{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T02:29:30Z","timestamp":1774924170126,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,18]],"date-time":"2025-01-18T00:00:00Z","timestamp":1737158400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Memristives provide a high degree of non-linearity to the model. This property has led to many studies focusing on developing memristive models to provide more non-linearity. This article studies a novel fractional discrete memristive system with incommensurate orders using \u03d1i-th Caputo-like operator. Bifurcation, phase portraits and the computation of the maximum Lyapunov Exponent (LEmax) are used to demonstrate their impact on the system\u2019s dynamics. Furthermore, we employ the sample entropy approach (SampEn), C0 complexity and the 0-1 test to quantify complexity and validate chaos in the incommensurate system. Studies indicate that the discrete memristive system with incommensurate fractional orders manifests diverse dynamical behaviors, including hidden chaos, symmetry, and asymmetry attractors, which are influenced by the incommensurate derivative values. Moreover, a 2D non-linear controller is presented to stabilize and synchronize the novel system. The work results are provided by numerical simulation obtained using MATLAB R2024a codes.<\/jats:p>","DOI":"10.3390\/sym17010143","type":"journal-article","created":{"date-parts":[[2025,1,20]],"date-time":"2025-01-20T04:04:12Z","timestamp":1737345852000},"page":"143","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["On Fractional Discrete Memristive Model with Incommensurate Orders: Symmetry, Asymmetry, Hidden Chaos and Control Approaches"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7644-9937","authenticated-orcid":false,"given":"Hussein","family":"Al-Taani","sequence":"first","affiliation":[{"name":"School of Electrical Engineering and Information Technology, German Jordanian University, Amman 11180, Jordan"}]},{"given":"Ma\u2019mon","family":"Abu Hammad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al-Zaytoonah University of Jordan, Amman 11733, Jordan"}]},{"given":"Mohammad","family":"Abudayah","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Information Technology, German Jordanian University, Amman 11180, Jordan"}]},{"given":"Louiza","family":"Diabi","sequence":"additional","affiliation":[{"name":"Laboratory of Dynamical Systems and Control, University of Larbi Ben M\u2019hidi, Oum El Bouaghi 04000, Algeria"}]},{"given":"Adel","family":"Ouannas","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Sciences, University of Larbi Ben M\u2019hidi, Oum El Bouaghi 04000, Algeria"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"115471","DOI":"10.1016\/j.chaos.2024.115471","article-title":"Dynamics analysis and FPGA implementation of discrete memristive cellular neural network with heterogeneous activation functions","volume":"187","author":"Wang","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"14463","DOI":"10.1007\/s11071-024-09791-6","article-title":"Symmetric multi-double-scroll attractors in Hopfield neural network under pulse controlled memristor","volume":"112","author":"Li","year":"2024","journal-title":"Nonlinear Dyn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.neunet.2023.12.008","article-title":"Memristor-induced hyperchaos, multiscroll and extreme multistability in fractional-order HNN: Image encryption and FPGA implementation","volume":"171","author":"Kong","year":"2024","journal-title":"Neural Netw."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1109\/TCSI.2024.3439869","article-title":"Memristive tabu learning neuron generated multi-wing attractor with FPGA implementation and application in encryption","volume":"72","author":"Deng","year":"2024","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Qian, K., Xiao, Y., Wei, Y., Liu, D., Wang, Q., and Feng, W. (2023). A robust memristor-enhanced polynomial hyper-chaotic map and its multi-channel image encryption application. Micromachines, 14.","DOI":"10.3390\/mi14112090"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"4501","DOI":"10.1007\/s11071-022-08086-y","article-title":"Dynamical analysis of a fractional discrete-time vocal system","volume":"111","author":"Vignesh","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Feng, W., Wang, Q., Liu, H., Ren, Y., Zhang, J., Zhang, S., Qian, K., and Wen, H. (2023). Exploiting newly designed fractional-order 3D Lorenz chaotic system and 2D discrete polynomial hyper-chaotic map for high-performance multi-image encryption. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7120887"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/j.neucom.2021.10.122","article-title":"Applications of fractional calculus in computer vision: A survey","volume":"489","author":"Arora","year":"2022","journal-title":"Neurocomputing"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Abdeljawad, T., Sher, M., Shah, K., Sarwar, M., Amacha, I., Alqudah, M., and Al-Jaser, A. (2024). Analysis of a class of fractal hybrid fractional differential equation with application to a biological model. Sci. Rep., 14.","DOI":"10.1038\/s41598-024-67158-8"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"121455","DOI":"10.1016\/j.ins.2024.121455","article-title":"Fuzzy Adaptive Control for Consensus Tracking in Multiagent Systems with Incommensurate Fractional-Order Dynamics: Application to Power Systems","volume":"689","author":"Sharafian","year":"2024","journal-title":"Inf. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Hamadneh, T., Abbes, A., Al-Tarawneh, H., Gharib, G.M., Salameh, W.M.M., Al Soudi, M.S., and Ouannas, A. (2023). On chaos and complexity analysis for a new sine-based memristor map with commensurate and incommensurate fractional orders. Mathematics, 11.","DOI":"10.3390\/math11204308"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"882","DOI":"10.1002\/asjc.2289","article-title":"Stabilization of a new commensurate\/incommensurate fractional order chaotic system","volume":"23","author":"Gholamin","year":"2021","journal-title":"Asian J. Control"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2675","DOI":"10.1140\/epjs\/s11734-023-00917-2","article-title":"Chaotic dynamics of fractional difference magnetic levitation model with application to image encryption","volume":"232","author":"Vignesh","year":"2023","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2330016","DOI":"10.1142\/S0218127423300161","article-title":"Extreme multistability and extreme events in a novel chaotic circuit with hidden attractors","volume":"33","author":"Ahmadi","year":"2022","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3187","DOI":"10.1140\/epjs\/s11734-022-00559-w","article-title":"Modeling different discrete memristive sine maps and its parameter identification","volume":"231","author":"Peng","year":"2022","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"106866","DOI":"10.1016\/j.rinp.2023.106866","article-title":"Hidden attractors in a new fractional-order Chua system with arctan nonlinearity and its DSP implementation","volume":"52","author":"Wu","year":"2023","journal-title":"Results Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"484","DOI":"10.1002\/acs.3207","article-title":"Control and synchronization of fractional-order chaotic satellite systems using feedback and adaptive control techniques","volume":"35","author":"Kumar","year":"2021","journal-title":"Int. J. Adapt. Control. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"031102","DOI":"10.1063\/5.0199236","article-title":"Offset boosting in a discrete system","volume":"34","author":"Li","year":"2024","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Hamadneh, T., Ahmed, S.B., Al-Tarawneh, H., Alsayyed, O., Gharib, G.M., Al Soudi, M.S., Abbes, A., and Ouannas, A. (2023). The new four-dimensional fractional chaotic map with constant and variable-order: Chaos, control and synchronization. Mathematics, 11.","DOI":"10.3390\/math11204332"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"124203","DOI":"10.1016\/j.physa.2020.124203","article-title":"On the synchronization and stabilization of fractional-order chaotic systems: Recent advances and future perspectives","volume":"551","author":"Balootaki","year":"2020","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zaqueros-Martinez, J., Rodriguez-Gomez, G., Tlelo-Cuautle, E., and Orihuela-Espina, F. (2023). Fuzzy synchronization of chaotic systems with hidden attractors. Entropy, 25.","DOI":"10.3390\/e25030495"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Elaskar, S. (2022). Symmetry in Nonlinear Dynamics and Chaos. Symmetry, 15.","DOI":"10.3390\/sym15010102"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"111992","DOI":"10.1016\/j.chaos.2022.111992","article-title":"A 3D memristive chaotic system with conditional symmetry","volume":"158","author":"Wang","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Karimov, T., Rybin, V., Kolev, G., Rodionova, E., and Butusov, D. (2021). Chaotic communication system with symmetry-based modulation. Appl. Sci., 11.","DOI":"10.3390\/app11083698"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Elbadri, M., Abdoon, M.A., Berir, M., and Almutairi, D.K. (2023). A symmetry chaotic model with fractional derivative order via two different methods. Symmetry, 15.","DOI":"10.3390\/sym15061151"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1016\/j.cjph.2020.11.007","article-title":"Asymmetric coexisting bifurcations and multi-stability in an asymmetric memristive diode-bridge-based jerk circuit","volume":"70","author":"Xu","year":"2021","journal-title":"Chin. J. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"112905","DOI":"10.1016\/j.chaos.2022.112905","article-title":"Memristor-coupled asymmetric neural networks: Bionic modeling, chaotic dynamics analysis and encryption application","volume":"166","author":"Lin","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Al-Taani, H., Abu Hammad, M.M., Abudayah, M., Diabi, L., and Ouannas, A. (2024). Asymmetry and Symmetry in New Three-Dimensional Chaotic Map with Commensurate and Incommensurate Fractional Orders. Symmetry, 16.","DOI":"10.3390\/sym16111447"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/s10203-020-00279-7","article-title":"An application of Sigmoid and Double-Sigmoid functions for dynamic policyholder behaviour","volume":"44","author":"Baione","year":"2021","journal-title":"Decis. Econ. Financ."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Mfungo, D.E., Fu, X., Wang, X., and Xian, Y. (2023). Enhancing image encryption with the Kronecker Xor product, the Hill Cipher, and the Sigmoid Logistic Map. Appl. Sci., 13.","DOI":"10.3390\/app13064034"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"8545","DOI":"10.1007\/s11071-023-08293-1","article-title":"Approximating piecewise nonlinearities in dynamic systems with sigmoid functions: Advantages and limitations","volume":"111","author":"Martinelli","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"012010","DOI":"10.1088\/1742-6596\/1471\/1\/012010","article-title":"Sigmoid activation function in selecting the best model of artificial neural networks","volume":"1471","author":"Pratiwi","year":"2020","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1007\/s12648-019-01512-9","article-title":"Chaos control for multi-scroll chaotic attractors generated by introducing a bipolar sigmoid function series","volume":"94","author":"Jiang","year":"2020","journal-title":"Indian J. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1602","DOI":"10.1016\/j.camwa.2011.03.036","article-title":"On Riemann and Caputo fractional differences","volume":"62","author":"Abdeljawad","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_35","first-page":"1","article-title":"Discrete fractional calculus with the nabla operator","volume":"62","author":"Atici","year":"2009","journal-title":"Electron. J. Qual. Theory Differ. Equ. [Electron. Only]"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1007\/s11071-013-1065-7","article-title":"Discrete fractional logistic map and its chaos","volume":"75","author":"Wu","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1655","DOI":"10.1016\/j.aej.2021.06.073","article-title":"Novel convenient conditions for the stability of nonlinear incommensurate fractional-order difference systems","volume":"61","author":"Shatnawi","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"4534","DOI":"10.1109\/TCSI.2021.3082895","article-title":"Discrete memristor hyperchaotic maps","volume":"68","author":"Bao","year":"2021","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"e27781","DOI":"10.1016\/j.heliyon.2024.e27781","article-title":"Assessing sigmoidal function on memristive maps","volume":"10","author":"Thoai","year":"2024","journal-title":"Heliyon"},{"key":"ref_40","unstructured":"Anastassiou, G.A. (2024, December 01). General Multiple Sigmoid Functions Relied Complex Valued Multivariate Trigonometric and Hyperbolic Neural Network Approximations. Available online: https:\/\/rgmia.org\/papers\/v26\/v26a43.pdf."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/j.cnsns.2014.06.042","article-title":"Jacobian matrix algorithm for Lyapunov exponents of the discrete fractional maps","volume":"22","author":"Wu","year":"2015","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Gottwald, G.A., and Melbourne, I. (2016). The 0-1 test for chaos: A review. Chaos Detection and Predictability, Springer.","DOI":"10.1007\/978-3-662-48410-4_7"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"H2039","DOI":"10.1152\/ajpheart.2000.278.6.H2039","article-title":"Physiological time-series analysis using approximate entropy and sample entropy","volume":"278","author":"Richman","year":"2000","journal-title":"Am. J. Physiol.-Heart Circ. Physiol."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"66723","DOI":"10.1109\/ACCESS.2018.2873782","article-title":"The entropy algorithm and its variants in the fault diagnosis of rotating machinery: A review","volume":"6","author":"Li","year":"2018","journal-title":"IEEE Access"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"1188","DOI":"10.1007\/BF02507729","article-title":"Mathematical foundation of a new complexity measure","volume":"26","author":"Shen","year":"2005","journal-title":"Appl. Math. Mech."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/143\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:31:29Z","timestamp":1759919489000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/143"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,18]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010143"],"URL":"https:\/\/doi.org\/10.3390\/sym17010143","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,18]]}}}