Basin Structure and Unpredictability in a Two-Dimensional Memristor-Based Cubic Map

Year 2026, Volume: 15 Issue: 2, 172 - 178, 29.01.2026

Serpil Yılmaz Kutluay

Abstract

This study investigates the dynamics of a two-dimensional memristor-based cubic map, with emphasis on the structure of its basins of attraction and the unpredictability of its long-term behavior. A recurrence-based automated method is employed to identify attractors without prior knowledge of their locations, which enables a comprehensive analysis of multistability. The resulting basin structures exhibit fractal boundaries and regions of divergence, reflecting a high sensitivity to initial conditions. To quantify the complexity of the attractor basins, basin entropy values are evaluated across various parameter sets as a measure of the system’s unpredictability. The results show regions of high basin entropy, which highlight the emergence of intricate fractal-like basin boundaries and robust chaotic behavior. These findings suggest that memristive elements can enhance complexity and unpredictability in discrete dynamical systems, with potential applications in the design of secure and resilient digital systems that exploit chaotic dynamics.

Keywords

Basins of attraction , Memristive systems , Multistability , Uncertainty exponent

İki Boyutlu Memristör Tabanlı Kübik Haritada Havza Yapısının ve Öngörülemezliğin Analizi

Abstract

Bu çalışma, iki boyutlu memristör tabanlı bir kübik haritanın dinamiklerini, özellikle çekici havzalarının yapısı ve uzun vadeli davranışının öngörülemezliği üzerinde durarak incelemektedir. Sistem parametreleri değiştirildikçe, periyodik, kaotik ve hiperkaotik rejimler arasındaki geçişler çatallanma diyagramları, Lyapunov üssü hesaplamaları ve sayısal benzetimler kullanılarak analiz edilmiştir. Çekicilerin konumlarına dair ön bilgi gerektirmeyen rekürans tabanlı otomatik bir yöntem kullanılarak çekiciler tanımlanmıştır. Bu yaklaşım, çoklu kararlılık durumlarının ayrıntılı olarak incelenmesini mümkün kılmıştır. Elde edilen havza yapıları, fraktal sınırlar ve sapma bölgeleri sergileyerek başlangıç koşullarına karşı yüksek hassasiyet göstermektedir.
Elde edilen bulgular, memristif elemanların ayrık dinamik sistemlerdeki karmaşıklığı ve öngörülemezliği artırabileceğini ortaya koymaktadır. Bu sonuçlar, kaotik davranışları temel alan güvenli ve dayanıklı sayısal sistemlerin geliştirilmesine yönelik çalışmalara katkı sunabilir.

Keywords

Çekici havzaları , Memristif sistemler , Çoklu kararlılık , Belirsizlik üssü

References

• [1] L. O. Chua, "Memristor—the missing circuit element" IEEE Transactions on Circuit Theory, vol. 18, no. 5, pp. 507–519, 1971. https://doi.org/10.1109/TCT.1971.1083337
• [2] L. O. Chua, "Memristors on ‘edge of chaos’", Nature Reviews Electrical Engineering vol.1, no. 9, pp. 614–627, 2024. https://doi.org/10.1038/s44287-024-00082-1
• [3] Y. Li et al., "Memristor-type chaotic mapping" Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 32, no. 2, 2022. https://doi.org/10.1063/5.0082983
• [4] L. Huang et al., "Design and analysis of a three-dimensional discrete memristive chaotic map with infinitely wide parameter range" Physica Scripta, vol. 97, no. 6, 065210, 2022. https://doi.org/10.1088/1402-4896/ac6e99
• [5] W. A. Yihyis et al., "A class of discrete memristor chaotic maps based on the internal perturbation" Symmetry, vol. 15, no. 8, 1574, 2023. https://doi.org/10.3390/sym15081574
• [6] L. Guodong et al., "Two modified chaotic maps based on discrete memristor model" Symmetry, vol. 14, no.4, 800, 2022. https://doi.org/10.3390/sym14040800
• [7] M. T. Shatnawi et al., "A multistable discrete memristor and its application to discrete-time FitzHugh–Nagumo model" Electronics, vol. 12, no. 13, 2929, 2023. https://doi.org/10.3390/electronics12132929
• [8] L. Fu et al., "A memristive Hénon map based on the state variable difference and its analog circuit implementation" IEEE Transactions on Industrial Electronics, vol. 71, no. 8, pp. 9668-9676, 2023. https://doi.org/10.1109/TIE.2023.3292857
• [9] J. Mou et al., "Discrete second-order memristor and its application to chaotic map" IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 71, no. 5, pp. 2824-2828, 2024. https://doi.org/10.1109/TCSII.2023.3348998
• [10] A. R. Nieto et. al., "A simple method to enlarge a basin of attraction using a memristive function" Journal of Difference Equations and Applications, vol. 1, no. 8, 2025. https://doi.org/10.1080/10236198.2025.2510551
• [11] S. Gao et al., "Design, hardware implementation, and application in video encryption of the 2D memristive cubic map" IEEE Internet of Things Journal, vol. 11, no. 12, pp. 21807 -21815, 2024. https://doi.org/10.1109/JIOT.2024.3376572
• [12] Q. Lai, L. Yang, and Y. Liu, "Design and realization of discrete memristive hyperchaotic map with application in image encryption" Chaos, Solitons & Fractals, vol. 165, 112781, 2022. https://doi.org/10.1016/j.chaos.2022.112781
• [13] M. Wang et al., "Two-dimensional memristive hyperchaotic maps with different coupling frames and its hardware implementation" Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 33, no. 7, 2023. https://doi.org/10.1063/5.0154516
• [14] H. Li et al., "Two-dimensional memristive hyperchaotic maps and application in secure communication" IEEE transactions on industrial electronics, vol. 68, no. 10, pp. 9931-9940, 2020. https://doi.org/10.1109/TIE.2020.3022539
• [15] B. Karakaya, A. Gülten, and M. Frasca, "A true random bit generator based on a memristive chaotic circuit: Analysis, design and FPGA implementation." Chaos, Solitons & Fractals 119 (2019): 143-149. https://doi.org/10.1016/j.chaos.2018.12.021
• [16] M. Şahin, H. Guler, and S. Hamamcı, "Design and realization of a hyperchaotic memristive system for communication system on FPGA" Traitement du Signal, vol. 37, no. 6, 2020. https://doi.org/10.18280/ts.370607
• [17] M. E. Sahin et al., "Application and modeling of a novel 4D memristive chaotic system for communication systems" Circuits, Systems, and Signal Processing, vol. 39, no. 7, pp. 3320-3349, 2020. https://doi.org/10.1007/s00034-019-01332-6
• [18] M. E. Sahin et al., "Design of a hyperchaotic memristive circuit based on wien bridge oscillator" Computers & Electrical Engineering, vol. 88, 106826, 2020. https://doi.org/10.1016/j.compeleceng.2020.106826
• [19] S. Gürsul and S. E. Hamamcı, "Investigation of power consumption effect of various memristor emulators on a logic gate" European Journal of Technique (EJT), vol. 11, no. 2, pp. 200-208, 2021. https://doi.org/10.36222/ejt.931338
• [20] S. M. Mohamed et al., "An encryption application and FPGA realization of a fractional memristive chaotic system" Electronics, vol. 12, no. 5, 1219, 2023. https://doi.org/10.3390/electronics12051219
• [21] S. Gao, et al. "A 3D memristive cubic map with dual discrete memristors: design, implementation, and application in image encryption" IEEE Transactions on Circuits and Systems for Video Technology, vol. 35, no. 8, pp. 7706 -7718, 2025. https://doi.org/10.1109/TCSVT.2025.3545868
• [22] G. Datseris and A. Wagemakers, "Effortless estimation of basins of attraction" Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 32, no.2, 2022. https://doi.org/10.1063/5.0076568
• [23] A. Daza et. al., “Basin entropy: a new tool to analyze uncertainty in dynamical systems” Scientific reports, vol. 6, no. 1, 31416, 2016. https://doi.org/10.1038/srep31416