{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:41:09Z","timestamp":1760211669459,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2016,6,22]],"date-time":"2016-06-22T00:00:00Z","timestamp":1466553600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Systems"],"abstract":"<jats:p>Chaotic dynamics are an interesting topic in nonlinear science that has been intensively studied during the last three decades due to its wide availability. Motivated by much researches on synchronization, the authors of this study have improved the time response of stabilization when parametrically excited \u03a66\u2014Van der Pol Oscillator (VDPO) and \u03a66\u2014Duffing Oscillator (DO) are synchronized identically as well as non-identically (with each other) using the Linear Active Control (LAC) technique using Mathematica. Furthermore, the authors have synchronized the same pairs of the oscillators using a more robust synchronization with faster time response of stability called Robust Adaptive Sliding Mode Control (RASMC). A comparative study has been done between the previous results of Njah\u2019s work and our results based on Mathematica via LAC. The time response of stabilization of synchronization using RASMC has been discussed.<\/jats:p>","DOI":"10.3390\/systems4020025","type":"journal-article","created":{"date-parts":[[2016,6,23]],"date-time":"2016-06-23T18:29:50Z","timestamp":1466706590000},"page":"25","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Improved Time Response of Stabilization in Synchronization of Chaotic Oscillators Using Mathematica"],"prefix":"10.3390","volume":"4","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2632-990X","authenticated-orcid":false,"given":"Mohammad","family":"Shahzad","sequence":"first","affiliation":[{"name":"College of Applied Sciences, Ministry of Higher Education, Nizwa 611, Sultanate of Oman"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Israr","family":"Ahmad","sequence":"additional","affiliation":[{"name":"College of Applied Sciences, Ministry of Higher Education, Nizwa 611, Sultanate of Oman"},{"name":"School of Quantitative Sciences, College of Arts &amp; Sciences, UUM, Keddha 06010, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Azizan","family":"Saaban","sequence":"additional","affiliation":[{"name":"School of Quantitative Sciences, College of Arts &amp; Sciences, UUM, Keddha 06010, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adyda","family":"Ibrahim","sequence":"additional","affiliation":[{"name":"School of Quantitative Sciences, College of Arts &amp; Sciences, UUM, Keddha 06010, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,6,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1637","DOI":"10.1140\/epjst\/e2015-02485-8","article-title":"Synchronization and circuit design of a chaotic system with coexisting hidden attractors","volume":"224","author":"Shahzad","year":"2015","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_2","first-page":"176","article-title":"Chaos Synchronization of a New Hyperchaotic Finance System Via a Novel Chatter Free Sliding Mode Control Strategy","volume":"17","author":"Yao","year":"2014","journal-title":"Int. J. Nonlinear Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"021011","DOI":"10.1115\/1.4031087","article-title":"A Novel Global Sliding Mode Control Based on Exponential Reaching Law for a Class of under Actuated Systems with External Disturbances","volume":"11","author":"Mobayen","year":"2015","journal-title":"J. Comput. Nonlinear Dyn."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Azar, A.T., and Zhu, Q. (2014). Advances and Applications in Sliding Mode Control Systems, Springer International Publishing.","DOI":"10.1007\/978-3-319-11173-5"},{"key":"ref_5","first-page":"123","article-title":"Synchronization Of Three Dimensional Cancer Model With Rosslar System Using A Robust Adaptive Sliding Mode Controller","volume":"6","author":"Shahzad","year":"2015","journal-title":"Int. J. Math. Arch."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1504\/IJAAC.2015.070957","article-title":"Adaptive control to synchronise and anti-synchronise two identical time delay Bhalekar-Gejji chaotic systems with unknown parameters","volume":"9","author":"Shahzad","year":"2015","journal-title":"Int. J. Autom. Control"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3318","DOI":"10.1016\/j.amc.2009.10.020","article-title":"Optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems","volume":"215","author":"Ma","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1016\/j.chaos.2009.02.028","article-title":"CMAC-based adaptive backstepping synchronization of uncertain chaotic systems","volume":"42","author":"Lin","year":"2009","journal-title":"Chaos Soliton Fract."},{"key":"ref_9","first-page":"1199","article-title":"Synchronization of two Lorenz system using Active control","volume":"64","author":"Bai","year":"1997","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1065","DOI":"10.1016\/j.chaos.2006.10.005","article-title":"Synchronization of Identical and Nonidentical 4-D Chaotic Systems using Active Control","volume":"37","author":"Vincent","year":"2008","journal-title":"Chaos Solitons Fract."},{"key":"ref_11","first-page":"243","article-title":"Synchronization and Anti-synchronization of double hump Duffing-Van der Pol Oscillators via Active Control","volume":"4","author":"Njah","year":"2009","journal-title":"J. Inf. Comput. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1177\/1077546309357024","article-title":"Synchronization via active control of parametrically and externally excited \u03a66\u2014Van der Pol and duffing oscillators and application to secure communications","volume":"17","author":"Njah","year":"2011","journal-title":"J. Vib. Cont."},{"key":"ref_13","first-page":"41","article-title":"Experimental study of synchronization & Anti-synchronization for spin orbit problem of Enceladus","volume":"3","author":"Shahzad","year":"2013","journal-title":"Int. J. of Cont. Sci. Eng."},{"key":"ref_14","first-page":"1235","article-title":"Synchronization of Shilnikov Chaos in CO2 Laser with Feedback","volume":"11","author":"Pisarchik","year":"2014","journal-title":"Laser Phys."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Hammami, S. (2015). State feedback-based secure image cryptosystem hyperchaotic synchronization. ISA Trans.","DOI":"10.1016\/j.isatra.2014.05.027"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1246","DOI":"10.1016\/j.cnsns.2006.12.011","article-title":"On control and synchronization in chaotic and hyperchaotic systems via linear feedback control","volume":"13","author":"Rafikov","year":"2008","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_17","first-page":"51","article-title":"Finite-Time Modified Projective Synchronization of Unknown Rossler and Coullet Systems","volume":"1","author":"Luo","year":"2013","journal-title":"Commun. Control Sci. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1613","DOI":"10.1016\/j.cnsns.2009.06.012","article-title":"Modified projective synchronization of chaotic systems with disturbances via active sliding mode control","volume":"15","author":"Cai","year":"2010","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2602","DOI":"10.1016\/j.cnsns.2011.09.033","article-title":"Robust adaptive modified function projective synchronization of different hyperchaotic systems subject to external disturbance","volume":"17","author":"Fu","year":"2012","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"739","DOI":"10.48084\/etasr.434","article-title":"A Research on the Synchronization of Two Novel Chaotic Systems Based on a Nonlinear Active Control Algorithm","volume":"5","author":"Ahmad","year":"2015","journal-title":"Eng. Tech. Appl. Sci. Res."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.chaos.2014.09.005","article-title":"Synchronization and anti-synchronization of Lu and bhalekar-Gejji chaotic systems using nonlinear active control","volume":"69","author":"Singh","year":"2014","journal-title":"Chaos Solitons Fract."},{"key":"ref_22","first-page":"141","article-title":"On Globally Exponential Stable Complete Synchronization of Nearly Identical Hyperchaotic Systems via Linear Active Control","volume":"30","author":"Ahmad","year":"2015","journal-title":"Ci\u00eanc. T\u00e9c. Vitivin\u00edc. Sci. Technol. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2037","DOI":"10.1007\/s11071-014-1793-3","article-title":"The Improved Results with Mathematica and Effects of External Uncertainty & Disturbances on Synchronization using a Robust Adaptive Sliding Mode Controller: A Comparative Study","volume":"79","author":"Shahzad","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2853","DOI":"10.1016\/j.cnsns.2010.09.038","article-title":"Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller","volume":"16","author":"Pourmahmood","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_25","first-page":"69","article-title":"Global Chaos Synchronization for WINDMI and Coullet Chaotic Systems Using Active Control","volume":"3","author":"Vaidyanathan","year":"2013","journal-title":"J. Cont. Eng. Tech."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ahmad, I., Saaban, A.B., Ibrahim, A.B., and Shahzad, M. (2016). A Research on Active Control to Synchronize a New 3D Chaotic System. Systems, 4.","DOI":"10.3390\/systems4010002"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"4859","DOI":"10.1016\/j.ijleo.2015.12.134","article-title":"The Synchronization of Chaotic Systems with Different Dimensions by a Robust Generalized Active Control","volume":"127","author":"Ahmad","year":"2016","journal-title":"Int. J. Light Electron. Opt."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Ahmad, I., Saaban, A., Ibrahim, A., and Shahzad, M. (2014). Global Chaos Identical and Nonidentical Synchronization of a New Chaotic System Using Linear Active Control. Complexity.","DOI":"10.1002\/cplx.21573"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.physa.2005.06.070","article-title":"Bifurcation and chaos in triple-well \u03a66\u2014Van der Pol oscillator driven by external and parametric excitations","volume":"357","author":"Siewe","year":"2005","journal-title":"Phys. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"362","DOI":"10.1016\/S0378-4371(01)00500-3","article-title":"Resonant oscillations and fractal basin boundaries of a particle in a \u03a66 potential","volume":"304","author":"Tchoukuegno","year":"2002","journal-title":"Phys. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1016\/S0020-7462(01)00081-6","article-title":"Linear feedback and parametric controls of vibrations and chaotic escape in a \u03a66 potential","volume":"38","author":"Tchoukuegno","year":"2003","journal-title":"Int. J. Nonlinear Mech."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"2086","DOI":"10.1016\/j.chaos.2007.06.067","article-title":"Robust synchronization of drive-response chaotic systems via adaptive sliding mode control","volume":"39","author":"Li","year":"2009","journal-title":"Chaos Soliton Fract."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"220","DOI":"10.1016\/j.physleta.2006.03.047","article-title":"Robust synchronization of chaotic systems via adaptive sliding mode control","volume":"356","author":"Yan","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1639","DOI":"10.1016\/j.apm.2011.09.023","article-title":"Chaos synchronization between two different chaotic systems with uncertainties, external disturbances, unknown parameters and input nonlinearities","volume":"36","author":"Aghababa","year":"2012","journal-title":"Appl. Math. Model."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"5757","DOI":"10.1016\/j.amc.2011.11.080","article-title":"A chattering-free robust adaptive sliding mode controller for synchronization of two different chaotic systems with unknown uncertainties and external disturbances","volume":"218","author":"Aghababa","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1002\/cplx.21459","article-title":"Synchronization of circular restricted three body problem with Lorenz hyper chaotic system using a robust adaptive sliding mode controller","volume":"18","author":"Khan","year":"2013","journal-title":"Complexity"}],"container-title":["Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-8954\/4\/2\/25\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:24:28Z","timestamp":1760210668000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-8954\/4\/2\/25"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,6,22]]},"references-count":36,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2016,6]]}},"alternative-id":["systems4020025"],"URL":"https:\/\/doi.org\/10.3390\/systems4020025","relation":{},"ISSN":["2079-8954"],"issn-type":[{"type":"electronic","value":"2079-8954"}],"subject":[],"published":{"date-parts":[[2016,6,22]]}}}