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<p>This paper develops a systematic framework for analyzing pattern formation arising from equivariant Turing bifurcations in reaction–diffusion systems on square domains. There exist various types of equivariant Turing bifurcations, including 2-fold, 3-fold, and 4-fold cases, each associated with distinct mode pairs. Explicit parameterizations for the third-order normal form coefficients are derived for every case based on the center manifold reduction. Through detailed normal form analysis, we characterize approximate expressions for the spatial patterns induced by these bifurcations and establish sufficient and critical conditions for their stability. Our study reveals that single-mode Turing patterns can give rise to various stripe and spot configurations, while mixed superposition patterns with multiple modes exhibit more intricate and diverse spatial structures. For the Schnakenberg model, we determine critical curves for constant steady-state instability and identify parameter conditions governing different types of bifurcations, encompassing standard Turing, equivariant Turing, and equivariant Turing–Turing bifurcations. Numerical simulations confirm the theoretical predictions, demonstrating strong agreement between analytically derived and observed pattern structures.</p>

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Pattern formation from equivariant Turing bifurcations in reaction–diffusion systems on a square domain

  • 期刊论文

  • 发表日期：

    2025年9月25日

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  • Chen Chen,
  • Hongbin Wang,
  • Weihua Jiang

摘要

This paper develops a systematic framework for analyzing pattern formation arising from equivariant Turing bifurcations in reaction–diffusion systems on square domains. There exist various types of equivariant Turing bifurcations, including 2-fold, 3-fold, and 4-fold cases, each associated with distinct mode pairs. Explicit parameterizations for the third-order normal form coefficients are derived for every case based on the center manifold reduction. Through detailed normal form analysis, we characterize approximate expressions for the spatial patterns induced by these bifurcations and establish sufficient and critical conditions for their stability. Our study reveals that single-mode Turing patterns can give rise to various stripe and spot configurations, while mixed superposition patterns with multiple modes exhibit more intricate and diverse spatial structures. For the Schnakenberg model, we determine critical curves for constant steady-state instability and identify parameter conditions governing different types of bifurcations, encompassing standard Turing, equivariant Turing, and equivariant Turing–Turing bifurcations. Numerical simulations confirm the theoretical predictions, demonstrating strong agreement between analytically derived and observed pattern structures.