{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T12:14:37Z","timestamp":1767183277616,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T00:00:00Z","timestamp":1703116800000},"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 behavior and bifurcations of solutions to three-dimensional (three-phase) quadratic polynomial dynamical systems (DSs) are considered. The integrability in elementary functions is proved for a class of autonomous polynomial DSs. The occurrence of bifurcations of the type-twisted fold is discovered on the basis and within the frames of the elements of the developed DS qualitative theory. The discriminant criterion applied originally to two-phase quadratic polynomial DSs is extended to three-phase DSs investigated in terms of their coefficient matrices. Specific classes of D- and S-vectors are introduced and a complete description of the symmetry relations inherent to the DS coefficient matrices is performed using the discriminant criterion.<\/jats:p>","DOI":"10.3390\/sym16010014","type":"journal-article","created":{"date-parts":[[2023,12,21]],"date-time":"2023-12-21T10:53:52Z","timestamp":1703156032000},"page":"14","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Bifurcations Associated with Three-Phase Polynomial Dynamical Systems and Complete Description of Symmetry Relations Using Discriminant Criterion"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2691-2820","authenticated-orcid":false,"given":"Yury","family":"Shestopalov","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Russian Technological University MIREA, 119454 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Azizaga","family":"Shakhverdiev","sequence":"additional","affiliation":[{"name":"Department of the Development and Operation of Oil and Gas Fields, Russian State Geological Prospecting University, 117485 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sergey V.","family":"Arefiev","sequence":"additional","affiliation":[{"name":"Department for the Development of Oil and Gas Fields in Western Siberia and the Perm Region, PJSC \u201cLUKOIL\u201d, 101000 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,21]]},"reference":[{"key":"ref_1","unstructured":"Gaiko, V.A. (2003, January 20\u201322). Global bifurcations and chaos in polynomial dynamical systems. Proceedings of the 2003 International Conference Physics and Control. Proceedings, St. Petersburg, Russia."},{"key":"ref_2","unstructured":"Gaiko, V.A. (2003). Book Series: Mathematics and Its Applications, Kluwer."},{"key":"ref_3","unstructured":"Luo, A. (2022). Polynomial Functional Dynamical Systems, E-Book, Springer International Publishing."},{"key":"ref_4","unstructured":"Bautin, N.N., and Leontovich, E.A. (1990). Methods and Examples of the Qualitative Analysis of Dynamical Systems in a Plane, Nauka."},{"key":"ref_5","unstructured":"Gaiko, V.A. (2000). Communications in Difference Equations, CRC Press."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Shestopalov, Y.V., and Shakhverdiev, A.K. (2021). Qualitative theory of two-dimensional polynomial dynamical systems. 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