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In recent years, the use of discrete memristors to enhance chaotic maps has received increasing attention. The introduction of memristors increases the complexity of chaotic maps, making them suitable for engineering applications based on chaotic systems. In this work, a fractional-order discrete memristor exhibiting local activity and controllable asymptotic stability points is constructed by using multiband nonlinear functions. The locally active property of this memristor is demonstrated by using the power-off plot and DC v - i plot. It is then introduced into the Henon map to construct a fractional-order memristive Henon map that can generate any number of coexisting attractors. Simulation results show that the number of fixed points in the system is controlled by the memristor parameters and related to the number of coexisting attractors, thus achieving controllable homogeneous multistability. The complex dynamical behaviors of this map are analyzed by using phase portraits, bifurcation diagrams, maximum Lyapunov exponent (MLE), and attractor basins. Numerical simulations show that the fractional-order map can generate various periodic orbits, chaotic attractors, and period-doubling bifurcations. The system is then implemented on an ARM digital platform. The experimental results are consistent with the simulation results, confirming the accuracy of the theoretical analysis and its physical feasibility. Finally, a parallel video encryption algorithm is designed by using the chaotic sequence iteratively generated by fraction-order memory Henon mapping, which mainly includes frame pixel scrambling and diffusion. Comprehensive security analyses are conducted, proving the robustness and reliability of the proposed encryption scheme. The results show that the encryption algorithm can effectively protect video information. In the future, we will explore other methods of constructing chaotic or hyperchaotic systems with controllable multistability and study their circuit implementation, synchronization control, and chaos-based engineering applications.
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Keywords:
- fractional-order Henon map /
- locally active discrete memristor /
- controllable multistability /
- video encryption
[1] Lorenz E N 1963 J. Atmos. Sci. 20 130Google Scholar
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Huang Z H, Li Y A, Chen Z, Liu L 2020 Acta Phys. Sin. 69 160501Google Scholar
[3] Hua Z Y, Zhou B H, Zhou Y C 2019 IEEE Trans. Ind. Electron. 66 1273Google Scholar
[4] Zhou S, Qiu Y Y, Wang X Y, Zhang Y Q 2023 Nonlinear Dyn. 111 9571Google Scholar
[5] Li H D, Li C L, Du J R 2023 Nonlinear Dyn. 111 2895Google Scholar
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[7] Lv Z W, Sun F Y, Cai C X 2022 Nonlinear Dyn. 109 3133Google Scholar
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[9] Zhang S, Li C B, Zheng J H, Wang X P, Zeng Z G, Peng X N 2022 IEEE Trans. Ind. Electron. 69 7202Google Scholar
[10] Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80Google Scholar
[11] 吴朝俊, 方礼熠, 杨宁宁 2024 物理学报 73 010501Google Scholar
Wu C J, Fang L Y, Yang N N 2024 Acta Phys. Sin. 73 010501Google Scholar
[12] Pratyusha N, Mandal S 2023 Circuits Syst. Signal Process. 42 3812Google Scholar
[13] Elsadany A A, Elsonbaty A, Hagras E A A 2023 Soft Comput. 27 4521Google Scholar
[14] Ji X Y, Dong Z K, Han Y F, Lai C S, Zhou G D, Qi D L 2022 IEEE Trans. Consum. Electron. 69 1005Google Scholar
[15] Ji X Y, Dong Z K, Han Y F, Lai C S, Qi D L 2023 IEEE Trans. Circuits Syst. Video Technol. 33 7928Google Scholar
[16] 郭慧朦, 梁燕, 董玉姣, 王光义 2023 物理学报 72 070501Google Scholar
Guo H M, Liang Y, Dong Y J, Wang G Y 2023 Acta Phys. Sin. 72 070501Google Scholar
[17] Ji X Y, Dong, Z K, Zhou G D, Lai C S, Qi D L 2024 IEEE Trans. Syst. Man. Cybern. Syst. 54 5137Google Scholar
[18] Chua L 2014 Semicond. Sci. Technol. 29 104001Google Scholar
[19] Ma M L, Yang Y, Qiu Z C, Peng Y X, Sun Y C, Li Z J, Wang M J 2022 Nonlinear Dyn. 107 2935Google Scholar
[20] Lai Q, Wan Z Q, Zhang H, Chen G R 2023 IEEE Trans. Neural Netw. Learn. Syst. 34 7824Google Scholar
[21] Zhang S, Li C B, Zheng J H, Wang X P, Zeng Z G, Chen G R 2021 IEEE Trans. Circuits Syst. I-Regul. Pap. 68 4945Google Scholar
[22] Li H Z, Hua Z Y, Bao H, Zhu L, Chen M, Bao B C 2021 IEEE Trans. Ind. Electron. 68 9931Google Scholar
[23] Abbes A, Ouannas A, Shawagfeh N, Khennaoui A A 2022 Eur. Phys. J. Plus 137 235Google Scholar
[24] Zhao L D 2021 Physica A 561 125150Google Scholar
[25] Zhao L D 2020 Circuits Syst. Signal Process. 39 6394Google Scholar
[26] Liu X G, Ma L 2020 Appl. Math. Comput. 385 125423Google Scholar
[27] Peng Y X, He S B, Sun K H 2021 Results Phys. 24 104106Google Scholar
[28] Liu X, Yu Y G 2021 Neural Comput. Appl. 33 10503Google Scholar
[29] Yang F F, Mou J, Ma C G, Cao Y H 2020 Opt. Lasers Eng. 129 106031Google Scholar
[30] Wang Y P, Liu S T, Li H 2020 Nonlinear Dyn. 102 579Google Scholar
[31] Ma C G, Mou J, Li P, Liu T M 2021 Eur. Phys. J. Spec. Top. 230 1945Google Scholar
[32] Hadjadj M A, Sadoudi S, Azzaz M S, Bendecheche H, Kaibou R 2022 J. Real- Time Image Process. 19 1049Google Scholar
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[35] Karmakar J, Pathak A, Nandi D, Mandal M K 2021 Digit. Signal Prog. 117 103143Google Scholar
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[38] Liu T M, Mou J, Banerjee S, Cao Y H, Han X T 2021 Nonlinear Dyn. 106 1011Google Scholar
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[40] Lin H, Wang C, Sun Y, Yao W 2020 Nonlinear Dyn. 100 3667Google Scholar
[41] 丁大为, 王谋媛, 王金, 杨宗立, 牛炎, 王威 2024 物理学报 73 100502Google Scholar
Ding D W, Wang M Y, Wang J, Yang Z L, Niu Y, Wang W 2024 Acta Phys. Sin. 73 100502Google Scholar
[42] 全旭, 邱达, 孙智鹏, 张贵重, 刘嵩 2023 物理学报 72 190502Google Scholar
Quan X, Qiu D, Sun Z P, Zhang G Z, Liu S 2023 Acta Phys. Sin. 72 190502Google Scholar
[43] 张贵重, 全旭, 刘嵩 2022 物理学报 71 240502Google Scholar
Zhang G Z, Quan X, Liu S 2022 Acta Phys. Sin. 71 240502Google Scholar
[44] 秦铭宏, 赖强, 吴永红 2022 物理学报 71 160502Google Scholar
Qin M H, Lai Q, Wu Y H 2022 Acta Phys. Sin. 71 160502Google Scholar
[45] El-Latif A A A, Abd-El-Atty B, Mazurczyk W, Fung C, Venegas-Andraca S E 2020 IEEE Trans. Netw. Serv. Manage. 17 118Google Scholar
[46] Jiang D, Chen T, Yuan Z, Li W X, Wang H T, Lu L L 2024 Inf. Sci. 666 120420Google Scholar
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图 20 密钥敏感性分析(News第91帧) (a)使用正确密钥解密的帧图像; (b)使用错误密钥解密的帧图像($ y(1) = 0.5 + {10^{ - 16}} $); (c)使用错误密钥解密的帧图像($ q = 0.95 + {10^{ - 16}} $)
Figure 20. Key sensitivity analysis (News frame 91): (a) Decrypted frame image with the correct key; (b) decrypted frame image with the wrong key ($ y(1) = 0.5 + {10^{ - 16}} $); (c) decrypted frame image with the wrong key ($ q = 0.95 + {10^{ - 16}} $).
表 1 原始帧图像和加密帧图像(News第91帧)在3个方向上的相关系数
Table 1. Correlation coefficients between the original frame image and the encrypted frame image (News frame 91) in three directions.
图像 方向 R G B 原始帧图像 水平 0.9536 0.9338 0.9408 垂直 0.9718 0.9618 0.9658 对角线 0.9346 0.9077 0.9173 加密后的帧图像 水平 0.0001 –0.0066 –0.0031 垂直 –0.0060 0.0014 –0.0001 对角线 –0.0004 0.0012 –0.0034 文献[45]加密帧图像 水平 0.0001 –0.0017 –0.0004 垂直 –0.0008 0.0009 0.0011 对角线 0.0001 0.0004 0.0007 表 2 原始帧图像和加密帧图像(News第91帧)的信息熵
Table 2. Information entropy of original frame image and encrypted frame image (News frame 91).
R G B 原始帧图像 7.2456 7.0573 6.9584 加密后的帧图像 7.9980 7.9986 7.9985 -
[1] Lorenz E N 1963 J. Atmos. Sci. 20 130Google Scholar
[2] 黄泽徽, 李亚安, 陈哲, 刘恋 2020 物理学报 69 160501Google Scholar
Huang Z H, Li Y A, Chen Z, Liu L 2020 Acta Phys. Sin. 69 160501Google Scholar
[3] Hua Z Y, Zhou B H, Zhou Y C 2019 IEEE Trans. Ind. Electron. 66 1273Google Scholar
[4] Zhou S, Qiu Y Y, Wang X Y, Zhang Y Q 2023 Nonlinear Dyn. 111 9571Google Scholar
[5] Li H D, Li C L, Du J R 2023 Nonlinear Dyn. 111 2895Google Scholar
[6] Araújo J, Gallas J A C 2021 Chaos Soliton. Fract. 150 111180Google Scholar
[7] Lv Z W, Sun F Y, Cai C X 2022 Nonlinear Dyn. 109 3133Google Scholar
[8] Fu L X, Wu X M, He S B, Wang H H, Sun K H 2023 IEEE Trans. Ind. Electron. 71 9668Google Scholar
[9] Zhang S, Li C B, Zheng J H, Wang X P, Zeng Z G, Peng X N 2022 IEEE Trans. Ind. Electron. 69 7202Google Scholar
[10] Strukov D B, Snider G S, Stewart D R, Williams R S 2008 Nature 453 80Google Scholar
[11] 吴朝俊, 方礼熠, 杨宁宁 2024 物理学报 73 010501Google Scholar
Wu C J, Fang L Y, Yang N N 2024 Acta Phys. Sin. 73 010501Google Scholar
[12] Pratyusha N, Mandal S 2023 Circuits Syst. Signal Process. 42 3812Google Scholar
[13] Elsadany A A, Elsonbaty A, Hagras E A A 2023 Soft Comput. 27 4521Google Scholar
[14] Ji X Y, Dong Z K, Han Y F, Lai C S, Zhou G D, Qi D L 2022 IEEE Trans. Consum. Electron. 69 1005Google Scholar
[15] Ji X Y, Dong Z K, Han Y F, Lai C S, Qi D L 2023 IEEE Trans. Circuits Syst. Video Technol. 33 7928Google Scholar
[16] 郭慧朦, 梁燕, 董玉姣, 王光义 2023 物理学报 72 070501Google Scholar
Guo H M, Liang Y, Dong Y J, Wang G Y 2023 Acta Phys. Sin. 72 070501Google Scholar
[17] Ji X Y, Dong, Z K, Zhou G D, Lai C S, Qi D L 2024 IEEE Trans. Syst. Man. Cybern. Syst. 54 5137Google Scholar
[18] Chua L 2014 Semicond. Sci. Technol. 29 104001Google Scholar
[19] Ma M L, Yang Y, Qiu Z C, Peng Y X, Sun Y C, Li Z J, Wang M J 2022 Nonlinear Dyn. 107 2935Google Scholar
[20] Lai Q, Wan Z Q, Zhang H, Chen G R 2023 IEEE Trans. Neural Netw. Learn. Syst. 34 7824Google Scholar
[21] Zhang S, Li C B, Zheng J H, Wang X P, Zeng Z G, Chen G R 2021 IEEE Trans. Circuits Syst. I-Regul. Pap. 68 4945Google Scholar
[22] Li H Z, Hua Z Y, Bao H, Zhu L, Chen M, Bao B C 2021 IEEE Trans. Ind. Electron. 68 9931Google Scholar
[23] Abbes A, Ouannas A, Shawagfeh N, Khennaoui A A 2022 Eur. Phys. J. Plus 137 235Google Scholar
[24] Zhao L D 2021 Physica A 561 125150Google Scholar
[25] Zhao L D 2020 Circuits Syst. Signal Process. 39 6394Google Scholar
[26] Liu X G, Ma L 2020 Appl. Math. Comput. 385 125423Google Scholar
[27] Peng Y X, He S B, Sun K H 2021 Results Phys. 24 104106Google Scholar
[28] Liu X, Yu Y G 2021 Neural Comput. Appl. 33 10503Google Scholar
[29] Yang F F, Mou J, Ma C G, Cao Y H 2020 Opt. Lasers Eng. 129 106031Google Scholar
[30] Wang Y P, Liu S T, Li H 2020 Nonlinear Dyn. 102 579Google Scholar
[31] Ma C G, Mou J, Li P, Liu T M 2021 Eur. Phys. J. Spec. Top. 230 1945Google Scholar
[32] Hadjadj M A, Sadoudi S, Azzaz M S, Bendecheche H, Kaibou R 2022 J. Real- Time Image Process. 19 1049Google Scholar
[33] Dolati N, Beheshti A, Azadegan H 2021 Multimed. Tools Appl. 80 2319Google Scholar
[34] Tabash F K, Izharuddin M 2019 Multimed. Tools Appl. 78 7365Google Scholar
[35] Karmakar J, Pathak A, Nandi D, Mandal M K 2021 Digit. Signal Prog. 117 103143Google Scholar
[36] Liu S C, Li Y X, Ge X Z, Li C B, Zhao Y B 2022 Phys. Scr. 97 085210Google Scholar
[37] Li X D, Yu H Y, Zhang H Y, Jin X, Sun H B, Liu J 2020 Multimed. Tools Appl. 79 23995Google Scholar
[38] Liu T M, Mou J, Banerjee S, Cao Y H, Han X T 2021 Nonlinear Dyn. 106 1011Google Scholar
[39] Lu Y M, Wang C H, Deng Q L, Xu C 2022 Chin. Phys. B 31 060502Google Scholar
[40] Lin H, Wang C, Sun Y, Yao W 2020 Nonlinear Dyn. 100 3667Google Scholar
[41] 丁大为, 王谋媛, 王金, 杨宗立, 牛炎, 王威 2024 物理学报 73 100502Google Scholar
Ding D W, Wang M Y, Wang J, Yang Z L, Niu Y, Wang W 2024 Acta Phys. Sin. 73 100502Google Scholar
[42] 全旭, 邱达, 孙智鹏, 张贵重, 刘嵩 2023 物理学报 72 190502Google Scholar
Quan X, Qiu D, Sun Z P, Zhang G Z, Liu S 2023 Acta Phys. Sin. 72 190502Google Scholar
[43] 张贵重, 全旭, 刘嵩 2022 物理学报 71 240502Google Scholar
Zhang G Z, Quan X, Liu S 2022 Acta Phys. Sin. 71 240502Google Scholar
[44] 秦铭宏, 赖强, 吴永红 2022 物理学报 71 160502Google Scholar
Qin M H, Lai Q, Wu Y H 2022 Acta Phys. Sin. 71 160502Google Scholar
[45] El-Latif A A A, Abd-El-Atty B, Mazurczyk W, Fung C, Venegas-Andraca S E 2020 IEEE Trans. Netw. Serv. Manage. 17 118Google Scholar
[46] Jiang D, Chen T, Yuan Z, Li W X, Wang H T, Lu L L 2024 Inf. Sci. 666 120420Google Scholar
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