{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T22:08:52Z","timestamp":1768342132160,"version":"3.49.0"},"reference-count":58,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,30]],"date-time":"2024-12-30T00:00:00Z","timestamp":1735516800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002383","name":"King Saud University, Saudi Arabia","doi-asserted-by":"publisher","award":["RSPD2024R920"],"award-info":[{"award-number":["RSPD2024R920"]}],"id":[{"id":"10.13039\/501100002383","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper provides a solution to the new fractional-order Lorenz\u2013Stenflo model using the adaptive predictor\u2013corrector approach and the \u03c1-Laplace New Iterative Method (L\u03c1NIM), representing an extensive comparison between both techniques with RK4 related to accuracy and error analysis. The results show that the suggested approaches allow one to be more accurate in analyzing the dynamics of the system. These techniques also produce results that are comparable to the results of other approximate techniques. The techniques can, thus, be used on a wider class of systems in order to provide more accurate results. These techniques also appropriately identify chaotic attractors in the system. These techniques can be applied to solve various numerical problems arising in science and engineering in the future.<\/jats:p>","DOI":"10.3390\/axioms14010020","type":"journal-article","created":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T07:51:57Z","timestamp":1735631517000},"page":"20","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Comparative Study and Numerical Solutions for the Fractional Modified Lorenz\u2013Stenflo System Using Two Methods"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8987-6161","authenticated-orcid":false,"given":"Mohamed","family":"Elbadri","sequence":"first","affiliation":[{"name":"Mathematics Department, College of Science, Jouf University, Sakaka 72388, Saudi Arabia"}]},{"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, P.O. Box 1142, Riyadh 12373, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-1046-7707","authenticated-orcid":false,"given":"Abdulrahman B. M.","family":"Alzahrani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6394-1452","authenticated-orcid":false,"given":"Rania","family":"Saadeh","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Zarqa University, Zarqa 13110, Jordan"}]},{"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, Al-Baha 61008, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,30]]},"reference":[{"key":"ref_1","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_2","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","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_3","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":"Microelectronics"},{"key":"ref_4","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_5","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_6","doi-asserted-by":"crossref","unstructured":"Bingi, K., Prusty, B.R., and Singh, A.P. (2023). A Review on Fractional-Order Modelling and Control of Robotic Manipulators. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7010077"},{"key":"ref_7","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_8","unstructured":"Oldham, K.B., and Spanier, J. (1974). The Fractional Calculus, Academic Press."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Carpinteri, A., and Mainardi, F. (1997). Fractional Calculus. Fractals and Fractional Calculus in Continuum Mechanics, International Centre for Mechanical Sciences; Springer.","DOI":"10.1007\/978-3-7091-2664-6"},{"key":"ref_10","unstructured":"Samko, S., Kilbas, A.A., and Marichev, O. (1993). Fractional Integrals and Derivatives, CRC Press."},{"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":"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_13","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 non-local and non-singular kernels","volume":"2020","author":"Ghanbari","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.46481\/jnsps.2023.1453","article-title":"Modeling and Analysis of a Fractional Visceral Leishmaniosis with Caputo and Caputo\u2013Fabrizio derivatives","volume":"5","author":"Almutairi","year":"2023","journal-title":"J. Niger. Soc. Phys. Sci."},{"key":"ref_15","unstructured":"Petras, I. (2000). Control of fractional-order Chua\u2019s system. arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.cjph.2019.11.003","article-title":"Traveling wave solutions of conformable time-fractional Zakharov\u2013Kuznetsov and Zoomeron equations","volume":"64","author":"Odabasi","year":"2020","journal-title":"Chin. J. Phys."},{"key":"ref_17","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_18","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_19","doi-asserted-by":"crossref","first-page":"135","DOI":"10.55579\/jaec.202153.340","article-title":"An Introductory Overview of Fractional-Calculus Operators Based Upon the Fox-Wright and Related Higher Transcendental Functions","volume":"5","author":"Srivastava","year":"2021","journal-title":"Adv. Eng. Comput."},{"key":"ref_20","first-page":"1501","article-title":"Some parametric and argument variations of the operators of fractional calculus and related special functions and integral transformations","volume":"22","author":"Srivastava","year":"2021","journal-title":"J. Nonlinear Convex Anal."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M. (2021). A Survey of Some Recent Developments on Higher Transcendental Functions of Analytic Number Theory and Applied Mathematics. Symmetry, 13.","DOI":"10.3390\/sym13122294"},{"key":"ref_22","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_23","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_24","doi-asserted-by":"crossref","unstructured":"Zhang, X., Boutat, D., and Liu, D. (2023). Applications of Fractional Operator in Image Processing and Stability of Control Systems. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7050359"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Lei, T., Mao, B., Zhou, X., and Fu, H. (2021). Dynamics Analysis and Synchronous Control of Fractional-Order Entanglement Symmetrical Chaotic Systems. Symmetry, 13.","DOI":"10.3390\/sym13111996"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Rahman, Z.-A.S.A., hlJasim, B.H., Al-Yasir, Y.I.A., and Abd-Alhameed, R.A. (2021). High-Security Image Encryption Based on a Novel Simple Fractional-Order Memristive Chaotic System with a Single Unstable Equilibrium Point. Electronics, 10.","DOI":"10.3390\/electronics10243130"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1016\/j.physa.2007.01.010","article-title":"On fractional order differential equations model for nonlocal epidemics","volume":"379","author":"Ahmed","year":"2007","journal-title":"Physica A"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Matouk, A.E., Abdelhameed, T.N., Almutairi, D.K., Abdelkawy, M.A., and Herzallah, M.A.E. (2023). Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems. Mathematics, 11.","DOI":"10.3390\/math11030591"},{"key":"ref_29","first-page":"237","article-title":"A New Homotopy Perturbation Method for Solving Laplace Equation","volume":"8","author":"ELbadri","year":"2013","journal-title":"Adv. Theor. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"130","DOI":"10.4236\/am.2018.92009","article-title":"Comparison between the Homotopy Perturbation Method and Homotopy Perturbation Transform Method","volume":"9","author":"Elbadri","year":"2018","journal-title":"Appl. Math."},{"key":"ref_31","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_32","doi-asserted-by":"crossref","first-page":"1","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_33","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_34","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2018\/3249720","article-title":"New Iterative Method for the Solution of Fractional Damped Burger and Fractional Sharma-Tasso-Olver Equations","volume":"2018","author":"Khan","year":"2018","journal-title":"Complexity"},{"key":"ref_35","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_36","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_37","doi-asserted-by":"crossref","first-page":"1850067","DOI":"10.1142\/S0218127418500670","article-title":"Matlab Code for Lyapunov Exponents of Fractional-Order Systems","volume":"28","author":"Danca","year":"2018","journal-title":"Int. J. Bifurc. Chaos Appl. Sci. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1007\/s00202-016-0444-8","article-title":"Dynamic analysis and chaos suppression in a fractional order brushless DC motor","volume":"99","author":"Rajagopal","year":"2016","journal-title":"Electr. Eng."},{"key":"ref_39","first-page":"1","article-title":"Hyperchaotic Chameleon: Fractional Order FPGA Implementation","volume":"2017","author":"Rajagopal","year":"2017","journal-title":"Complexity"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2020\/9845031","article-title":"Analysis of the Financial Chaotic Model with the Fractional Derivative Operator","volume":"2020","author":"Diouf","year":"2020","journal-title":"Complexity"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1305","DOI":"10.1016\/j.chaos.2006.07.051","article-title":"Nonlinear dynamics and chaos in a fractional-order financial system","volume":"36","author":"Chen","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"7469","DOI":"10.1007\/s00500-019-04373-w","article-title":"An exponential jerk system, its fractional-order form with dynamical analysis and engineering application","volume":"24","author":"Rajagopal","year":"2019","journal-title":"Soft Comput."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"101275","DOI":"10.1016\/j.jksus.2020.101275","article-title":"Analysis of a fractional-order chaotic system in the context of the Caputo fractional derivative via bifurcation and Lyapunov exponents","volume":"33","author":"Sene","year":"2021","journal-title":"J. King Saud Univ. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Owolabi, K.M., G\u00f3mez-Aguilar, J.F., Fern\u00e1ndez-Anaya, G., Lav\u00edn-Delgado, J.E., and Hern\u00e1ndez-Castillo, E. (2020). Modelling of Chaotic Processes with Caputo Fractional Order Derivative. Entropy, 22.","DOI":"10.3390\/e22091027"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"91","DOI":"10.31349\/RevMexFis.67.91","article-title":"Mathematical views of the fractional Chua\u2019s electrical circuit described by the Caputo-Liouville derivative","volume":"67","author":"Sene","year":"2021","journal-title":"Rev. Mex. F\u00edsica"},{"key":"ref_46","first-page":"860","article-title":"New approach to a generalized fractional integral","volume":"218","author":"Katugampola","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2022\/4975104","article-title":"Qualitative Analysis of a Hyperchaotic Lorenz-Stenflo Mathematical Model via the Caputo Fractional Operator","volume":"2022","author":"Deressa","year":"2022","journal-title":"J. Funct. Spaces"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1088\/0031-8949\/53\/1\/015","article-title":"Generalized Lorenz equations for acoustic-gravity waves in the atmosphere","volume":"53","author":"Stenflo","year":"1996","journal-title":"Phys. Scr."},{"key":"ref_49","first-page":"1","article-title":"Analysis of a Generalized Lorenz\u2013Stenflo Equation","volume":"2017","author":"Zhang","year":"2017","journal-title":"Complexity"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"014002","DOI":"10.1088\/1402-4896\/aca1e8","article-title":"From chaos to encryption using fractional order Lorenz-Stenflo model with flux-controlled feedback memristor","volume":"98","author":"Khan","year":"2022","journal-title":"Phys. Scr."},{"key":"ref_51","unstructured":"Redhwan, S.S., Shaikh, S.L., and Abdo, M.S. (2020). Theory of Nonlinear Caputo-Katugampola Fractional Differential Equations. arXiv."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.apnum.2020.04.015","article-title":"Numerical simulation of initial value problems with generalized Caputo-type fractional derivatives","volume":"156","author":"Odibat","year":"2020","journal-title":"Appl. Numer. Math."},{"key":"ref_53","first-page":"88","article-title":"A modified Laplace transform for certain generalized fractional operators","volume":"1","author":"Jarad","year":"2018","journal-title":"Results Nonlinear Anal."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1023\/A:1016592219341","article-title":"A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations","volume":"29","author":"Diethelm","year":"2002","journal-title":"Nonlinear Dyn."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1907","DOI":"10.1108\/EC-09-2017-0347","article-title":"Quasilinearized Scale-3 Haar wavelets-based algorithm for numerical simulation of fractional dynamical systems","volume":"35","author":"Mittal","year":"2018","journal-title":"Eng. Comput."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"081001","DOI":"10.1115\/1.4043565","article-title":"A numerical algorithm to capture spin patterns of fractional Bloch nuclear magnetic resonance flow models","volume":"14","author":"Mittal","year":"2019","journal-title":"J. Comput. Nonlinear Dyn."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Elbadri, M., Abdoon, M.A., Almutairi, D.K., Almutairi, D.M., and Berir, M. (2024). Numerical Simulation and Solutions for the Fractional Chen System via Newly Proposed Methods. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8120709"},{"key":"ref_58","doi-asserted-by":"crossref","unstructured":"Abdoon, M.A., Elgezouli, D.E., Halouani, B., Abdelaty, A.M.Y., Elshazly, I.S., Ailawalia, P., and El-Qadeem, A.H. (2024). Novel Dynamic Behaviors in Fractional Chaotic Systems: Numerical Simulations with Caputo Derivatives. Axioms, 13.","DOI":"10.3390\/axioms13110791"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/20\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:57:14Z","timestamp":1760115434000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/20"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,30]]},"references-count":58,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010020"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010020","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,12,30]]}}}