Abstract In this paper, we consider a four-neuron bi-directional associative memory (BAM, for short) neural network with two delays. We choose connection weights and the sum of delays as bifurcation parameters and derive the critical values where a double Hopf bifurcation may occur by analyzing the associated characteristic equation which is a fourth-degree polynomial exponential equation. Meanwhile, we obtain some parameter conditions on the existence of invariant 2-tori of the truncated normal form near the bifurcation point by the center manifold theorem and normal form method. Despite the fact that the higher-degree terms may destroy the invariant 2-tori of the truncated normal form, we prove that the neural network model has quasi-periodic invariant 2-tori for most of the parameter set where the truncated normal form possesses invariant 2-tori in a sufficiently small neighborhood of the bifurcation point. Numerical examples and simulations are given to support the theoretical analysis.

中文翻译：

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延迟 BAM 神经网络中的准周期不变 2-tori

摘要 在本文中，我们考虑了一个具有两个延迟的四神经元双向联想记忆（BAM，简称 BAM）神经网络。我们选择连接权重和延迟总和作为分岔参数，并通过分析相关的特征方程，即四次多项式指数方程，推导出可能发生双 Hopf 分岔的临界值。同时，通过中心流形定理和范式方法，我们得到了分岔点附近截断范式不变2-tori存在的一些参数条件。尽管更高阶项可能会破坏截断范式的不变 2-tori，我们证明了神经网络模型对于大部分参数集具有准周期不变的 2-tori，其中截断的范式在分岔点的足够小的邻域内具有不变的 2-tori。给出了数值例子和模拟来支持理论分析。