In this paper, a three-dimensional fractional-order (FO) discrete Hopfield neural networks (FODHNN) in the left Caputo discrete delta’s sense is proposed, the dynamic behavior and synchronization of FODHNN are studied, and the system is applied to image encryption. First, FODHNN is shown to exhibit rich nonlinear dynamics behaviors. Phase portraits, bifurcation diagrams and Lyapunov exponents are carried out to verify chaotic dynamics in this system. Moreover, by using stability theorem of FO discrete linear systems, a suitable control scheme is designed to achieve synchronization of the FODHNN. Finally, image encryption system based on the chaotic FODHNN is presented. Some security analysis and tests are given to show the effective of the encryption system.

本文提出了一种在左Caputo离散增量意义上的三维分数阶（FO）离散Hopfield神经网络（FODHNN），研究了FODHNN的动态行为和同步，并将该系统应用于图像加密。首先，FODHNN被证明具有丰富的非线性动力学行为。执行相图，分叉图和李雅普诺夫指数以验证该系统中的混沌动力学。此外，通过使用FO离散线性系统的稳定性定理，设计了一种合适的控制方案来实现FODHNN的同步。最后，提出了一种基于混沌FODHNN的图像加密系统。进行了一些安全分析和测试，以证明加密系统的有效性。