{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:37:49Z","timestamp":1760240269981,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,19]],"date-time":"2019-04-19T00:00:00Z","timestamp":1555632000000},"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>This paper investigates the asymptotic tracking control problem of the chaotic system. Firstly, a reference system is presented, the output of which can asymptotically track a given command. Then, a both physically implementable and simple controller is designed, by which the given chaotic system synchronizes the reference system, and thus the output of such chaotic systems can asymptotically track the given command. It should be pointed out that the output of the given chaotic system can asymptotically track arbitrary desired periodic orbits. Finally, several illustrative examples are taken as example to show the validity and effectiveness of the obtained results.<\/jats:p>","DOI":"10.3390\/sym11040568","type":"journal-article","created":{"date-parts":[[2019,4,22]],"date-time":"2019-04-22T03:15:53Z","timestamp":1555902953000},"page":"568","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["Tracking Control of a Class of Chaotic Systems"],"prefix":"10.3390","volume":"11","author":[{"given":"Anqing","family":"Yang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Science), Jinan 250353, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Linshan","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Science), Jinan 250353, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zuoxun","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering and Automation, Qilu University of Technology (Shandong Academy of Science), Jinan 250353, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2393-0767","authenticated-orcid":false,"given":"Rongwei","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Science), Jinan 250353, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1196","DOI":"10.1103\/PhysRevLett.64.1196","article-title":"Controlling Chaos","volume":"64","author":"Ott","year":"1990","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"1804","DOI":"10.1103\/PhysRevLett.76.1804","article-title":"Phase synchronization of chaotic oscillators","volume":"76","author":"Rosenblum","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4193","DOI":"10.1103\/PhysRevLett.78.4193","article-title":"From phase to lag synchronization in coupled chaotic oscillators","volume":"78","author":"Rosenblum","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"455","DOI":"10.1016\/j.physleta.2005.04.002","article-title":"Adaptive control for anti-synchronization of Chua\u2019s chaotic system","volume":"339","author":"Hu","year":"2005","journal-title":"Phys. Lett. A"},{"key":"ref_6","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_7","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1515\/acsc-2016-0004","article-title":"Coexistence of synchronization and anti-synchronization in chaotic systems","volume":"26","author":"Ren","year":"2016","journal-title":"Arch. Control Sci."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"040205","DOI":"10.1088\/0256-307X\/28\/4\/040205","article-title":"Simultaneous synchronization and anti-synchronization of two identical new 4D chaotic systems","volume":"28","author":"Guo","year":"2011","journal-title":"Chin. Phys. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5593","DOI":"10.1016\/j.physleta.2008.07.016","article-title":"A simple adaptive controller for chaos and hyperchaos synchronization","volume":"372","author":"Guo","year":"2008","journal-title":"Phys. Lett. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1488","DOI":"10.1016\/j.physleta.2010.01.048","article-title":"Stabilization of three-dimensional chaotic systems via single state feedback controller","volume":"374","author":"Yu","year":"2010","journal-title":"Phys. Lett. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s11071-008-9452-1","article-title":"Anti-synchronization of liu system and Lorenz system with known and unknown parameters","volume":"57","author":"Wang","year":"2009","journal-title":"Nonlinear Dyn."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1569","DOI":"10.1007\/s11071-015-2261-4","article-title":"Projective lag synchronization in drive-response dynamical networks with delay coupling via hybrid feedback control","volume":"82","author":"Bakar","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/s11071-016-2690-8","article-title":"A new robust adaptive control method for modified function projective synchronization with unknown bounded parametric uncertainties and external disturbances","volume":"85","author":"Du","year":"2016","journal-title":"Nonlinear Dyn."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1007\/s11071-016-3056-y","article-title":"Control problems of Chen-Lee system by adaptive control method","volume":"87","author":"Liu","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s11071-017-3645-4","article-title":"Projective synchronization of a class of chaotic systems by dynamic feedback control method","volume":"90","author":"Guo","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1109\/9.847098","article-title":"Deadzone compensation in motion control systems using neural networks","volume":"45","author":"Selmic","year":"2000","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1269","DOI":"10.1016\/S0005-1098(01)00066-8","article-title":"Neural net backlash compensation with Hebbian tuning using dynamic inversion","volume":"37","author":"Selmic","year":"2001","journal-title":"Automatica"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1109\/TAC.2006.890473","article-title":"Robust adaptive output control of uncertain nonlinear plants with unknown backlash nonlinearity","volume":"52","author":"Zhou","year":"2007","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"914","DOI":"10.1088\/1009-1963\/14\/5\/010","article-title":"A novel sliding mode nonlinear proportional-integral control scheme for controlling chaos","volume":"14","author":"Yu","year":"2005","journal-title":"Chin. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"306","DOI":"10.1088\/1009-1963\/15\/2\/012","article-title":"A simple asymptotic trajectory control of full states of a unified chaotic system","volume":"15","author":"Yu","year":"2006","journal-title":"Chin. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"475","DOI":"10.2478\/AMNS.2018.2.00037","article-title":"Intermittent transition to chaos in vibroimpact system","volume":"3","author":"Bazhenov","year":"2018","journal-title":"Appl. Math. Nonlinear Sci."},{"key":"ref_22","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_23","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1016\/j.chaos.2006.09.067","article-title":"Parametric study of the fractional order Chen-Lee system","volume":"37","author":"Tam","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1671","DOI":"10.1016\/S0960-0779(04)00431-X","article-title":"On a four-dimensional chaotic system","volume":"23","author":"Qi","year":"2005","journal-title":"Chaos Solitons Fractals"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/568\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:46:45Z","timestamp":1760186805000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/568"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,19]]},"references-count":24,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,4]]}},"alternative-id":["sym11040568"],"URL":"https:\/\/doi.org\/10.3390\/sym11040568","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,4,19]]}}}