Memristors are widely considered to be promising candidates to mimic biological synapses. In this paper, by introducing a non-ideal flux-controlled memristor model into a Hopfield neural network (HNN), a novel memristive HNN model with multi-double-scroll attractors is constructed. The parity of the number of double scrolls can be flexibly controlled by the internal parameters of the memristor. Through theoretical analysis and numerical simulation, various coexisting attractors and amplitude control are observed. Particularly, the interesting and rare phenomenon of the memristor initial offset boosting coexisting dynamics is discovered, in which the initial offset boosting coexisting double-scroll attractors with banded attraction basins are distributed in a line along the boosting route with the variation of the memristor initial condition. In addition, it is also found that the number of the initial offset boosting coexisting double-scroll attractors is closely related to the total number of scrolls and ultimately tends to infinity with increasing the total number of scrolls, meaning the emergence of extreme multistability. Then, the random performance of the initial offset boosting coexisting double-scroll attractors is tested by the NIST test suite. Moreover, an encryption scheme based on them is also proposed. The obtained results show that they have excellent randomness and are suitable for image encryption application. Finally, numerical simulation results are well demonstrated by circuit experiments, showing the feasibility of the designed memristive multi-double-scroll HNN model.

忆阻器被广泛认为是模仿生物突触的有前途的候选人。本文通过将非理想的磁通控制忆阻器模型引入Hopfield神经网络（HNN），构建了具有多双滚动吸引子的新型忆阻HNN模型。双滚动数的奇偶性可以通过忆阻器的内部参数灵活控制。通过理论分析和数值模拟，观察到各种并存吸引子和振幅控制。特别地，发现了忆阻器初始偏置助推并存动力学的有趣且罕见的现象，其中，随着忆阻器初始条件的变化，初始偏置助推并存的双涡卷吸引器与带状吸引盆沿激励路径成线分布。 。另外，还发现，初始偏移增强共存的双涡卷吸引子的数量与涡卷的总数紧密相关，并且随着涡卷总数的增加最终趋于无穷大，这意味着出现了极端的多稳定性。然后，由NIST测试套件测试初始偏置增强共存双滚动吸引子的随机性能。此外，还提出了基于它们的加密方案。所得结果表明它们具有优良的随机性，适合图像加密应用。最后，通过电路实验很好地证明了数值模拟结果，表明了设计的忆阻多双滚动HNN模型的可行性。还发现，初始补偿增强共存的双涡卷吸引子的数量与涡卷的总数密切相关，并且随着涡卷总数的增加最终趋于无穷大，这意味着出现了极端的多重稳定性。然后，由NIST测试套件测试初始偏置增强共存双滚动吸引子的随机性能。此外，还提出了基于它们的加密方案。所得结果表明它们具有优良的随机性，适合图像加密应用。最后，通过电路实验很好地证明了数值模拟结果，表明了设计的忆阻多双滚动HNN模型的可行性。还发现，初始补偿增强共存的双涡卷吸引子的数量与涡卷的总数密切相关，并且随着涡卷总数的增加最终趋于无穷大，这意味着出现了极端的多重稳定性。然后，由NIST测试套件测试初始偏置增强共存双滚动吸引子的随机性能。此外，还提出了基于它们的加密方案。所得结果表明它们具有优良的随机性，适合图像加密应用。最后，通过电路实验很好地证明了数值模拟结果，表明了设计的忆阻多双滚动HNN模型的可行性。