{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T11:56:37Z","timestamp":1768478197641,"version":"3.49.0"},"reference-count":25,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,6]],"date-time":"2024-10-06T00:00:00Z","timestamp":1728172800000},"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 research article introduces a novel chaotic satellite system based on fractional derivatives. The study explores the characteristics of various fractional derivative satellite systems through detailed phase portrait analysis and computational simulations, employing fractional calculus. We provide illustrations and tabulate the phase portraits of these satellite systems, highlighting the influence of different fractional derivative orders and parameter values. Notably, our findings reveal that chaos can occur even in systems with fewer than three dimensions. To validate our results, we utilize a range of analytical tools, including equilibrium point analysis, dissipative measures, Lyapunov exponents, and bifurcation diagrams. These methods confirm the presence of chaos and offer insights into the system\u2019s dynamic behavior. Additionally, we demonstrate effective control of chaotic dynamics using feedback active control techniques, providing practical solutions for managing chaos in satellite systems.<\/jats:p>","DOI":"10.3390\/sym16101319","type":"journal-article","created":{"date-parts":[[2024,10,7]],"date-time":"2024-10-07T06:23:18Z","timestamp":1728282198000},"page":"1319","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Synchronization of Chaotic Satellite Systems with Fractional Derivatives Analysis Using Feedback Active Control Techniques"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2943-4825","authenticated-orcid":false,"given":"Sanjay","family":"Kumar","sequence":"first","affiliation":[{"name":"Amity School of Engineering and Technology, Amity University, Patna 801503, India"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-1257-5886","authenticated-orcid":false,"given":"Amit","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Atma Ram Sanatan Dharma College, University of Delhi, New Delhi 110021, India"}]},{"given":"Pooja","family":"Gupta","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Gargi College, University of Delhi, New Delhi 110049, India"}]},{"given":"Ram Pravesh","family":"Prasad","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hansraj College, University of Delhi, New Delhi 110007, India"}]},{"given":"Praveen","family":"Kumar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Ramjas College, University of Delhi, New Delhi 110007, India"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1016\/j.chaos.2004.02.035","article-title":"Chaos in the fractional order Chen system and its control","volume":"22","author":"Li","year":"2004","journal-title":"Chaos Solit. Fract."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"2166","DOI":"10.1016\/j.physleta.2009.04.032","article-title":"Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system","volume":"373","author":"Matouk","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_4","first-page":"8693","article-title":"A nonlinear viscoelastic fractional derivative model of infant hydrocephalus","volume":"217","author":"Wilkie","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jtbi.2011.01.034","article-title":"A mathematical model on fractional Lotka-Volterra equations","volume":"277","author":"Das","year":"2011","journal-title":"J. Theor. Biol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/s11071-009-9609-6","article-title":"Adaptive feedback control and synchronization of non-identical chaotic fractional order systems","volume":"60","author":"Odibat","year":"2010","journal-title":"Nonlinear Dyn."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"832","DOI":"10.1016\/j.camwa.2012.11.015","article-title":"A note on stability of sliding mode dynamics in suppression of fractional-order chaotic systems","volume":"66","author":"Faieghi","year":"2013","journal-title":"Comput. Math. Appl."},{"key":"ref_8","first-page":"307","article-title":"Synchronization of fractional order Rabinovich-Fabrikant systems using sliding mode control techniques","volume":"29","author":"Khan","year":"2019","journal-title":"Arch. Control Sci."},{"key":"ref_9","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_10","doi-asserted-by":"crossref","first-page":"1097","DOI":"10.1016\/j.chaos.2004.09.090","article-title":"Chaos in the fractional order periodically forced complex Duffing\u2019s oscillators","volume":"24","author":"Gao","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/j.physleta.2006.01.068","article-title":"Chaotic dynamics of the fractional-order L\u00fc system and its synchronization","volume":"354","author":"Lu","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1125","DOI":"10.1016\/j.chaos.2005.02.023","article-title":"Chaotic dynamics and synchronization of fractional-order Arneodo\u2019s systems","volume":"26","author":"Lu","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1938","DOI":"10.1016\/j.aml.2011.05.025","article-title":"Dynamical behaviors and synchronization in the fractional order hyperchaotic chen system","volume":"24","author":"Hegazi","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_14","first-page":"734","article-title":"Synchronization of a perturbed satellite attitude motion","volume":"8","author":"Djaouida","year":"2014","journal-title":"Int. J. Mech. Aerosp. Ind. Mechatron. Manuf. Eng."},{"key":"ref_15","unstructured":"Duan, G.R., and Yu, H.H. (2013). LMI in Control Systems Analysis, Design and Applications, CRC Press."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Kong, L.Y., Zhoul, F.Q., and Zou, I. (2006, January 7\u201311). The control of chaotic attitude motion of a perturbed spacecraft. Proceedings of the 25th Chinese Control Conference, Harbin, China.","DOI":"10.1109\/CHICC.2006.280796"},{"key":"ref_17","unstructured":"Liu, T., and Zhao, J. (2003). Dynamics of Spacecraft, Harbin Institute of Technology Press. (In Chinese)."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1007\/s40435-014-0089-2","article-title":"Generalized projective synchronization of chaotic satellites problem using linear matrix inequality","volume":"2","author":"Farid","year":"2014","journal-title":"Int. J. Dynam. Control"},{"key":"ref_19","unstructured":"Goeree, B.B., and Fasse, E.D. (2000, January 28\u201330). Sliding mode attitude control of a small satellite for ground tracking maneuvers. Proceedings of the American Control Conference, Chicago, IL, USA."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1597","DOI":"10.1002\/oca.2428","article-title":"Measuring chaos and synchronization of chaotic satellite systems using sliding mode control","volume":"39","author":"Khan","year":"2018","journal-title":"Optim. Control Appl. Methods"},{"key":"ref_21","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_22","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1965","DOI":"10.1016\/j.automatica.2009.04.003","article-title":"Mittag-leffler stability of fractional order nonlinear dynamic systems","volume":"45","author":"Li","year":"2009","journal-title":"Automatica"},{"key":"ref_24","unstructured":"Zhang, R.W. (1998). Satellite Orbit and Attitude Dynamics and Control, Beihang University Press. (In Chinese)."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Sidi, M.J. (1997). Spacecraft Dynamics and Control\u2014A Practical Engineering Approach, Cambridge University Press.","DOI":"10.1017\/CBO9780511815652"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1319\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:11:44Z","timestamp":1760112704000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/10\/1319"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,10,6]]},"references-count":25,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2024,10]]}},"alternative-id":["sym16101319"],"URL":"https:\/\/doi.org\/10.3390\/sym16101319","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,10,6]]}}}