Abstract This paper examines the dynamical analysis of the Lengyel-Epstein system with a discrete delay in detail. Under the assumption that the unique positive equilibrium of the model is locally asymptotically stable in the absence of the delay, the effect of the increase of delay on the stability of the unique positive equilibrium is analyzed in detail. It is found that under suitable conditions on the other parameters, the delay doesn’t affect the stability of the equilibrium, namely, the equilibrium is absolutely stable while under the other conditions on the other parameters, the equilibrium will become ultimately unstable after passing through multiple stability switches and Hopf bifurcations at some certain critical values of delay. Particularly, by means of the normal form method and the center manifold reduction for retarded functional differential equations, the explicit formulae determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions are obtained. To verify our theoretical conclusions, some numerical simulations for specific examples are also included at the end of this article.

中文翻译：

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Lengyel-Epstein 化学反应系统延迟引起的多重稳定性开关和 Hopf 分岔

摘要 本文详细研究了具有离散延迟的 Lengyel-Epstein 系统的动力学分析。在假设模型的唯一正平衡在没有延迟的情况下局部渐近稳定的情况下，详细分析了延迟的增加对唯一正平衡稳定性的影响。发现在其他参数合适的条件下，延迟不影响平衡的稳定性，即平衡是绝对稳定的，而在其他参数的其他条件下，平衡通过后最终会变得不稳定在某些特定延迟临界值下的多个稳定性开关和 Hopf 分岔。特别，利用范式方法和延迟泛函微分方程的中心流形约简，得到了确定Hopf分岔方向和分岔周期解稳定性的显式公式。为了验证我们的理论结论，本文最后还附上了一些具体例子的数值模拟。