In this paper, a fractional-order chaotic circuit with different coupled memristors is established. The dimensionality of the system is reduced by the flux-charge analysis method and the stability of the equilibrium points is analyzed by the fractional-order stability theory. Then, the complex dynamic behaviors, including periodic and chaotic attractors, period doubling bifurcation orbit, coexistence bifurcation, and asymmetric coexisting attractors, are studied by phase diagrams, bifurcation portraits, Lyapunov exponent spectra, and attractive basins. Moreover, the analog circuit of the fractional-order coupled system is constructed and the results validate the correctness of the theoretical analysis. Finally, a novel encryption scheme based on the fractional-order coupled memristive system combined with Josephus traversal and DNA operations is proposed. The simulation results show that this algorithm has a good effect.

本文建立了具有不同耦合忆阻器的分数阶混沌电路。采用通量-电荷分析法对系统进行降维，采用分数阶稳定性理论分析平衡点的稳定性。然后，通过相图、分岔图、Lyapunov 指数谱和吸引盆研究了复杂的动力学行为，包括周期和混沌吸引子、倍周期分岔轨道、共存分岔和不对称共存吸引子。此外，构建了分数阶耦合系统的模拟电路，结果验证了理论分析的正确性。最后，提出了一种基于分数阶耦合忆阻系统结合约瑟夫斯遍历和DNA操作的新型加密方案。仿真结果表明，该算法具有良好的效果。