This work principally probes into the stability, the existence, and control of Hopf bifurcation of new established fractional-order delayed bidirectional associative memory (BAM) neural networks with four neurons. Firstly, based on the earlier study, we set up a class of new fractional-order multiple delayed BAM neural networks with four neurons. Secondly, applying an appropriate variable transformation, we get a new equivalent fractional-order single delayed BAM neural networks with four neurons. With the aid of the stability theory and bifurcation knowledge of fractional-order differential dynamical systems, a novel sufficient criterion to guarantee the stability and the appearance of Hopf bifurcation of the addressed fractional-order delayed BAM neural networks is set up. Thirdly, designing a suitable time-delayed feedback controller, the stability region and the time of appearance of Hopf bifurcation for the involved neural networks have been adjusted. Finally, two simulation examples are presented to illustrate the rationality of the mathematical derivation results. The study vindicates that the time delay has great effect on the stability, bifurcation, and its control for involved network models. The obtained fruits of this work can be effectively applied to control and design neural networks. Also, some research ideas will play a key role in studying the related fractional-order dynamical models in actual world.

中文翻译：

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进一步研究分叉及其对涉及四个神经元和多个延迟的分数阶双向联想记忆神经网络的控制

这项工作主要探讨了新建立的具有四个神经元的分数阶延迟双向联想记忆 (BAM) 神经网络的 Hopf 分岔的稳定性、存在性和控制。首先，在前期研究的基础上，我们建立了一类新的具有四个神经元的分数阶多重延迟 BAM 神经网络。其次，应用适当的变量变换，我们得到了一个新的具有四个神经元的等效分数阶单延迟 BAM 神经网络。借助分数阶微分动力系统的稳定性理论和分岔知识，建立了一个新的充分准则来保证所提出的分数阶时滞 BAM 神经网络的稳定性和 Hopf 分岔的出现。第三，设计合适的延时反馈控制器，调整了所涉及神经网络的稳定区域和 Hopf 分叉出现时间。最后给出两个仿真实例说明数学推导结果的合理性。研究表明，时间延迟对所涉及网络模型的稳定性、分叉及其控制有很大影响。这项工作取得的成果可以有效地应用于控制和设计神经网络。此外，一些研究思路将对研究现实世界中相关的分数阶动力学模型起到关键作用。研究表明，时间延迟对所涉及网络模型的稳定性、分叉及其控制有很大影响。这项工作取得的成果可以有效地应用于控制和设计神经网络。此外，一些研究思路将对研究现实世界中相关的分数阶动力学模型起到关键作用。研究表明，时间延迟对所涉及网络模型的稳定性、分叉及其控制有很大影响。这项工作取得的成果可以有效地应用于控制和设计神经网络。此外，一些研究思路将对研究现实世界中相关的分数阶动力学模型起到关键作用。