{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:05:09Z","timestamp":1760238309687,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,4]],"date-time":"2020-08-04T00:00:00Z","timestamp":1596499200000},"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>We provide a general method for identifying real quadratic polynomial dynamical systems that can be transformed to symmetric ones by a bijective polynomial map of degree one, the so-called affine map. We mainly focus on symmetry groups generated by rotations, in other words, we treat equivariant and reversible equivariant systems. The description is given in terms of affine varieties in the space of parameters of the system. A general algebraic approach to find subfamilies of systems having certain symmetries in polynomial differential families depending on many parameters is proposed and computer algebra computations for the planar case are presented.<\/jats:p>","DOI":"10.3390\/sym12081300","type":"journal-article","created":{"date-parts":[[2020,8,4]],"date-time":"2020-08-04T10:45:17Z","timestamp":1596537917000},"page":"1300","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On Some Symmetries of Quadratic Systems"],"prefix":"10.3390","volume":"12","author":[{"given":"Maoan","family":"Han","sequence":"first","affiliation":[{"name":"Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tatjana","family":"Petek","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering and Computer Science, University of Maribor, SI-2000 Maribor, Slovenia"},{"name":"Institute of Mathematics, Physics and Mechanics, SI-1000 Ljubljana, Slovenia"},{"name":"Center for Applied Mathematics and Theoretical Physics, SI-2000 Maribor, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2775-2953","authenticated-orcid":false,"given":"Valery G.","family":"Romanovski","sequence":"additional","affiliation":[{"name":"Faculty of Electrical Engineering and Computer Science, University of Maribor, SI-2000 Maribor, Slovenia"},{"name":"Center for Applied Mathematics and Theoretical Physics, SI-2000 Maribor, Slovenia"},{"name":"Faculty of Natural Science and Mathematics, University of Maribor, SI-2000 Maribor, Slovenia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1142\/S1230161208000249","article-title":"Time-reversibility in 2-dimensional systems","volume":"15","author":"Romanovski","year":"2008","journal-title":"Open Syst. Inf. Dyn."},{"key":"ref_2","first-page":"55","article-title":"Reversibility in polynomial sytems of ODE\u2019s","volume":"338","author":"Han","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_3","first-page":"925","article-title":"Reversing Symmetries in dynamical systems","volume":"25","author":"Lamb","year":"1992","journal-title":"Physica A"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1006\/jdeq.1999.3632","article-title":"Reversible Equivariant systems","volume":"159","author":"Lamb","year":"1999","journal-title":"J. Differ. Equ."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Horn, R.A., and Johnson, C.R. (1991). Topics in Matrix Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511840371"},{"key":"ref_6","first-page":"25","article-title":"Rotation symmetry groups of planar Hamiltonian systems","volume":"5","author":"Li","year":"1989","journal-title":"Ann. Differ. Equ."},{"key":"ref_7","first-page":"1","article-title":"General form of Zq-reversible-equivariant planar systems and limit cycle bifurcations","volume":"40","author":"Han","year":"2011","journal-title":"J. Shanghai Norm. Univ."},{"key":"ref_8","unstructured":"Cox, D., Little, J., and O\u2019Shea, D. (1997). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer."},{"key":"ref_9","unstructured":"Romanovski, V.G., and Shafer, D.S. (2009). The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhauser."},{"key":"ref_10","unstructured":"Decker, W., Greuel, G.-M., Pfister, G., and Sh\u00f6nemann, H. (2020, August 02). Singular 3-1-6\u2013A Computer Algebra System for Polynomial Computations (2012). Available online: http:\/\/www.singular.uni-kl.de."},{"key":"ref_11","unstructured":"Decker, W., Laplagne, S., Pfister, G., and Schonemann, H.A. (2020, August 02). primdec.lib. A Singular 3-1 Library for Computing the Prime Decomposition and Radical of Ideals. Available online: http:\/\/www.singular.uni-kl.de."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/S0747-7171(88)80040-3","article-title":"Gr\u00f6bner bases and primary decomposition of polynomials","volume":"6","author":"Gianni","year":"1988","journal-title":"J. Symb. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Brjuno, A.D. (1989). Local Methods in Nonlinear Differential Equations, Springer.","DOI":"10.1007\/978-3-642-61314-2"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1300\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:54:13Z","timestamp":1760176453000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1300"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,4]]},"references-count":13,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["sym12081300"],"URL":"https:\/\/doi.org\/10.3390\/sym12081300","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,8,4]]}}}