Multistability and offset boosting in a fractional-order discrete memristive neuron model with application to image encryption

doi:10.1088/1674-1056/ae1454

Abstract This paper presents a three-dimensional discrete fractional-order memristor-coupled neuron (DFOMCN) map derived from coupling a two-dimensional neuron model through a discrete memristor, exhibiting unique offset-boosting dynamics. Unlike integer-order chaotic systems, the offset-boosting behavior in this fractional-order map achieved through parameter offset demonstrates explicit dependence on initial conditions, attributable to the memory effects of fractional-order operators. Rigorous dynamical analysis is conducted using Lyapunov exponent spectra, bifurcation diagrams, and phase portraits, revealing significantly richer hyperchaotic behavior compared to its integer-order counterpart. The system’s multistability and conditional symmetry are rigorously investigated, leading to initial-induced heterogeneous multistability and initialboosted homogeneous multistability. Furthermore, permutation entropy (PE) complexity analysis confirms exceptional unpredictability and pseudo-randomness in generated sequences. The proposed DFOMCN map is successfully applied to a novel color image encryption scheme, demonstrating outstanding security performance with high resistance to statistical, differential, and brute-force attacks.

Keywords: fractional-order hyperchaotic map memristor-coupled neuron offset boosting multistability image encryption

Received: 08 September 2025
Revised: 14 October 2025
Accepted manuscript online: 17 October 2025

PACS:	05.45.-a	(Nonlinear dynamics and chaos)
	84.30.-r	(Electronic circuits)
	45.10.Hj	(Perturbation and fractional calculus methods)
	89.20.Ff	(Computer science and technology)

Cite this article:

Haneche Nabil and Hamaizia Tayeb Multistability and offset boosting in a fractional-order discrete memristive neuron model with application to image encryption 2026 Chin. Phys. B 35 080504

[1] Mandelbrot B B 1983 The Fractal Geometry of Nature/Revised and Enlarged Edition (New York)
[2] Boulaaras S, Jan R and Pham V T 2023 Eur. Phys. J. Spec. Top. 232 2347
[3] Al-Raeei M 2021 Chaos Solitons Fractals 150 111209
[4] Deogan A S, Dilz R and Caratelli D 2024 Mathematics 12 932
[5] Buedo-Fernandez S and Nieto J J 2020 Front. Phys. 8 377
[6] Cai S M, Chen Y M and Liu Q X 2024 Appl. Math. Model. 132 645
[7] Barros L C D, Lopes M M, Pedro F S, Esmi E, Santos J P C dos and Sanchez D E 2021 Comput. Appl. Math. 40 72
[8] Yusuf A, Qureshi S and Shah S F 2020 Chaos Solitons Fractals 132 109552
[9] Alkhateeb R, Abu Hammad M, AL-Shutnawi B, Laiche N and Chikr El Mezouar Z 2025 Symmetry 17 764
[10] Fan E Y, Li C and Stynes M 2023 Math. Comput. Simul. 208 501
[11] Allogmany R and Alzahrani S S 2025 Fractal Fractional 9 289
[12] Diaz J B and Osler T J 1974 Math. Comput. 28 185
[13] Abdeljawad T 2011 Comput. Math. Appl. 62 1602
[14] Danca M F 2022 Chaos Solitons Fractals 157 111851
[15] Wang H, Li G, Sun K and He S 2024 Eur. Phys. J. Plus 139 154
[16] Wang Y, Liu S and Li H 2020 Nonlinear Dyn. 102 579
[17] Yang F, Mou J, Liu J, Ma C and Yan H 2020 Signal Process. 169 107373
[18] Shukla M K and Sharma B B 2017 AEU-Int. J. Electron. Commun. 78 265
[19] Alexan W, Alexan N and Gabr M 2023 Fractal Fractional 7 287
[20] Zarebnia M and Parvaz R 2021 Chaos Solitons Fractals 152 111402
[21] Huang W, Xia X, Zhu C, Steichen P, Quan W, Mao W, Yang J, Chu L and Li X 2021 Nano-Micro Lett. 13 85
[22] Sung S H, Kim T J, Shin H, Im T H and Lee K J 2022 Nat. Commun. 13 2811
[23] Zheng L, Sarkar A, Noorwali A and Mohammad A 2025 Neural Process. Lett. 57 50
[24] Ping P, Xu F, Mao Y and Wang Z 2018 Neurocomputing 283 53
[25] Xu G, Xu J, Xiu C, Liu F and Zang Y 2017 Neurocomputing 227 108
[26] Zhou X, Sun K, Wang H and Yao Z 2024 Nonlinear Dyn. 112 9547
[27] Yu F, Xu S, Lin Y, He T, Xiao X, Cai S and Li Y 2024 Eur. Phys. J. Spec. Top. 1
[28] Xie Y and Lai Q 2024 Integration 97 102196
[29] Shim Y and Husbands P 2018 Adapt. Behav. 26 165
[30] Xiu C, Fang J and Liu Y 2022 Chaos Solitons Fractals 158 112040
[31] Xiu C, Zhou R and Liu Y 2020 Chaos Solitons Fractals 141 110316
[32] Chen T, Wang L and Duan S 2020 Neurocomputing 380 36
[33] Parastesh F, Rajagopal K, Jafari S and Perc M 2025 Fractal Fractional 9 276
[34] Luo D, Wang C, Deng Q and Sun Y 2025 Nonlinear Dyn. 113 5811
[35] Lai Q and Yang L 2024 Chaos 34 013145
[36] Wang M, Peng J, Zhang X, Iu H H C and Li Z 2023 Nonlinear Dyn. 111 15397
[37] Yu F, Wang X, Guo R, Ying Z, He Y and Zou Q 2025 Chin. Phys. B
[38] Lai Q and Liu Y 2025 Sci. China Technol. Sci. 68 1
[39] Zhang X, Li C, Iu H H, Zhao L and Yang Y 2024 Chaos Solitons Fractals 185 115150
[40] Bao B, Zhao Q, Yu X, Wu H and Xu Q 2023 Chaos Solitons Fractals 173 113748
[41] Huang K, Li C, Li Y, Lei T and Fu H 2024 Electronics 13 2138
[42] Bao H, Chen Z, Cai J, Xu Q and Bao B 2022 Sci. China Technol. Sci. 65 2582
[43] Zhiyuan F, Yan L, Guangyi W and Yana G 2024 Analog Integr. Circuits Signal Process. 118 329
[44] Lai Q and Hua H 2025 Expert Syst. Appl. 264 125854
[45] He T, Yu F, Lin Y, He S, Yao W, Cai S and Jin J 2025 Chin. Phys. B
[46] Lai Q, Yang L and Chen G 2025 IEEE Trans. Ind. Electron. 72 969
[47] Atici F M and Eloe P 2009 Electron. J. Qual. Theory Differ. Equ. 2009 1
[48] Deshpande A and Daftardar-Gejji V 2016 Pramana 87 49
[49] Cermak J, Gy ori I and Nechvatal L 2015 Fract. Calc. Appl. Anal. 18 651
[50] Chua L 2014 Semicond. Sci. Technol. 29 104001
[51] Huang L, Zheng L, Yang Y, Yu X and Li C 2024 Phys. Scr. 99 095244
[52] Ma T, Mou J, Banerjee S and Cao Y 2023 Chaos Solitons Fractals 176 114113
[53] Wu G C and Baleanu D 2015 Commun. Nonlinear Sci. Numer. Simul. 22 95
[54] Bandt C and Pompe B 2002 Phys. Rev. Lett. 88 174102
[55] Peng Y, Sun K, Peng D and Ai W 2019 Physica A 525 96
[56] Natiq H, Al-Saidi N M G, Said M R M and Kilicman A 2018 Eur. Phys. J. Plus 133 6
[57] Lone P N, Singh D and Mir U H 2022 Multimed. Tools Appl. 81 5669
[58] Cao W, Mao Y and Zhou Y 2020 Signal Process. 171 107457
[59] Wang P, Wang Q, Sang H, Li K, Yu X and Xiong W 2025 Sci. Rep. 15 5952
[60] Valandar M Y, Barani M J and Ayubi P 2019 Optik 193 162921
[61] Dua M, Makhija D, Manasa P Y L and Mishra P 2022 Open Comput. Sci. 12 37
[62] Liu C and Ding Q 2020 Complexity 2020 3837209
[63] Sharma M 2020 Multimed. Tools Appl. 79 355
[64] Bao H, Hua Z, Li H, Chen M and Bao B 2021 IEEE Trans. Circuits Syst. I 68 4534
[65] An T, Gao T, Chen T and Jiang D 2025 Complex Intell. Syst. 11 319
[66] Chen X, Gong M, Gan Z, Lu Y, Chai X and He X 2023 Complex Intell. Syst. 9 927
[67] Wang Q, Sang H, Wang P, Yu X and Yang Z 2024 Sci. Rep. 14 29615
[68] Wu Y, Noonan J P and Agaian S 2011 Cyber J. Multidiscip. J. Sci. Technol. J. Sel. Areas Telecommun. 1 31
[69] Islam Y, Li C, Sun K and He S 2024 Multimed. Tools Appl. 83 67327
[70] Teng L, Wang X, Yang F and Xian Y 2021 Nonlinear Dyn. 105 1859
[71] Zhou J, Zhou N R and Gong L H 2020 Opt. Laser Technol. 131 106437
[72] Yu F, Wang X, Guo R, Ying Z, Cai S and Jin J 2025 Integration 102475
[73] Yu F, He S, Yao W, Cai S and Xu Q 2025 IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst.
[74] Alvarez G and Li S 2006 Int. J. Bifurcation Chaos 16 2129
[75] Wang X Y, Li P, Zhang Y Q, Liu L Y, Zhang H and Wang X 2018 Multimed. Tools Appl. 77 6243
[76] Lone P N, Singh D and Mir U H 2022 Multimed. Tools Appl. 81 5669
[77] Xuejing K and Zihui G 2020 Signal Process. Image Commun. 80 115670
[78] Khan M, Alanazi A S, Khan L S and Hussain I 2021 Complex Intell. Syst. 7 2751
[79] He S, Yu F, Guo R, Zheng M, Tang T, Jin J and Wang C 2025 Fractal Fractional 9 561
[80] Pareschi F, Rovatti R and Setti G 2012 IEEE Trans. Inf. Forensics Security 7 491

[1]	A visually meaningful medical image encryption scheme based on image steganography and memristive Hopfield neural networks Wei Yao(姚卫), Xiangyun Huang(黄翔云), Jianhua Xiao(肖捡花), Fei Yu(余飞), and Yichuang Sun(孙义闯). Chin. Phys. B, 2026, 35(6): 068702.
[2]	Color image encryption algorithm using a new four-dimensional hyperchaotic system Bo Lin(林博), Min Liu(刘敏), and Guodong Ye(叶国栋). Chin. Phys. B, 2026, 35(5): 050502.
[3]	Dynamic analysis and DNA coding-based image encryption of memristor synapse-coupled hyperchaotic IN-HNN network Shuang Zhao(赵双), Yunzhen Zhang(张云贞), Xiangjun Chen(陈湘军), Bin Gao(高彬), and Chengjie Chen(陈成杰). Chin. Phys. B, 2026, 35(1): 010502.
[4]	Synchronization of neuromorphic memristive Josephson junction network and its application Dejun Yan(严德军), Fuqiang Wu(吴富强), and Wenshuai Wang(汪文帅). Chin. Phys. B, 2026, 35(1): 010505.
[5]	An enhanced fingerprint template protection scheme based on four-dimensional superchaotic system and dynamic DNA coding Baiqiang Hu(胡百强), Jiahui Liu(刘嘉辉), and Zhe Liu(刘喆). Chin. Phys. B, 2025, 34(7): 070505.
[6]	Resonant tunneling diode cellular neural network with memristor coupling and its application in police forensic digital image protection Fei Yu(余飞), Dan Su(苏丹), Shaoqi He(何邵祁), Yiya Wu(吴亦雅), Shankou Zhang(张善扣), and Huige Yin(尹挥戈). Chin. Phys. B, 2025, 34(5): 050502.
[7]	Analysis and image encryption of memristive chaotic system with coexistence bubble Da Qiu(邱达), Bo Zhang(张博), Tingting Zhang(张婷婷), Song Liu(刘嵩), and Peiyu He(何培宇). Chin. Phys. B, 2025, 34(4): 040203.
[8]	Hybrid image encryption scheme based on hyperchaotic map with spherical attractors Zhitang Han(韩智堂), Yinghong Cao(曹颖鸿), Santo Banerjee, and Jun Mou(牟俊). Chin. Phys. B, 2025, 34(3): 030503.
[9]	Multi-scroll hopfield neural network excited by memristive self-synapses and its application in image encryption Ting He(何婷), Fei Yu(余飞), Yue Lin(林越), Shaoqi He(何邵祁), Wei Yao(姚卫), Shuo Cai(蔡烁), and Jie jin(金杰). Chin. Phys. B, 2025, 34(12): 120506.
[10]	Quantum color image encryption: Dual scrambling scheme based on DNA codec and quantum Arnold transform Tao Cheng(程涛), Run-Sheng Zhao(赵润盛), Shuang Wang(王爽), Kehan Wang(王柯涵), and Hong-Yang Ma(马鸿洋). Chin. Phys. B, 2025, 34(1): 010305.
[11]	A color image encryption scheme based on a 2D coupled chaotic system and diagonal scrambling algorithm Jingming Su(苏静明), Shihui Fang(方士辉), Yan Hong(洪炎), and Yan Wen(温言). Chin. Phys. B, 2024, 33(7): 070502.
[12]	Novel self-embedding holographic watermarking image encryption protection scheme Linian Wang(王励年), Nanrun Zhou(周楠润), Bo Sun(孙博), Yinghong Cao(曹颖鸿), and Jun Mou(牟俊). Chin. Phys. B, 2024, 33(5): 050501.
[13]	Efficient single-pixel imaging encrypted transmission based on 3D Arnold transformation Zhen-Yu Liang(梁振宇), Chao-Jin Wang(王朝瑾), Yang-Yang Wang(王阳阳), Hao-Qi Gao(高皓琪), Dong-Tao Zhu(朱东涛), Hao-Li Xu(许颢砾), and Xing Yang(杨星). Chin. Phys. B, 2024, 33(3): 034204.
[14]	A lightweight symmetric image encryption cryptosystem in wavelet domain based on an improved sine map Baichi Chen(陈柏池), Linqing Huang(黄林青), Shuting Cai(蔡述庭), Xiaoming Xiong(熊晓明), and Hui Zhang(张慧). Chin. Phys. B, 2024, 33(3): 030501.
[15]	A novel variable-order fractional chaotic map and its dynamics Zhouqing Tang(唐周青), Shaobo He(贺少波), Huihai Wang(王会海), Kehui Sun(孙克辉), Zhao Yao(姚昭), and Xianming Wu(吴先明). Chin. Phys. B, 2024, 33(3): 030503.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Multistability and offset boosting in a fractional-order discrete memristive neuron model with application to image encryption

Cite this article:

Online attention