{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,21]],"date-time":"2026-02-21T07:31:21Z","timestamp":1771659081309,"version":"3.50.1"},"reference-count":28,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,13]],"date-time":"2023-08-13T00:00:00Z","timestamp":1691884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deputyship for Research &amp; Innovation, Ministry of Education in Saudi Arabia","award":["IFP-2022-31"],"award-info":[{"award-number":["IFP-2022-31"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, complex dynamics are found in a fractional-order multi-scroll chaotic system based on the extended Gamma function. Firstly, the extended left and right Caputo fractional differential operators are introduced. Then, the basic features of the extended left Caputo fractional differential operator are outlined. The proposed operator is shown to have a new fractional parameter (higher degree of freedom) that increases the system\u2019s ability to display more varieties of complex dynamics than the corresponding case of the Caputo fractional differential operator. Numerical results are performed to show the effectiveness of the proposed fractional operators. Then, rich complex dynamics are obtained such as coexisting one-scroll chaotic attractors, coexisting two-scroll chaotic attractors, or approximate periodic cycles, which are shown to persist in a shorter range as compared with the corresponding states of the integer-order counterpart of the multi-scroll system. The bifurcation diagrams, basin sets of attractions, and Lyapunov spectra are used to confirm the existence of the various scenarios of complex dynamics in the proposed systems.<\/jats:p>","DOI":"10.3390\/sym15081582","type":"journal-article","created":{"date-parts":[[2023,8,14]],"date-time":"2023-08-14T10:20:14Z","timestamp":1692008414000},"page":"1582","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Symmetry in a Fractional-Order Multi-Scroll Chaotic System Using the Extended Caputo Operator"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5834-4234","authenticated-orcid":false,"given":"A. E.","family":"Matouk","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia"},{"name":"College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1275-2384","authenticated-orcid":false,"given":"D. K.","family":"Almutairi","sequence":"additional","affiliation":[{"name":"College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M. A. E.","family":"Herzallah","sequence":"additional","affiliation":[{"name":"Faculty of Science, Zagazig University, Zagazig 44519, Egypt"},{"name":"Faculty of Science and Humanities, Shaqra University, Al-Dawadmi 11911, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M. A.","family":"Abdelkawy","sequence":"additional","affiliation":[{"name":"Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt"},{"name":"Department of Mathematics and Statistics, Faculty of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"T. N.","family":"Abdelhameed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Majmaah 11952, Saudi Arabia"},{"name":"College of Engineering, Majmaah University, Al-Majmaah 11952, Saudi Arabia"},{"name":"Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef 62514, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1016\/j.physa.2007.01.010","article-title":"On fractional order differential equations model for non-local epidemics","volume":"379","author":"Ahmed","year":"2007","journal-title":"Phys. A"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Abbas, S., Nazar, M., Nisa, Z.U., Amjad, M., El Din, S.M., and Alanzi, A.M. (2022). Heat and Mass Transfer Analysis of MHD Jeffrey Fluid over a Vertical Plate with CPC Fractional Derivative. Symmetry, 14.","DOI":"10.3390\/sym14122491"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Kumar, A., Alzaid, S.S., Alkahtani, B.S.T., and Kumar, S. (2022). Complex Dynamic Behaviour of Food Web Model with Generalized Fractional Operator. Mathematics, 10.","DOI":"10.3390\/math10101702"},{"key":"ref_4","first-page":"269","article-title":"Fractional thermoelasticity model of a 2D problem of mode-I crack in a fibre-reinforced thermal environment","volume":"5","author":"Zenkour","year":"2019","journal-title":"J. Appl. Comput. Mech."},{"key":"ref_5","first-page":"1676","article-title":"A Modified Thermoelastic Fractional Heat Conduction Model with A Single-Lag and Two Different Fractional-Orders","volume":"7","author":"Abouelregal","year":"2021","journal-title":"J. Appl. Comput. Mech."},{"key":"ref_6","unstructured":"Matignon, D. (1996, January 9\u201312). Stability results for fractional differential equations with applications to control processing. Proceedings of the Computational Engineering in Systems and Application Multi-Conference, IMACS, IEEE-SMC Proceedings, Lille, France."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.jare.2022.03.008","article-title":"Chaotic attractors that exist only in fractional-order case","volume":"45","author":"Matouk","year":"2023","journal-title":"J. Adv. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"e18645","DOI":"10.1016\/j.heliyon.2023.e18645","article-title":"Applications of the generalized gamma function to a fractional-order biological system","volume":"9","author":"Matouk","year":"2023","journal-title":"Heliyon"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"482","DOI":"10.1016\/S0378-4371(00)00387-3","article-title":"Fractional market dynamics","volume":"287","author":"Laskin","year":"2000","journal-title":"Phys. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","article-title":"Linear Models of Dissipation whose Q is almost Frequency Independent\u2014II","volume":"13","author":"Caputo","year":"1967","journal-title":"Geophys. J. R. Astron. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic nonperiodic flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"J. Atmos. Sci."},{"key":"ref_12","first-page":"397","article-title":"An equation for continuous chaos","volume":"57","year":"1976","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1109\/31.55064","article-title":"Canonical realization of Chua's circuit family","volume":"37","author":"Chua","year":"1990","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1142\/S0218127499001024","article-title":"Yet another chaotic attractor","volume":"9","author":"Chen","year":"1999","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1016\/j.chaos.2004.02.060","article-title":"A new chaotic attractor","volume":"22","author":"Liu","year":"2004","journal-title":"Chaos Solitons Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"034101","DOI":"10.1103\/PhysRevLett.91.034101","article-title":"Chaotic Dynamics of the Fractional Lorenz System","volume":"91","author":"Grigorenko","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1109\/81.404062","article-title":"Chaos in a fractional order Chua\u2019s system","volume":"42","author":"Hartley","year":"1995","journal-title":"IEEE Trans. Circuits Syst. I"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1016\/j.chaos.2004.02.013","article-title":"Chaos in Chen\u2019s system with a fractional order","volume":"22","author":"Li","year":"2004","journal-title":"Chaos Solitons Fractals"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.physa.2004.04.113","article-title":"Chaos and hyperchaos in the fractional-order R\u00f6ssler equations","volume":"341","author":"Li","year":"2004","journal-title":"Phys. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"945","DOI":"10.1016\/j.automatica.2010.02.023","article-title":"A note on fractional-order derivatives of periodic functions","volume":"46","author":"Tavazoei","year":"2010","journal-title":"Automatica"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1507","DOI":"10.1142\/S021812740401014X","article-title":"A new chaotic system and beyond: The generalized lorenz-like system","volume":"14","author":"Chen","year":"2004","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_22","unstructured":"Vanecek, A., and Celikovsky, S. (1996). Control Systems: From Linear Analysis to Synthesis of Chaos, Prentice-Hall."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/0375-9601(83)90209-8","article-title":"At least one Lyapunov exponent vanishes if the trajectory of an attractor does not contain a fixed point","volume":"94","author":"Haken","year":"1983","journal-title":"Phys. Lett. A"},{"key":"ref_24","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":"Phys. D Nonlinear Phenom."},{"key":"ref_25","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"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1080\/03610929608831688","article-title":"A generalized gamma distribution and its application in reliability","volume":"25","author":"Agarwal","year":"1996","journal-title":"Commun. Statist. Theory Meth."},{"key":"ref_27","first-page":"100458","article-title":"Rich complex dynamics in new fractional-order hyperchaotic systems using a modified Caputo operator based on the extended Gamma function","volume":"6","author":"Matouk","year":"2022","journal-title":"Part. Diff. Equat. Appl. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"112946","DOI":"10.1016\/j.chaos.2022.112946","article-title":"Chaotic dynamics in some fractional predator\u2013prey models via a new Caputo operator based on the generalised Gamma function","volume":"166","author":"Matouk","year":"2023","journal-title":"Chaos Solitons Fractals"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1582\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:32:47Z","timestamp":1760128367000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/8\/1582"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,13]]},"references-count":28,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["sym15081582"],"URL":"https:\/\/doi.org\/10.3390\/sym15081582","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,13]]}}}