{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T07:30:13Z","timestamp":1773905413460,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,5,25]],"date-time":"2023-05-25T00:00:00Z","timestamp":1684972800000},"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>In this article, we have investigated solutions to a symmetry chaotic system with fractional derivative order using two different methods\u2014the numerical scheme for the ABC fractional derivative, and the Laplace decomposition method, with help from the MATLAB and Mathematica platforms. We have explored progressive and efficient solutions to the chaotic model through the successful implementation of two mathematical methods. For the phase portrait of the model, the profiles of chaos are plotted by assigning values to the attached parameters. Hence, the offered techniques are relevant for advanced studies on other models. We believe that the unique techniques that have been proposed in this study will be applied in the future to build and simulate a wide range of fractional models, which can be used to address more challenging physics and engineering problems.<\/jats:p>","DOI":"10.3390\/sym15061151","type":"journal-article","created":{"date-parts":[[2023,5,26]],"date-time":"2023-05-26T01:52:43Z","timestamp":1685065963000},"page":"1151","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["A Symmetry Chaotic Model with Fractional Derivative Order via Two Different Methods"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8987-6161","authenticated-orcid":false,"given":"Mohamed","family":"Elbadri","sequence":"first","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\/0000-0003-0509-1506","authenticated-orcid":false,"given":"Mohamed A.","family":"Abdoon","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences (Mathematics) Deanship of Common First Year, Shaqra University, Riyadh 15342, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Bakht Al-Ruda University, Duwaym 999129, 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, Bakht Al-Ruda University, Duwaym 999129, Sudan"},{"name":"Department of Mathematics, Faculty of Science and Arts, Al-Baha University, Baljurashi 1988, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1275-2384","authenticated-orcid":false,"given":"Dalal Khalid","family":"Almutairi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Education (Majmaah), Majmaah University, Al-Majmaah 11952, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,5,25]]},"reference":[{"key":"ref_1","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_2","unstructured":"Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus, Academic Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1515\/anona-2022-0256","article-title":"On Cauchy Problem for Fractional Parabolic-Elliptic Keller-Segel Model","volume":"12","author":"Nguyen","year":"2022","journal-title":"Adv. Nonlinear Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1080\/07362994.2021.1906274","article-title":"Stochastic Pseudo-Parabolic Equations with Fractional Derivative and Fractional Brownian Motion","volume":"40","author":"Thach","year":"2022","journal-title":"Stoch. Anal. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"347","DOI":"10.3934\/dcdsb.2022079","article-title":"New Well-Posedness Results for Stochastic Delay Rayleigh-Stokes Equations","volume":"28","author":"Tuan","year":"2023","journal-title":"Discret. Contin. Dyn. Systems. Ser. B"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7759","DOI":"10.3934\/math.2022436","article-title":"Chebyshev Fifth-Kind Series Approximation for Generalized Space Fractional Partial Differential Equations","volume":"7","author":"Ali","year":"2022","journal-title":"AIMS Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"8622","DOI":"10.3934\/math.2022482","article-title":"Spectral Collocation Approach with Shifted Chebyshev Sixth-Kind Series Approximation for Generalized Space Fractional Partial Differential Equations","volume":"7","author":"Mukheimer","year":"2022","journal-title":"AIMS Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"28","DOI":"10.22436\/jmcs.027.01.03","article-title":"Optical Solitons for Conformable Space-Time Fractional Nonlinear Model","volume":"27","author":"Asjad","year":"2022","journal-title":"J. Math. Comput. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"770","DOI":"10.1515\/phys-2021-0085","article-title":"Novel Soliton Solutions to the Atangana\u2013Baleanu Fractional System of Equations for the ISALWs","volume":"19","author":"Asjad","year":"2021","journal-title":"Open Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"17252","DOI":"10.3934\/math.2022950","article-title":"A Novel Finite Difference Based Numerical Approach for Modified Atangana- Baleanu Caputo Derivative","volume":"7","author":"Chawla","year":"2022","journal-title":"AIMS Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"674","DOI":"10.3389\/fphy.2022.959168","article-title":"Numerical Analysis of Hydrothermal Flow and Heat Transfer inside a Cavity Formed Due to Faults Causing Earthquakes","volume":"10","author":"Nadeem","year":"2022","journal-title":"Front. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"101420","DOI":"10.1016\/j.jksus.2021.101420","article-title":"Bright and Singular Soliton Solutions to the Atangana-Baleanu Fractional System of Equations for the ISALWs","volume":"33","author":"Rezazadeh","year":"2021","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_13","first-page":"2174806","article-title":"New Soliton Solutions for the Space-Time Fractional Modified Third Order Korteweg\u2013de Vries Equation","volume":"2022","author":"Asjad","year":"2022","journal-title":"J. Ocean. Eng. Sci."},{"key":"ref_14","first-page":"359","article-title":"Computational Examples of a Class of Fractional Order Nonlinear Evolution Equations Using Modified Extended Direct Algebraic Method","volume":"15","author":"Younis","year":"2015","journal-title":"J. Comput. Methods Sci. Eng."},{"key":"ref_15","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_16","unstructured":"Samko, G., Kilbas, A., and Marichev, O. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach."},{"key":"ref_17","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_18","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_19","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, Circuit Realization and S-Box Design. Entropy, 20.","DOI":"10.3390\/e20010012"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"140","DOI":"10.1016\/j.chaos.2006.10.054","article-title":"A note on the fractional-order chua\u2019s system","volume":"38","author":"Petras","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2162356","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_22","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1016\/j.mejo.2019.05.005","article-title":"FPGA implementation of sound encryption system based on fractional-order chaotic systems","volume":"90","author":"Hassan","year":"2019","journal-title":"Microelectron. J."},{"key":"ref_23","first-page":"88","article-title":"A modified Laplace transform for certain generalized fractional operators","volume":"1","author":"Fahd","year":"2018","journal-title":"Res. Nonl. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"171","DOI":"10.22436\/jnsa.013.03.06","article-title":"Global Asymptotic Stability of the Fractional Differential Equations","volume":"13","author":"Sene","year":"2019","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_25","first-page":"504","article-title":"A Fractional Epidemiological Model for Computer Viruses Pertaining to a New Fractional Derivative","volume":"316","author":"Singh","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"109811","DOI":"10.1016\/j.chaos.2020.109811","article-title":"Chaotic Behaviour of Fractional Predator-Prey Dynamical System","volume":"135","author":"Kumar","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_27","first-page":"1450","article-title":"Numerical Simulations for a Variable Order Fractional Schnakenberg Model","volume":"1637","author":"Hammouch","year":"2014","journal-title":"AIP Publ. LLC"},{"key":"ref_28","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_29","first-page":"489","article-title":"Traveling-wave solutions of the generalized Zakharov equation with time-space fractional derivatives","volume":"5","author":"Hammouch","year":"2014","journal-title":"J. MESA"},{"key":"ref_30","unstructured":"Belgacem, F.B.M., Silambarasan, R., Zakia, H., and Mekkaoui, T. (2017). Trends in Mathematics, Springer."},{"key":"ref_31","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_32","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1186\/s13662-020-02890-9","article-title":"Some New Edge Detecting Techniques Based on Fractional Derivatives with Nonlocal and Non-Singular Kernels","volume":"2020","author":"Ghanbari","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"109631","DOI":"10.1016\/j.chaos.2020.109631","article-title":"Second-Grade Fluid Model with Caputo\u2013Liouville Generalized Fractional Derivative","volume":"133","author":"Sene","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_34","first-page":"219","article-title":"Control of fractional-order Chua\u2019s system","volume":"53","author":"Petras","year":"2002","journal-title":"J. Electr. Eng."},{"key":"ref_35","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_36","doi-asserted-by":"crossref","first-page":"1799","DOI":"10.1016\/j.aml.2011.04.037","article-title":"Application of the Laplace Decomposition Method for Solving Linear and Nonlinear Fractional Diffusion\u2013Wave Equations","volume":"24","author":"Jafari","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_37","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_38","doi-asserted-by":"crossref","first-page":"1101","DOI":"10.1016\/j.ijleo.2017.11.124","article-title":"New Optical Solitons of Cubic-Quartic Nonlinear Schr\u00f6dinger Equation","volume":"157","author":"Hosseini","year":"2017","journal-title":"Optik"},{"key":"ref_39","first-page":"1213","article-title":"The Generalized (2 + 1) and (3 + 1)-Dimensional with Advanced Analytical Wave Solutions via Computational Applications","volume":"12","author":"Faeza","year":"2021","journal-title":"Int. J. Nonlinear Anal. Appl."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1080\/17455030.2017.1362133","article-title":"Bright and Singular Soliton Solutions of the Conformable Time-Fractional Klein\u2013Gordon Equations with Different Nonlinearities","volume":"28","author":"Hosseini","year":"2017","journal-title":"Waves Random Complex Media"},{"key":"ref_41","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_42","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_43","doi-asserted-by":"crossref","first-page":"023116","DOI":"10.1063\/1.5086771","article-title":"Numerical Solutions of the Fractional Fisher\u2019s Type Equations with Atangana-Baleanu Fractional Derivative by Using Spectral Collocation Methods","volume":"29","author":"Saad","year":"2019","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1140\/epjp\/i2019-13003-7","article-title":"Strange Chaotic Attractors under Fractal-Fractional Operators Using Newly Proposed Numerical Methods","volume":"134","author":"Qureshi","year":"2019","journal-title":"Eur. Phys. J. Plus"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1151\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:42:14Z","timestamp":1760125334000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/6\/1151"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,25]]},"references-count":44,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["sym15061151"],"URL":"https:\/\/doi.org\/10.3390\/sym15061151","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,25]]}}}