{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T19:11:54Z","timestamp":1648926714645},"reference-count":27,"publisher":"World Scientific Pub Co Pte Lt","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2016,7]]},"abstract":"<jats:p> Invariant algebraic surfaces are commonly observed in differential systems arising in mathematical modeling of natural phenomena. In this paper, we study the integrability and dynamics of quadratic polynomial differential systems defined in [Formula: see text] having an elliptic paraboloid as an invariant algebraic surface. We obtain the normal form for these kind of systems and, by using the invariant paraboloid, we prove the existence of first integrals, exponential factors, Darboux invariants and inverse Jacobi multipliers, for suitable choices of parameter values. We characterize all the possible configurations of invariant parallels and invariant meridians on the invariant paraboloid and give necessary conditions for the invariant parallel to be a limit cycle and for the invariant meridian to have two orbits heteroclinic to a point at infinity. We also study the dynamics of a particular class of the quadratic polynomial differential systems having an invariant paraboloid, giving information about localization and local stability of finite singular points and, by using the Poincar\u00e9 compactification, we study their dynamics on the Poincar\u00e9 sphere (at infinity). Finally, we study the well-known Rabinovich system in the case of invariant paraboloids, performing a detailed study of its dynamics restricted to these invariant algebraic surfaces. <\/jats:p>","DOI":"10.1142\/s0218127416501340","type":"journal-article","created":{"date-parts":[[2016,8,5]],"date-time":"2016-08-05T09:03:24Z","timestamp":1470387804000},"page":"1650134","source":"Crossref","is-referenced-by-count":5,"title":["Integrability and Dynamics of Quadratic Three-Dimensional Differential Systems Having an Invariant Paraboloid"],"prefix":"10.1142","volume":"26","author":[{"given":"Marcelo","family":"Messias","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica e Computa\u00e7\u00e3o, Faculdade de Ci\u00eancias e Tecnologia, UNESP Univ Estadual Paulista, Presidente Prudente, S\u00e3o Paulo, Brazil"}]},{"given":"Alisson C.","family":"Reinol","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Instituto de Bioci\u00eancias, Letras e Ci\u00eancias Exatas, UNESP Univ Estadual Paulista, S\u00e3o Jos\u00e9 do Rio Preto, S\u00e3o Paulo, Brazil"}]}],"member":"219","published-online":{"date-parts":[[2016,8,5]]},"reference":[{"key":"S0218127416501340BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2012.03.009"},{"key":"S0218127416501340BIB002","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/21\/2\/002"},{"key":"S0218127416501340BIB003","doi-asserted-by":"publisher","DOI":"10.2307\/2001320"},{"key":"S0218127416501340BIB004","author":"Dias F. 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