图 1  施加$ v_n=A\sin (\omega n) $时忆阻器的磁滞回线　(a)固定A = 0.03, ω分别取11.25, 18, 37.5; (b) 固定ω = 11.25, A分别取0.03, 0.05, 0.1

Fig. 1.  Pinched hysteresis loops of the memristor when applying $ v_n=A\sin (\omega n) $: (a) Fixed A = 0.03, ω selected as 11.25, 18 and 37.5; (b) fixed ω = 11.25, A selected as 0.03, 0.05 and 0.1.

图 2  忆阻器复值高斯混沌系统的构成框图

Fig. 2.  Structure diagram of the memristive complex-valued Gaussian map

图 3  依赖于实数参数的动力学行为仿真　(a) 前两个李雅普诺夫指数; (b) $ z_{\mathrm{r}} $(绿色)与$ z_{\mathrm{i}} $(粉色)的分岔图

Fig. 3.  The real-valued parameter a-dependent dynamical behavior simulations: (a) First two Lyapunov exponents; (b) bifurcation of $ z_{\mathrm{r}} $ (green colored) and $ z_{\mathrm{i}}$ (pink colored)

图 4  依赖于耦合参数b的动力学行为仿真　(a1), (a2) 固定$ b_{\mathrm{i}} = 0.1 $, $ b_{\mathrm{r}}\in[0, 1] $; (b1), (b2)固定$ b_{\mathrm{r}} = 0.1 $,$ b_{\mathrm{i}} \in [-0.8, 0.8]$

Fig. 4.  The coupling parameter b-dependent dynamical behavior simulations: (a1), (a2) $ b_{\mathrm{r}}\in[0, 1] $ with $ b_{\mathrm{i}}=0.1 $; (b1), (b2) $ b_{\mathrm{i}}\in[-0.8, 0.8] $ with $ b_{\mathrm{r}}=0.1 $

图 5  在$ y{\text{-}}z_{\mathrm{r}} $(红色)和$ y{\text{-}}z_{\mathrm{i}} $(蓝色)平面上的相图　(a) 混沌吸引子, $ b_{\mathrm{r}} $ = 0.1; (b) 周期吸引子, $ b_{\mathrm{r}} $ = 0.3; (c) 混沌吸引子, $ b_{\mathrm{r}} $ = 0.4; (d) 准周期吸引子, $ b_{\mathrm{r}} $ = 0.9

Fig. 5.  Phase portrait in $ y \text{-}z_{\mathrm{r}} $ (red) and $ y \text{-}z_{\mathrm{i}} $ (blue) plane: (a) Chaotic attractor with $ b_{\mathrm{r}} $ = 0.1; (b) periodic attractor with $ b_{\mathrm{r}} $ = 0.3; (c) chaotic attractor with $ b_{\mathrm{r}} $ = 0.4; (d) quasi-periodic attractor with $ b_{\mathrm{r}} $ = 0.9

图 6  状态变量的分岔图　(a) 原始系统; (b) 施加变量旋转后; (c) 施加参数复化后; (d) 施加复共轭后

Fig. 6.  Bifurcation diagram of $ z_{\mathrm{r }}$: (a) Original system; (b) under variable rotation; (c) under parameter complexification; (d) under complex conjugation.

图 7  李雅普诺夫指数　(a) 原始系统; (b) 施加变量旋转后; (c) 施加参数复化后; (d) 施加复共轭后

Fig. 7.  Lyapunov exponents: (a) Original system; (b) under variable rotation; (c) under parameter complexification; (d) under complex conjugation.

图 8  当$ b_{\mathrm{r}}=1 $时的相图　(a) 原始系统; (b) 施加变量旋转后; (c) 施加参数复化后; (d) 施加复共轭后

Fig. 8.  Phase portraits with $ b_{\mathrm{r}}= 1 $: (a) Original system; (b) under variable rotation; (c) under parameter complexification; (d) under complex conjugation.

图 9  基于FPGA实现MCVGM的结构框图

Fig. 9.  Block diagram for the FPGA-based implementation of MCVGM

图 10  基于FPGA硬件实验产生的混沌吸引子　(a) $ b_{\mathrm{r}} $ = 0.1; (b) $ b_{\mathrm{r}} $ = 0.4

Fig. 10.  FPGA-based experimental chaotic attractor: (a) $ b_{\mathrm{r}} $=0.1; (b) $ b_{\mathrm{r}} $=0.4

图 11  基于FPGA产生混沌序列与数值仿真之间的误差统计　(a)绝对误差分布; (b)绝对误差累积分布函数; (c)误差指标对比; (d)绝对误差分布直方图

Fig. 11.  Error statistics between FPGA-generated chaotic sequence and numerical simulation: (a) Absolute error distribution; (b) cumulative distribution function of absolute errors; (c) comparison of error metrics; (d) histogram of absolute error distribution.

图 12  双图像复值加密系统流程图

Fig. 12.  Flowchart of the dual-image complex-valued encryption system

图 13  像素直方图测试　(a1)—(d1) 原始图像; (a2)—(d2) 原始图像的像素直方图; (a3)—(d3) 加密图像的像素直方图

Fig. 13.  Histogram test of cryptosystem: (a1)—(d1) Original image; (a2)—(d2) histogram of original image; (a3)—(d3) histogram of encrypted image.

图 14  解密结果　(a1), (b1)使用正确密钥得到的还原图像; (a2), (b2)初始条件$ z_0 $仅有微小偏差时的解密图像

Fig. 14.  Decrypted images: (a1), (b1) With correct secret keys; (a2), (b2) with tiny variation in the initial condition $ z_0 $

图 15  图像的像素相关性分布　(a1)水平方向, (a2)垂直方向, (a3)对角方向的原始图像; (b1)水平方向, (b2)垂直方向, (b3)对角方向的加密图像

Fig. 15.  Distribution of pixel correlations in images: Original image in (a1) horizontal direction, (a2) vertical direction, (a3) diagonal direction; encrypted image in (b1) horizontal direction, (b2) vertical direction, (b3) diagonal direction.

图 16  数据丢失与噪声攻击测试结果　(a1) 图像I₁密文遭受30%数据丢失后的图像; (a2) 从数据丢失后的密文恢复出的图像; (b1) 图像I₁密文添加强度为30%的椒盐噪声后的图像; (b2) 从被添加噪声后的密文中恢复的图像

Fig. 16.  Test results of data loss and noise attacks: (a1) Ciphertext images of I₁ with 30% loss; (a2) recovered images from ciphertext after data loss; (b1) ciphertext images of I₁ with 30% intensity of pepper and salt noise; (b2) recovered images from the ciphertext after adding noise.