{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:37:39Z","timestamp":1760146659490,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2024,11,26]],"date-time":"2024-11-26T00:00:00Z","timestamp":1732579200000},"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>The aim of present paper is to obtain approximate semi-analytical solutions for the Qi-type dynamical system, while neglecting its chaotic behaviors. These solutions are derived using the Optimal Auxiliary Functions Method (OAFM). The impact of the system\u2019s physical parameters is also investigated. A special case, involving a constant of motion, is considered for which closed-form solutions are obtained. The dynamical system is reduced to a second-order nonlinear differential equation, which is analytically solved through the OAFM procedure. The influence of initial conditions on the system is explored, specifically regarding the presence or absence of symmetries. An exact parametric solution is obtained for a particular case. A good agreement between the analytical and corresponding numerical results is demonstrated, highlighting the accuracy of the proposed method. A comparative analysis underlines the advantages of the OAFM compared to other analytical methods. These findings have numerous technological applications, such as in nonlinear circuits with three channels that involve adapted physical parameters to ensure effective functioning of electronic circuits, as well as in information storage, encryption, and communication systems.<\/jats:p>","DOI":"10.3390\/sym16121578","type":"journal-article","created":{"date-parts":[[2024,11,26]],"date-time":"2024-11-26T04:02:18Z","timestamp":1732593738000},"page":"1578","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Semi-Analytical Solutions for the Qi-Type Dynamical System"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1679-1498","authenticated-orcid":false,"given":"Remus-Daniel","family":"Ene","sequence":"first","affiliation":[{"name":"Department of Mathematics, Politehnica University of Timisoara, 300006 Timisoara, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7051-6753","authenticated-orcid":false,"given":"Nicolina","family":"Pop","sequence":"additional","affiliation":[{"name":"Department of Physical Foundations of Engineering, Politehnica University of Timisoara, 2 Vasile Parvan Blvd., 300223 Timisoara, Romania"}]},{"given":"Rodica","family":"Badarau","sequence":"additional","affiliation":[{"name":"Department of Mechanical Machines, Equipment and Transportation, Politehnica University of Timisoara, 1 Mihai Viteazul Blvd., 300222 Timisoara, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.cja.2024.05.031","article-title":"Identification of aircraft longitudinal aerodynamic parameters using an online corrective test for wind tunnel virtual flight","volume":"37","author":"Tai","year":"2024","journal-title":"Chin. J. Aeronaut."},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1080\/00029890.1975.11994008","article-title":"Period Three Implies Chaos","volume":"82","author":"Li","year":"1975","journal-title":"Am. Math. Mon."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/0375-9601(76)90101-8","article-title":"An Equation for Continuous Chaos","volume":"57","year":"1976","journal-title":"Phys. Lett. A"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"R\u00f6ssler, O.E. (1977, January 2\u20137). Continuous chaos, Synergetics: A Workshop. Proceedings of the International Workshop on Synergetics, Schloss Elmau, Bavaria.","DOI":"10.1007\/978-3-642-66784-8_17"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.1103\/PhysRevA.30.1155","article-title":"Simplest chaotic nonautonomous circuit","volume":"30","author":"Matsumoto","year":"1984","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1142\/S0218127499001024","article-title":"Yet another chaotic attractor","volume":"9","author":"Chen","year":"1999","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1142\/S0218127402004851","article-title":"Dynamical analysis of a new chaotic attractor","volume":"12","author":"Chen","year":"2002","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2917","DOI":"10.1142\/S021812740200631X","article-title":"Bridge the gap between the Lorenz system and the Chen system","volume":"12","author":"Chen","year":"2002","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"R647","DOI":"10.1103\/PhysRevE.50.R647","article-title":"Some simple chaotic flows","volume":"50","author":"Sprott","year":"1994","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1016\/S0375-9601(97)00088-1","article-title":"Simplest dissipative chaotic flow","volume":"228","author":"Sprott","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1031","DOI":"10.1016\/j.chaos.2004.02.060","article-title":"A new chaotic attractor","volume":"22","author":"Liu","year":"2004","journal-title":"Chaos Solitons Fractals"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1088\/0253-6102\/55\/4\/17","article-title":"Analytical Hopf bifurcation and stability analysis of T system","volume":"55","author":"Choudhury","year":"2011","journal-title":"Commun. Theor. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"102210","DOI":"10.1016\/j.vlsi.2024.102210","article-title":"Analysis of a new three-dimensional Jerk chaotic system with transient chaos and its adaptive backstepping synchronous control","volume":"98","author":"Yan","year":"2024","journal-title":"Integration"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1650237","DOI":"10.1142\/S0218127416502370","article-title":"Force Analysis of Qi Chaotic System","volume":"26","author":"Qi","year":"2016","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"575","DOI":"10.1016\/j.ijleo.2013.07.013","article-title":"A new hyperchaotic system and its generalized synchronization","volume":"125","author":"Li","year":"2014","journal-title":"Optik"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1007\/s12346-021-00454-0","article-title":"A New Five-Dimensional Hyperchaotic System with Six Coexisting Attractors","volume":"20","author":"Yang","year":"2021","journal-title":"Qual. Theory Dyn. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Zghair, H.K., Mehdi, S.A., and Sadkhan, S.B. (2020, January 12\u201313). Design and Analytic of A Novel Seven Dimension Hyper Chaotic Systems. Proceedings of the 1st International Conference of Information Technology to Enhance E-Learning and Other Application 2020, Baghdad, Iraq.","DOI":"10.1109\/IT-ELA50150.2020.9253077"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"111931","DOI":"10.1016\/j.automatica.2024.111931","article-title":"Semi-global stabilization of parabolic PDE\u2013ODE systems with input saturation","volume":"171","author":"Xu","year":"2025","journal-title":"Automatica"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"111845","DOI":"10.1016\/j.automatica.2024.111845","article-title":"PDE-based observation and predictor-based control for linear systems with distributed infinite input and output delays","volume":"170","author":"Xu","year":"2024","journal-title":"Automatica"},{"key":"ref_21","first-page":"169","article-title":"Approximate analytical solutions to Jerk equation","volume":"Volume 182","author":"Marinca","year":"2016","journal-title":"Springer Proceedings in Mathematics 399 & Statistics: Proceedings of the Dynamical Systems: Theoretical and Experimental Analysis, Lodz, Poland, 7\u201310 December 2015"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1515\/eng-2018-0028","article-title":"Optimal Auxiliary Functions Method for viscous flow due to a stretching surface with partial slip","volume":"8","author":"Marinca","year":"2018","journal-title":"Open Eng."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Ene, R.-D., Pop, N., and Lapadat, M. (2022). Approximate Closed-Form Solutions for the Rabinovich System via the Optimal Auxiliary Functions Method. Symmetry, 14.","DOI":"10.20944\/preprints202209.0484.v1"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ene, R.-D., Pop, N., Lapadat, M., and Dungan, L. (2022). Approximate closed-form solutions for the Maxwell-Bloch equations via the Optimal Homotopy Asymptotic Method. Mathematics, 10.","DOI":"10.20944\/preprints202209.0474.v1"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1016\/j.jmaa.2005.05.009","article-title":"An iterative method for solving nonlinear functional equations","volume":"316","author":"Jafari","year":"2006","journal-title":"J. Math. Anal. Appl."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1578\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:39:33Z","timestamp":1760114373000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/12\/1578"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,26]]},"references-count":25,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2024,12]]}},"alternative-id":["sym16121578"],"URL":"https:\/\/doi.org\/10.3390\/sym16121578","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,11,26]]}}}