{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:23:16Z","timestamp":1760242996090,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2015,4,27]],"date-time":"2015-04-27T00:00:00Z","timestamp":1430092800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A projective synchronization scheme for a kind of n-dimensional discrete dynamical system is proposed by means of a linear feedback control technique. The scheme consists of master and slave discrete dynamical systems coupled by linear state error variables. A kind of novel 3-D chaotic discrete system is constructed, to which the test for chaos is applied. By using the stability principles of an upper or lower triangular matrix, two controllers for achieving projective synchronization are designed and illustrated with the novel systems. Lastly some numerical simulations are employed to validate the effectiveness of the proposed projective synchronization scheme.<\/jats:p>","DOI":"10.3390\/e17052677","type":"journal-article","created":{"date-parts":[[2015,4,27]],"date-time":"2015-04-27T12:16:51Z","timestamp":1430137011000},"page":"2677-2687","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Projective Synchronization of Chaotic Discrete Dynamical Systems via Linear State Error Feedback Control"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7739-4808","authenticated-orcid":false,"given":"Baogui","family":"Xin","sequence":"first","affiliation":[{"name":"Nonlinear Science Center, School of Economics and Management, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiheng","family":"Wu","sequence":"additional","affiliation":[{"name":"Nonlinear Science Center, School of Economics and Management, Shandong University of Science and Technology, Qingdao 266590, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,4,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1098\/rsta.2009.0230","article-title":"On chaos synchronization and secure communication","volume":"368","author":"Kinzel","year":"2010","journal-title":"Phil. Trans. R. Soc. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4479","DOI":"10.1016\/j.cnsns.2011.01.021","article-title":"Projective synchronization of chaotic fractional-order energy resources demand\u2013supply systems via linear control","volume":"16","author":"Xin","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Xin, B., Chen, T., and Liu, Y. (2010). Synchronization of chaotic fractional-order WINDMI systems via linear state error feedback control. Math. Probl. Eng, 2010.","DOI":"10.1155\/2010\/859685"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4334","DOI":"10.3390\/e15104334","article-title":"Synchronization of a novel hyperchaotic complex-variable system based on finite-time stability theory","volume":"15","author":"Zhou","year":"2013","journal-title":"Entropy"},{"key":"ref_5","first-page":"29","article-title":"Finite-time synchronization of chaotic complex networks with stochastic disturbance","volume":"17","author":"Li","year":"2015","journal-title":"Entropy"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3265","DOI":"10.3390\/e15083355","article-title":"Synchronization of a class of fractional-order chaotic neural networks","volume":"15","author":"Chen","year":"2013","journal-title":"Entropy"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1143\/PTP.69.32","article-title":"Stability theory of synchronized motion in coupled-oscillator systems","volume":"69","author":"Fujisaka","year":"1983","journal-title":"Prog. Theor. Phys"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1103\/PhysRevLett.64.821","article-title":"Synchronization in chaotic systems","volume":"64","author":"Pecora","year":"1990","journal-title":"Phys. Rev. Lett"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3042","DOI":"10.1103\/PhysRevLett.82.3042","article-title":"Projective synchronization in three-dimensional chaotic systems","volume":"82","author":"Mainieri","year":"1999","journal-title":"Phys. Rev. Lett"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"420","DOI":"10.1016\/j.physleta.2004.10.072","article-title":"Observer-based control for full-state projective synchronization of a general class of chaotic maps in any dimension","volume":"333","author":"Wen","year":"2004","journal-title":"Phys. Lett. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/j.chaos.2004.09.117","article-title":"Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems","volume":"26","author":"Wen","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"954","DOI":"10.1016\/j.jsv.2009.02.049","article-title":"Line spectra reduction and vibration isolation via modified projective synchronization for acoustic stealth of submarines","volume":"324","author":"Wen","year":"2009","journal-title":"J. Sound Vib"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1504\/IJVD.2012.045926","article-title":"Vibration control for active seat suspension system based on projective chaos synchronisation","volume":"58","author":"Wen","year":"2012","journal-title":"Int. J. Veh. Des"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s11071-010-9810-7","article-title":"Designing Hopf limit circle to dynamical systems via modified projective synchronization","volume":"63","author":"Wen","year":"2011","journal-title":"Nonlinear Dyn"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3167","DOI":"10.1016\/j.cnsns.2013.04.007","article-title":"Designing torus-doubling solutions to discrete time systems by hybrid projective synchronization","volume":"18","author":"Xie","year":"2013","journal-title":"Commun. Nonlinear Sci. Numer. Simul"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1088\/0253-6102\/51\/2\/17","article-title":"Adaptive control and function projective synchronization in 2D discrete-time chaotic systems","volume":"51","author":"Yin","year":"2009","journal-title":"Commun. Theor. Phys"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1029","DOI":"10.1016\/j.chaos.2005.08.123","article-title":"Adaptive fuzzy synchronization of discrete-time chaotic systems","volume":"28","author":"Vasegh","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1007\/s11071-009-9559-z","article-title":"Reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems","volume":"59","author":"Zhang","year":"2010","journal-title":"Nonlinear Dyn"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"905","DOI":"10.1007\/s11071-013-1009-2","article-title":"The synchronization between two discrete-time chaotic systems using active robust model predictive control","volume":"74","author":"Zhang","year":"2013","journal-title":"Nonlinear Dyn"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1142\/S0218127410025429","article-title":"Synchronization of chaotic fractional-order systems via linear control","volume":"20","author":"Odibat","year":"2010","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1745","DOI":"10.1007\/s11071-011-0102-7","article-title":"Analysis and control of a hyperchaotic system with only one nonlinear term","volume":"67","author":"Chen","year":"2012","journal-title":"Nonlinear Dyn"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1549","DOI":"10.1007\/s11071-012-0555-3","article-title":"Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control","volume":"70","author":"Chen","year":"2012","journal-title":"Nonlinear Dyn"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Xin, B., and Chen, T. (2012). Projective synchronization of N-dimensional chaotic fractional-order systems via linear state error feedback control. Discrete Dyn. Nat. Soc, 2012.","DOI":"10.1155\/2012\/191063"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1137\/080718851","article-title":"On the implementation of the 0-1 test for chaos","volume":"8","author":"Gottwald","year":"2009","journal-title":"SIAM J. Appl. Dyn. Sys"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1088\/0951-7715\/22\/6\/006","article-title":"On the validity of the 0-1 test for chaos","volume":"22","author":"Gottwald","year":"2009","journal-title":"Nonlinearity"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Xin, B., and Li, Y. (2013). Bifurcation and Chaos in a Price Game of Irrigation Water in a Coastal Irrigation District. Discrete Dyn. Nat. Soc, 2013.","DOI":"10.1155\/2013\/408904"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1495","DOI":"10.1007\/s11071-013-0880-1","article-title":"Application of Takagi\u2013Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization","volume":"73","author":"Chen","year":"2013","journal-title":"Nonlinear Dyn"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/5\/2677\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:45:24Z","timestamp":1760215524000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/5\/2677"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,4,27]]},"references-count":27,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2015,5]]}},"alternative-id":["e17052677"],"URL":"https:\/\/doi.org\/10.3390\/e17052677","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2015,4,27]]}}}