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<p>This study conducts a systematic analysis of the double Hopf bifurcation induced by dual memory delays in a diffusive model, with the aim of elucidating their influence on the dynamical interactions between two intelligent species. Using the crossing curves method, we first identify stable regions and double Hopf bifurcation points in the two-dimensional memory delay parameter spaces. Next, we derive the normal form of the Hopf-Hopf bifurcation in terms of the original system parameters, creatively providing an approach to characterizing local dynamics near bifurcation points. By means of a numerical example, we reveal that perturbations in memory delays near the double Hopf bifurcation point can induce transitions among distinct spatiotemporal patterns. In particular, the emergence of bistable periodic solutions with different wave numbers suggests a potential manifestation of resilience in the interaction dynamics of the two intelligent species.</p>

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Double Hopf bifurcation generated by dual memory delays in a diffusive two-species model

  • 期刊论文

  • 发表日期：

    2026年1月12日

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  • Shuai Li,
  • Qian Wei,
  • Xinyu Song,
  • Lihua Dai,
  • Chengdai Huang

摘要

This study conducts a systematic analysis of the double Hopf bifurcation induced by dual memory delays in a diffusive model, with the aim of elucidating their influence on the dynamical interactions between two intelligent species. Using the crossing curves method, we first identify stable regions and double Hopf bifurcation points in the two-dimensional memory delay parameter spaces. Next, we derive the normal form of the Hopf-Hopf bifurcation in terms of the original system parameters, creatively providing an approach to characterizing local dynamics near bifurcation points. By means of a numerical example, we reveal that perturbations in memory delays near the double Hopf bifurcation point can induce transitions among distinct spatiotemporal patterns. In particular, the emergence of bistable periodic solutions with different wave numbers suggests a potential manifestation of resilience in the interaction dynamics of the two intelligent species.