{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T19:45:09Z","timestamp":1769283909150,"version":"3.49.0"},"reference-count":54,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2023,10,28]],"date-time":"2023-10-28T00:00:00Z","timestamp":1698451200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Saudi Arabia","award":["RSPD2023R920"],"award-info":[{"award-number":["RSPD2023R920"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This study aims to find a solution to the symmetry chaotic jerk system by using a new ABC-FD scheme and the NILM method. The findings of the supplied methods have been compared to Runge\u2013Kutta\u2019s fourth order (RK4). It was discovered that the suggested techniques gave results comparable to the RK4 method. Our primary goal is to develop effective methods for addressing symmetrical, chaotic systems. Using ABC-FD and NILM presents innovative approaches for comprehending and effectively handling intricate dynamics. The findings of this study have significant significance for addressing the occurrence of chaotic behavior in diverse scientific and engineering contexts. This research significantly contributes to fractional calculus and its various applications. The application of ABC-FD, which can identify chaotic behavior, makes our work stand out. This novel approach contributes to advancing research in nonlinear dynamics and fractional calculus. The present study not only offers a resolution to the problem of symmetric chaotic jerk systems but also presents a framework that may be applied to tackle analogous challenges in several domains. The techniques outlined in this paper facilitate the development and computational analysis of prospective fractional models, thereby contributing to the progress of scientific and engineering disciplines.<\/jats:p>","DOI":"10.3390\/sym15111991","type":"journal-article","created":{"date-parts":[[2023,10,30]],"date-time":"2023-10-30T09:27:28Z","timestamp":1698658048000},"page":"1991","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["A Comparative Numerical Study of the Symmetry Chaotic Jerk System with a Hyperbolic Sine Function via Two Different Methods"],"prefix":"10.3390","volume":"15","author":[{"given":"Abdulrahman B. M.","family":"Alzahrani","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0509-1506","authenticated-orcid":false,"given":"Mohamed A.","family":"Abdoon","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8987-6161","authenticated-orcid":false,"given":"Mohamed","family":"Elbadri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences and Arts, Jouf University, Tubarjal 74713, Saudi Arabia"},{"name":"Department of Mathematic, University of Gezira, Wad Madani 21111, Sudan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5762-3474","authenticated-orcid":false,"given":"Mohammed","family":"Berir","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, Al-Baha University, Baljurashi 65622, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Diaa Eldin","family":"Elgezouli","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,28]]},"reference":[{"key":"ref_1","unstructured":"Samko, G., Kilbas, A., and Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Carpinteri, A., and Mainardi, F. (1997). Fractals and Fractional Calculus in Continuum Mechanics, Springer.","DOI":"10.1007\/978-3-7091-2664-6"},{"key":"ref_3","unstructured":"Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus, Academic Press."},{"key":"ref_4","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"110201","DOI":"10.1088\/1402-4896\/acfe73","article-title":"Advanced fractional calculus, differential equations and neural networks: Analysis, modeling and numerical computations","volume":"98","author":"Baleanu","year":"2023","journal-title":"Phys. Scr."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2912","DOI":"10.1177\/1077546315623887","article-title":"Second-order fast terminal sliding mode control design based on LMI for a class of non-linear uncertain systems and its application to chaotic systems","volume":"23","author":"Mobayen","year":"2017","journal-title":"J. Vib. Control."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1016\/j.physa.2016.05.045","article-title":"Chaos synchronization of fractional chaotic maps based on the stability condition","volume":"460","author":"Wu","year":"2016","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_8","first-page":"140","article-title":"A Note on the Fractional-Order Chua\u2019s System","volume":"38","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Lai, Q., Akgul, A., Li, C., Xu, G., and \u00c7avu\u015fo\u011flu, \u00dc. (2017). A New Chaotic System with Multiple Attractors: Dynamic Analysis, CircuitRealization and S-Box Design. Entropy, 20.","DOI":"10.3390\/e20010012"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Xu, G., Shekofteh, Y., Akg\u00fcl, A., Li, C., and Panahi, S. (2018). A New Chaotic System with a Self-Excited Attractor: Entropy Measurement, Signal Encryption, and Parameter Estimation. Entropy, 20.","DOI":"10.3390\/e20020086"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2016.05.002","article-title":"Hidden Attractors in Dynamical Systems","volume":"637","author":"Dudkowski","year":"2016","journal-title":"Phys. Rep."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Alexan, W., Alexan, N., and Gabr, M. (2023). Multiple-Layer Image Encryption Utilizing Fractional-Order Chen Hyperchaotic Map and Cryptographically Secure PRNGs. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7040287"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Stamov, G., and Stamova, I. (2023). Extended Stability and Control Strategies for Impulsive and Fractional Neural Networks: A Review of the Recent Results. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7040289"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"7688","DOI":"10.1002\/mma.7083","article-title":"Discrete Generalized Fractional Operators Defined Using H-Discrete Mit-tag-Leffler Kernels and Applications to AB Fractional Difference Systems","volume":"46","author":"Mohammed","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1582","DOI":"10.3390\/sym15081582","article-title":"Symmetry in a Fractional-Order Multi-Scroll Chaotic System Using the Extended Caputo Operator","volume":"15","author":"Matouk","year":"2023","journal-title":"Symmetry"},{"key":"ref_16","unstructured":"Belgacem, F.B.M., Silambarasan, R., Zakia, H., and Mekkaoui, T. (2017). Trends in Mathematics, Springer."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.aej.2023.05.011","article-title":"Analysis, Modeling and Simulation of a Fractional-Order Influenza Model","volume":"74","author":"Abdoon","year":"2023","journal-title":"Alex. Eng. J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"268","DOI":"10.18280\/mmep.090133","article-title":"Advantages of the Differential Equations for Solving Problems in Mathematical Physics with Symbolic Computation","volume":"9","author":"Abdoon","year":"2022","journal-title":"Math. Model. Eng. Probl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2022\/2162356","article-title":"Computational Technique to Study Analytical Solutions to the Fractional Modified KDV-Zakharov-Kuznetsov Equation","volume":"2022","author":"Abdoon","year":"2022","journal-title":"Abstr. Appl. Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"113039","DOI":"10.1016\/j.chaos.2022.113039","article-title":"A High-Speed Pseudo-Random Bit Generator Driven by 2D-Discrete Hyperchaos","volume":"167","author":"Yang","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"106753","DOI":"10.1016\/j.rinp.2023.106753","article-title":"An Efficient Method for the Fractional Electric Circuits Based on Fibonacci Wavelet","volume":"52","author":"Ahmed","year":"2023","journal-title":"Results Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"100560","DOI":"10.1016\/j.padiff.2023.100560","article-title":"An approximate solution of a time fractional Burgers\u2019 equation involving the Caputo-Katugampola fractional derivative","volume":"8","author":"Elbadri","year":"2023","journal-title":"Partial. Differ. Equ. Appl. Math."},{"key":"ref_23","first-page":"741","article-title":"A Numerical Confirmation of a Fractional SEITR for Influenza Model Efficiency","volume":"175","author":"Almutairi","year":"2023","journal-title":"Appl. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1459","DOI":"10.1109\/TCSI.2005.851717","article-title":"Design and Implementation of N-Scroll Chaotic Attractors from a General Jerk Circuit","volume":"52","author":"Yu","year":"2005","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1016\/j.isatra.2019.08.004","article-title":"A Novel Fractional Variable-Order Equivalent Circuit Model and Parameter Identification of Electric Vehicle Li-Ion Batteries","volume":"97","author":"Zhang","year":"2020","journal-title":"ISA Trans."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"9","DOI":"10.24018\/ejmath.2023.4.3.190","article-title":"Transfer Function and Z-Transform of an Electrical System in MATLAB\/Simulink","volume":"4","author":"Abdullah","year":"2023","journal-title":"Eur. J. Math. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1016\/j.proeng.2012.01.071","article-title":"Research on the Multi-Scroll Chaos Generation Based on Jerk Mode","volume":"29","author":"Chunxia","year":"2012","journal-title":"Procedia Eng."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1659","DOI":"10.1016\/j.chaos.2006.04.058","article-title":"Multi-Scroll and Hypercube Attractors from a General Jerk Circuit Using Josephson Junctions","volume":"34","year":"2007","journal-title":"Chaos Solitons Fractals"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1119\/1.18585","article-title":"Some Simple Chaotic Jerk Functions","volume":"65","author":"Sprott","year":"1997","journal-title":"Am. J. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/S0375-9601(97)00088-1","article-title":"Simplest Dissipative Chaotic Flow","volume":"228","author":"Sprott","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_31","first-page":"273","article-title":"Simple Formula for Designing the PID Controller of a DC-DC Buck Converter","volume":"14","author":"Samosir","year":"2023","journal-title":"Int. J. Power Electron. Drive Syst."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1049\/el.2013.0029","article-title":"Current-tunable Chaotic Jerk Oscillator","volume":"49","author":"Srisuchinwong","year":"2013","journal-title":"Electron. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"100014","DOI":"10.1016\/j.nwnano.2023.100014","article-title":"Emperor\u2019s New Clothes: Novel Textile-Based Supercapacitors Using Sheep Wool Fiber as Electrode Substrate","volume":"3","author":"Grube","year":"2023","journal-title":"Nano Trends"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1567","DOI":"10.1142\/S0218127410027076","article-title":"Simplest Chaotic Circuit","volume":"20","author":"Muthuswamy","year":"2010","journal-title":"Int. J. Bifurc. Chaos Appl. Sci. Eng."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1016\/j.jmaa.2005.05.009","article-title":"An Iterative Method for Solving Nonlinear Functional Equations","volume":"316","author":"Jafari","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2125","DOI":"10.1007\/s00366-020-01202-9","article-title":"A New Iterative Method with \u03c1-Laplace Transform for Solving Fractional Differential Equations with Caputo Generalized Fractional Derivative","volume":"38","author":"Bhangale","year":"2022","journal-title":"Eng. Comput"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3249720","DOI":"10.1155\/2018\/3249720","article-title":"New Iterative Method for the Solution of Fractional Damped Burger and Fractional Sharma-TassoOlver Equations","volume":"2018","author":"Khan","year":"2018","journal-title":"Complexity"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1016\/j.rinp.2018.07.004","article-title":"New Transform Iterative Method for Solving Some Klein-Gordon Equations","volume":"10","author":"Alderremy","year":"2018","journal-title":"Results Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1140\/epjp\/i2017-11717-0","article-title":"New Numerical Approximation of Fractional Derivative with Non-Local and Non-Singular Kernel:Application to Chaotic Models","volume":"132","author":"Toufik","year":"2017","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Golmankhaneh, A.K. (2023). Fractal Calculus and its Applications: F\u03b1-Calculus, World Scientific Publishing Co., Ltd.","DOI":"10.1142\/12988"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"763","DOI":"10.2298\/TSCI160111018A","article-title":"New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model","volume":"20","author":"Atangana","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_42","first-page":"1055434","article-title":"Fractional Integral Inequalities via Atangana-Baleanu Operators for Convex and Concave Functions","volume":"2021","author":"Akdemir","year":"2021","journal-title":"J. Funct. Spaces"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1128","DOI":"10.29020\/nybg.ejpam.v16i2.4769","article-title":"A New Scheme for Solving a Fractional Differential Equation and a Chaotic System","volume":"2023","author":"Qazza","year":"2023","journal-title":"Eur. J. Pure Appl. Math"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Argyris, J., Faust, G., Haase, M., and Friedrich, R. (2015). An Exploration of Dynamical Systems and Chaos, Springer.","DOI":"10.1007\/978-3-662-46042-9"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","article-title":"Determining Lyapunov exponents from a time series","volume":"16","author":"Wolf","year":"1985","journal-title":"Physical D"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Kumar, S., Bhagwan, J., and J\u00e4ntschi, L. (2022). Optimal Derivative-Free One-Point Algorithms for Computing Multiple Zeros of Nonlinear Equations. Symmetry, 14.","DOI":"10.3390\/sym14091881"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1007\/s11075-015-0023-5","article-title":"An optimal fourth-order family of methods for multiple roots and its dynamics","volume":"71","author":"Behl","year":"2016","journal-title":"Numer. Algorithms"},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Sharma, J.R., Kumar, S., and J\u00e4ntschi, L. (2020). On derivative free multiple-root finders with optimal fourth order convergence. Mathematics, 8.","DOI":"10.3390\/math8071091"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Sharma, J.R., Kumar, D., and J\u00e4ntschi, L. (2019). On a reduced cost higher order Traub-Steffensen-Like method for nonlinear systems. Symmetry, 11.","DOI":"10.3390\/sym11070891"},{"key":"ref_50","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_51","doi-asserted-by":"crossref","unstructured":"Saadeh, R., Abdoon, M., Qazza, A., and Berir, M. (2023). A Numerical Solution of Generalized Caputo Fractional Initial Value Problems. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7040332"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"3950816","DOI":"10.1155\/2020\/3950816","article-title":"A New Solution of Time-Fractional Coupled KdV Equation by Using Natural Decomposition Method","volume":"2020","author":"Elbadri","year":"2020","journal-title":"Abstr. Appl. Anal."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"3586802","DOI":"10.1155\/2022\/3586802","article-title":"Initial Value Problems with Generalized Fractional Derivatives and Their Solutions via Generalized Laplace Decomposition Method","volume":"2022","author":"Elbadri","year":"2022","journal-title":"Adv. Math. Phys."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Elbadri, M. (2023). A Numerical Solution and Comparative Study of the Symmetric Rossler Attractor with the Generalized Caputo Fractional Derivative via Two Different Methods. Mathematics, 11.","DOI":"10.3390\/math11132997"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/11\/1991\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:13:23Z","timestamp":1760130803000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/11\/1991"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,28]]},"references-count":54,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2023,11]]}},"alternative-id":["sym15111991"],"URL":"https:\/\/doi.org\/10.3390\/sym15111991","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,28]]}}}