In the past several decades, many papers involving the stability and Hopf bifurcation of delayed neural networks have been published. However, the results on the stability and Hopf bifurcation for fractional order neural networks with delays and fractional order neural networks with leakage delays are very rare. This paper is concerned with the stability and the existence of Hopf bifurcation of fractional order BAM neural networks with or without leakage delay. The Laplace transform, stability and bifurcation theory of fractional-order differential equations and Matlab software will be applied. The stability condition and the sufficient criterion of existence of Hopf bifurcation for fractional order BAM neural networks with delay (leakage delay) are established. It is found that when the sum of two delays (leakage delay) crosses a critical value, then a Hopf bifurcation will appear. The obtained results play an important role in designing neural networks. Also the derived results are new and enrich the bifurcation theory of fractional order delayed differential equations.

在过去的几十年中，已经发表了许多涉及延迟神经网络的稳定性和Hopf分支的论文。但是，具有延迟的分数阶神经网络和具有泄漏延迟的分数阶神经网络的稳定性和Hopf分支的结果很少。本文关注具有或不具有泄漏延迟的分数阶BAM神经网络的Hopf分支的稳定性和存在性。将应用分数阶微分方程的Laplace变换，稳定性和分叉理论以及Matlab软件。建立了具有时滞（泄漏时滞）的分数阶BAM神经网络的Hopf分支的稳定性条件和存在准则。发现当两个延迟之和（泄漏延迟）超过临界值时，然后出现Hopf分叉。获得的结果在设计神经网络中起重要作用。导出的结果也是新的，并且丰富了分数阶时滞微分方程的分叉理论。