><searchLink fieldCode="DE" term="%22Diffusion+kinetics%22">Diffusion kinetics</searchLink><br /><searchLink fieldCode="DE" term="%22Stability+of+linear+systems%22">Stability of linear systems</searchLink><br /><searchLink fieldCode="DE" term="%22Bifurcation+theory%22">Bifurcation theory</searchLink>
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Besides self-diffusion, cross-diffusion also plays an important role in biochemical reactions. This paper focuses on the 2D Brusselator model with cross-diffusion, and investigates the effects of cross-diffusion on spatial pattern formation (e.g., hexagonal patterns arising from resonant Turing–Turing bifurcation). Firstly, with the aid of linear stability analysis, we establish the conditions for the occurrence of (equivariant) Turing/Hopf bifurcations and their interactions (e.g., (equivariant) Turing-(equivariant) Turing bifurcation, (equivariant) Turing–Hopf bifurcation), and finally determine the first (Turing/Hopf) bifurcation curves and stability regions of the unique coexistence equilibrium. Roughly speaking, the emergence of cross-diffusion renders the coexistence equilibrium more prone to losing stability through (equivariant) Turing bifurcation, but exerts no influence on the occurrence of Hopf bifurcation. Then, with the aid of the normal form method and center manifold theory, we establish the third-order normal form of resonant Turing–Turing bifurcation for the 2D Brusselator model with cross-diffusion, and then investigate spatial hexagonal pattern formation arising from the interacting modes (0 , 2) / (2 , 1) , and finally theoretically predict and numerically display bistable spatial hexagonal patterns , their coexistence with uniform/stripe patterns , as well as transient flipped hexagonal patterns. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Mathematics & Computers in Simulation is the property of Elsevier B.V. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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RecordInfo BibRecord:
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    Identifiers:
      – Type: doi
        Value: 10.1016/j.matcom.2026.04.013
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      – Code: eng
        Text: English
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        PageCount: 28
        StartPage: 42
    Subjects:
      – SubjectFull: Pattern formation (Physical sciences)
        Type: general
      – SubjectFull: Reaction-diffusion equations
        Type: general
      – SubjectFull: Hopf bifurcations
        Type: general
      – SubjectFull: Diffusion kinetics
        Type: general
      – SubjectFull: Stability of linear systems
        Type: general
      – SubjectFull: Bifurcation theory
        Type: general
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      – TitleFull: Hexagonal patterns and resonant Turing–Turing bifurcation in the Brusselator model with cross-diffusion on a 2D domain.
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            NameFull: Cao, Xun
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            NameFull: Jiang, Weihua
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          Dates:
            – D: 01
              M: 10
              Text: Oct2026
              Type: published
              Y: 2026
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              Value: 248
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            – TitleFull: Mathematics & Computers in Simulation
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