A Fractional-Order Memristive Neural Network with Infinitely Many Butterfly Attractors and Its Application in Industrial Image Security

Memristors, whose magnetic flux is inherently dependent on external excitation, have been widely employed to model electromagnetic induction effects in neural systems. However, when such induction mechanisms are incorporated into fractional-order neurons, the resulting nonlinear dynamics remain largely unexplored. This paper proposes a novel fractional-order memristive neural network (FO-MNN) by embedding two memristors into a single Hopfield-type neuron, both serving to characterize electromagnetic induction behavior. The complex nonlinear dynamics induced by the two memristive modules are systematically investigated. Numerical simulations reveal that, by tuning the parameters of the first memristive module, Lorenz-like double-wing butterfly attractors can be generated. When both memristive modules act simultaneously, the network exhibits highly complex multi-double-wing butterfly chaotic attractors, whose wing numbers can be flexibly adjusted via the control parameter of the second memristive module. Moreover, variations in the initial state of the second memristor lead to initial-condition-dependent coexistence of multiple double-wing butterfly attractors. These rich dynamical behaviors highlight the strong potential of the proposed FO-MNN for chaos-based engineering and security applications. Finally, a novel privacy-protection scheme for the Industrial Internet of Things (IIoT) is constructed based on the FO-MNN, and its effectiveness is validated through encryption experiments.