{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:56:27Z","timestamp":1740149787939,"version":"3.37.3"},"reference-count":46,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,2,23]],"date-time":"2022-02-23T00:00:00Z","timestamp":1645574400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11702194;11702195"],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"In this paper, a novel fractional-order discrete map with a sinusoidal function possessing typical nonlinear features, including chaos and bifurcations, is proposed. Firstly, the basic properties involving the stability of the equilibrium points and the symmetry of the map are studied by theoretical analysis. Secondly, the dynamics of the map in commensurate-order and incommensurate-order cases with initial conditions belonging to different basins of attraction is investigated by numerical simulations. The bifurcation types and influential parameters of the map are analyzed via nonlinear tools. Hopf, period-doubling, and symmetry-breaking bifurcations are observed when a parameter or an order is varied. Bifurcation diagrams and maximum Lyapunov exponent spectrums, with both a variation in a system parameter and an order or two orders, are shown in a three-dimensional space. A comparison of the bifurcations in fractional-order and integral-order cases shows that the variation in an order has no effect on the symmetry-breaking bifurcation point. Finally, the heterogeneous hybrid synchronization of the map is realized by designing suitable controllers. It is worth noting that the increase in a derivative order can promote the synchronization speed for the fractional-order discrete map.<\/jats:p>","DOI":"10.3390\/e24030320","type":"journal-article","created":{"date-parts":[[2022,2,23]],"date-time":"2022-02-23T14:34:38Z","timestamp":1645626878000},"page":"320","source":"Crossref","is-referenced-by-count":5,"title":["A Fractional-Order Sinusoidal Discrete Map"],"prefix":"10.3390","volume":"24","author":[{"given":"Xiaojun","family":"Liu","sequence":"first","affiliation":[{"name":"School of Sciences, Xi\u2019an University of Posts and Telecommunications, Xi\u2019an 710061, China"}]},{"given":"Dafeng","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Automation, Xi\u2019an University of Posts and Telecommunications, Xi\u2019an 710061, China"}]},{"given":"Ling","family":"Hong","sequence":"additional","affiliation":[{"name":"State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1038\/261459a0","article-title":"Simple mathematical models with very complicated dynamics","volume":"261","author":"May","year":"1976","journal-title":"Nature"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1007\/BF01608556","article-title":"A two-dimensional mapping with a strange attractor","volume":"50","year":"1976","journal-title":"Commun. Math. Phys."},{"key":"ref_3","first-page":"9","article-title":"Un atracteur \u00e9trange du type attracteur de H\u00e9non","volume":"39","author":"Lozi","year":"1978","journal-title":"J. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0167-2789(85)90092-2","article-title":"An exploration of the H\u00e9non quadratic map","volume":"14","author":"Hitzl","year":"1985","journal-title":"Phys. D"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"R2712","DOI":"10.1103\/PhysRevE.51.R2712","article-title":"Design of hyperchaotic flows","volume":"51","author":"Baier","year":"1995","journal-title":"Phys. Rev. E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1650222","DOI":"10.1142\/S0218127416502229","article-title":"Dynamical Analysis and Circuit Simulation of a New Fractional-Order Hyperchaotic System and Its Discretization","volume":"26","author":"Elsonbaty","year":"2016","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"685","DOI":"10.1016\/j.chaos.2005.04.037","article-title":"A note on the fractional-order Chen system","volume":"27","author":"Lu","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1007\/s11071-010-9904-2","article-title":"Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci","volume":"65","author":"Zeng","year":"2010","journal-title":"Nonlinear Dyn."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1016\/j.isatra.2018.07.014","article-title":"Synchronization of different fractional order chaotic systems with time-varying parameter and orders","volume":"80","author":"Behinfaraz","year":"2018","journal-title":"ISA Trans."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.nonrwa.2015.05.014","article-title":"The effect of vaccines on backward bifurcation in a fractional order HIV model","volume":"26","author":"Huo","year":"2015","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_11","unstructured":"Miller, K.S., and Ross, B. (1989). Univalent Functions, Fractional Calculus and Their Applications, Ellis Howard."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"435101","DOI":"10.1088\/1751-8113\/41\/43\/435101","article-title":"Fractional equations of kicked systems and discrete maps","volume":"41","author":"Tarasov","year":"2008","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"4573","DOI":"10.1016\/j.cnsns.2011.02.007","article-title":"Fractional standard map: Riemann-Liouville vs. Caputo","volume":"16","author":"Edelman","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"106070","DOI":"10.1016\/j.aml.2019.106070","article-title":"Stability criteria about discrete fractional maps","volume":"101","author":"Wang","year":"2020","journal-title":"Appl. Math. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14232\/ejqtde.2009.4.3","article-title":"Discrete fractional calculus with the nabla operator","volume":"4","author":"Atici","year":"2009","journal-title":"Electron. J. Qual. Theory Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Goodrich, C., and Peterson, A.C. (2015). Discrete Fractional Calculus, Springer.","DOI":"10.1007\/978-3-319-25562-0"},{"key":"ref_17","first-page":"1","article-title":"Fractional Sums and Differences with Binomial Coefficients","volume":"2013","author":"Abdeljawad","year":"2013","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1016\/j.cnsns.2017.01.002","article-title":"Stability analysis of Caputo\u2013like discrete fractional systems","volume":"48","author":"Baleanu","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.cnsns.2018.04.019","article-title":"A new collection of real world applications of fractional calculus in science and engineering","volume":"64","author":"Sun","year":"2018","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_20","first-page":"779","article-title":"Necessary and sufficient conditions for stability of fractional discrete-time linear state-space systems","volume":"61","author":"Ruszewski","year":"2013","journal-title":"Bull. Pol. Acad. Sci. Technol. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2243","DOI":"10.4236\/am.2014.515218","article-title":"Discrete Chaos in Fractional Henon Map","volume":"05","author":"Hu","year":"2014","journal-title":"Appl. Math."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.sigpro.2014.02.022","article-title":"Chaos synchronization of the discrete fractional logistic map","volume":"102","author":"Wu","year":"2014","journal-title":"Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1945","DOI":"10.1140\/epjs\/s11734-021-00133-w","article-title":"Dynamic analysis of a new two-dimensional map in three forms: Integer-order, fractional-order and improper fractional-order","volume":"230","author":"Ma","year":"2021","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s12648-015-0742-4","article-title":"Chaotic synchronization between linearly coupled discrete fractional H\u00e9non maps","volume":"90","author":"Liu","year":"2015","journal-title":"Indian J. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1519","DOI":"10.1007\/s11071-017-3743-3","article-title":"A new contribution for the impulsive synchronization of fractional-order discrete-time chaotic systems","volume":"90","author":"Megherbi","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2150064","DOI":"10.1142\/S0218348X2150064X","article-title":"Image encryption technology based on fractional two-dimensional discrete chaotic map accompanied with menezes-vanstone elliptic curve cryptosystem","volume":"29","author":"Liu","year":"2021","journal-title":"Fractals"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2092","DOI":"10.1177\/1077546315574649","article-title":"Image encryption technique based on fractional chaotic time series","volume":"22","author":"Wu","year":"2016","journal-title":"J. Vib. Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"7242791","DOI":"10.1155\/2019\/7242791","article-title":"Secure Communication of Fractional Complex Chaotic Systems Based on Fractional Difference Function Synchronization","volume":"2019","author":"Liu","year":"2019","journal-title":"Complexity"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1697","DOI":"10.1007\/s11071-014-1250-3","article-title":"Discrete chaos in fractional delayed logistic maps","volume":"80","author":"Wu","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_30","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":"2021","journal-title":"Alex. Eng. J."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"104106","DOI":"10.1016\/j.rinp.2021.104106","article-title":"Chaos in the discrete memristor-based system with fractional-order difference","volume":"24","author":"Peng","year":"2021","journal-title":"Results Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2021\/6768215","article-title":"An Unprecedented 2-Dimensional Discrete-Time Fractional-Order System and Its Hidden Chaotic Attractors","volume":"2021","author":"Khennaoui","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Almatroud, A.O., Khennaoui, A.-A., Ouannas, A., and Pham, V.-T. (2021). Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting. Int. J. Nonlinear Sci. Numer. Simul.","DOI":"10.1515\/ijnsns-2020-0180"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"083131","DOI":"10.1063\/5.0005059","article-title":"The discrete fractional duffing system: Chaos, 0\u20131 test, C0 complexity, entropy, and control","volume":"30","author":"Ouannas","year":"2020","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"126100","DOI":"10.1016\/j.physa.2021.126100","article-title":"On chaos and projective synchronization of a fractional difference map with no equilibria using a fuzzy-based state feedback control","volume":"578","author":"Bekiros","year":"2021","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3913","DOI":"10.1140\/epjs\/s11734-021-00331-6","article-title":"A new fractional-order 2D discrete chaotic map and its DSP implement","volume":"230","author":"Han","year":"2021","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"240503","DOI":"10.7498\/aps.62.240503","article-title":"Bifurcation and attractor of two-dimensional sinusoidal discrete map","volume":"62","author":"Chuang","year":"2013","journal-title":"Acta Phys. Sin."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Bi, C., Zhang, Q., Xiang, Y., and Wang, J. (2013, January 15\u201317). Nonlinear dynamics of two-dimensional sinusoidal discrete map. Proceedings of the 2013 International Conference on Communications, Circuits and Systems (ICCCAS), Chengdu, China.","DOI":"10.1109\/ICCCAS.2013.6765377"},{"key":"ref_39","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_40","doi-asserted-by":"crossref","first-page":"513","DOI":"10.1090\/S0025-5718-1988-0929549-2","article-title":"On a new definition of the fractional difference","volume":"50","author":"Gray","year":"1988","journal-title":"Math. Compu."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.chaos.2019.04.002","article-title":"On the dynamics, control and synchronization of fractional-order Ikeda map","volume":"123","author":"Ouannas","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"651","DOI":"10.1515\/fca-2015-0040","article-title":"On explicit stability conditions for a linear fractional difference system","volume":"18","year":"2015","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1299","DOI":"10.1016\/j.na.2007.06.030","article-title":"Limitations of frequency domain approximation for detecting chaos in fractional order systems","volume":"69","author":"Tavazoei","year":"2008","journal-title":"Nonlinear Anal. Theory Methods Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1887","DOI":"10.1140\/epjs\/s11734-021-00136-7","article-title":"Symmetry-breaking, amplitude control and constant Lyapunov exponent based on single parameter snap flows","volume":"230","author":"Leutcho","year":"2021","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_45","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_46","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1016\/j.chaos.2018.12.019","article-title":"On fractional\u2013order discrete\u2013time systems: Chaos, stabilization and synchronization","volume":"119","author":"Khennaoui","year":"2019","journal-title":"Chaos Solitons Fractals"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/320\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,26]],"date-time":"2024-07-26T14:46:31Z","timestamp":1722005191000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/320"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,23]]},"references-count":46,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["e24030320"],"URL":"https:\/\/doi.org\/10.3390\/e24030320","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,2,23]]}}}