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Abstract: The modeling and dynamical analysis of discrete chaotic systems is a vital research field, and various chaotic maps have been developed using mathematical and control-theoretic approaches. However, physical circuit design of mathematically defined discrete chaotic systems and the computation of their energy functions remain challenging and open problems. In this study, a two-dimensional (2D) chaotic map is constructed using an open-loop modulation coupling method, and its dynamical characteristics are analyzed using bifurcation diagrams. Lyapunov exponents (LEs) and spectral entropy (SE) complexity are also inspected under different parameter configurations. Furthermore, the proposed chaotic map is expressed using two distinct physical memristive circuits: one is composed of a magnetic flux-controlled memristor, a nonlinear resistor, and a capacitor; the other utilizes a charge-controlled memristor, a nonlinear resistor, and an inductor. Moreover, two energy functions are derived from the two memristor-coupled circuits for the proposed chaotic map. The results demonstrate that the mathematical model of the discrete chaotic system can be effectively expressed through these two nonlinear circuits. Our study offers a theoretical foundation and viable methodology for the physical circuit representation of discrete chaotic systems and determination of their energy functions.

一个二维混沌映射的忆阻耦合电路设计与能量函数确定

[1]AkraamM, RashidT, ZafarS, 2023. A chaos-based image encryption scheme is proposed using multiple chaotic maps. Mathematical Problems in Engineering, 2023(1):2003724.

[2]AlexanW, ChenYL, PorLY, et al., 2023a. Hyperchaotic maps and the single neuron model: a novel framework for chaos-based image encryption. Symmetry, 15(5):1081.

[3]AlexanW, AlexanN, GabrM, 2023b. Multiple-layer image encryption utilizing fractional-order chen hyperchaotic map and cryptographically secure prngs. Fractal and Fractional, 7(4):287.

[4]BaoH, HuaZY, LiHZ, et al., 2021. Discrete memristor hyperchaotic maps. IEEE Transactions on Circuits and Systems I: Regular Papers, 68(11):4534-4544.

[5]ChenYX, YangFF, WangCN, 2025. Coherence resonance in a memristive map neuron and adaptive energy regulation. Modern Physics Letters B, 39(17):2550008.

[6]GabrM, KorayemY, ChenYL, et al., 2023. R³—rescale, rotate, and randomize: a novel image cryptosystem utilizing chaotic and hyper-chaotic systems. IEEE Access, 11:119284-119312.

[7]GabrM, DiabA, ElshoushHT, et al., 2024. Data security utilizing a memristive coupled neural network in 3D models. IEEE Access, 12:116457-116477.

[8]GaoS, ZhangZY, IuHHC, et al., 2025. A parallel color image encryption algorithm based on a 2-D Logistic-Rulkov neuron map. IEEE Internet of Things Journal, 12(11):18115-18124.

[9]GuoYT, XieY, MaJ, 2023. How to define energy function for memristive oscillator and map. Nonlinear Dynamics, 111(23):21903-21915.

[10]GuoYT, MaJ, ZhangXF, et al., 2024. Memristive oscillator to memristive map, energy characteristic. Science China Technological Sciences, 67(5):1567-1578.

[11]HuaZY, ZhouYC, PunCM, et al., 2015. 2D sine logistic modulation map for image encryption. Information Sciences, 297:80-94.

[12]IbarzB, CasadoJM, SanjuánMAF, 2011. Map-based models in neuronal dynamics. Physics Reports, 501(1-2):1-74.

[13]JacksonJ, PerumalR, 2025. A robust image encryption technique based on an improved fractional order chaotic map. Nonlinear Dynamics, 113(7):7277-7296.

[14]JiaJN, WangCN, ZhangXF, et al., 2024. Energy and self-adaption in a memristive map neuron. Chaos, Solitons & Fractals, 182:114738.

[15]LaiQ, LiuY, 2023. A cross-channel color image encryption algorithm using two-dimensional hyperchaotic map. Expert Systems with Applications, 223:119923.

[16]LeiZ, MaJ, 2025. Coherence resonance and energy dynamics in a memristive map neuron. Chaos, 35(2):023158.

[17]LiKS, WangQ, HuCY, et al., 2024. Dynamical analysis of a novel 2D Lyapunov exponent controllable memristive chaotic map. Chaos, 34(8):083135.

[18]LiLZ, 2024. A novel chaotic map application in image encryption algorithm. Expert Systems with Applications, 252:124316.

[19]LiYN, LvM, MaJ, et al., 2024. A discrete memristive neuron and its adaptive dynamics. Nonlinear Dynamics, 112(9):7541-7553.

[20]LiuXC, MouJ, ZhangYS, et al., 2024. A new hyperchaotic map based on discrete memristor and meminductor: dynamics analysis, encryption application, and DSP implementation. IEEE Transactions on Industrial Electronics, 71(5):5094-5104.

[21]LuoLQ, FlanaganJG, 2007. Development of continuous and discrete neural maps. Neuron, 56(2):284-300.

[22]MaML, YangY, QiuZC, et al., 2022. A locally active discrete memristor model and its application in a hyperchaotic map. Nonlinear Dynamics, 107(3):2935-2949.

[23]MaYJ, TianY, ZhangL, et al., 2024. Two-dimensional hyperchaotic effect coupled mapping lattice and its application in dynamic S-box generation. Nonlinear Dynamics, 112(19):17445-17476.

[24]MuniSS, FatoyinboHO, GhoshI, 2022. Dynamical effects of electromagnetic flux on Chialvo neuron map: nodal and network behaviors. International Journal of Bifurcation and Chaos, 32(9):2230020.

[25]NarayananR, JohnstonD, 2012. Functional maps within a single neuron. Journal of Neurophysiology, 108(9):2343-2351.

[26]PengYX, SunKH, HeSB, 2020. Dynamics analysis of chaotic maps: from perspective on parameter estimation by meta-heuristic algorithm. Chinese Physics B, 29(3):030502.

[27]PengYX, HeSB, SunKH, 2021. A higher dimensional chaotic map with discrete memristor. AEU-International Journal of Electronics and Communications, 129:153539.

[28]PengYX, LanZX, SunKH, et al., 2023. A simple color image encryption algorithm based on a discrete memristive hyperchaotic map and time-controllable operation. Optics & Laser Technology, 165:109543.

[29]RamakrishnanB, MehrabbeikM, ParasteshF, et al., 2022. A new memristive neuron map model and its network’s dynamics under electrochemical coupling. Electronics, 11(1):153.

[30]SamehSM, MoustafaHED, AbdelhayEH, et al., 2024. An effective chaotic maps image encryption based on metaheuristic optimizers. The Journal of Supercomputing, 80(1):141-201.

[31]TutuevaAV, NepomucenoEG, KarimovAI, et al., 2020. Adaptive chaotic maps and their application to pseudo-random numbers generation. Chaos, Solitons & Fractals, 133:109615.

[32]UmarT, NadeemM, AnwerF, 2024. A new modified Skew Tent Map and its application in pseudo-random number generator. Computer Standards & Interfaces, 89:103826.

[33]VermaV, KumarS, 2025. Quantum image encryption algorithm based on 3D-BNM chaotic map. Nonlinear Dynamics, 113(4):3829-3855.

[34]WangBC, ZhangXF, ZhuZG, et al., 2024. A new memristive map neuron, self-regulation and coherence resonance. The European Physical Journal B, 97(8):124.

[35]WangC, ChongZL, ZhangHL, et al., 2024. Color image encryption based on discrete memristor logistic map and DNA encoding. Integration, 96:102138.

[36]WangP, WangQ, SangHW, et al., 2025. Dynamic analysis of a novel 3D chaotic map with two internal frequencies. Scientific Reports, 15(1):5952.

[37]WangZ, ParasteshF, NatiqH, et al., 2024. Synchronization patterns in a network of diffusively delay-coupled memristive Chialvo neuron map. Physics Letters A, 514-515:129607.

[38]WuWQ, ZhouJP, 2024. Constructing new high-order polynomial chaotic maps and application in pseudorandom number generator. Physica Scripta, 99(3):035238.

[39]XiangQ, ShenYZ, PengSS, et al., 2024. A two-dimensional discrete memristor map: analysis and implementation. International Journal of Bifurcation and Chaos, 34(10):2450124.

[40]XuQ, HuangLP, WangN, et al., 2023. Initial-offset-boosted coexisting hyperchaos in a 2D memristive Chialvo neuron map and its application in image encryption. Nonlinear Dynamics, 111(21):20447-20463.

[41]YangFF, ZhouP, MaJ, 2024a. An adaptive energy regulation in a memristive map linearized from a circuit with two memristive channels. Communications in Theoretical Physics, 76(3):035004.

[42]YangFF, SongXL, MaJ, 2024b. A memristive map neuron under noisy electric field. Chinese Journal of Physics, 91:287-298.

[43]YangFF, RenLJ, MaJ, et al., 2024c. Two simple memristive maps with adaptive energy regulation and digital signal process verification. Journal of Zhejiang University-SCIENCE A, 25(5):382-394.

[44]ZhangSH, ZhangHL, WangC, 2023. Dynamical analysis and applications of a novel 2-D hybrid dual-memristor hyperchaotic map with complexity enhancement. Nonlinear Dynamics, 111(16):15487-15513.