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 SM ISO690:2012COZMA, Dumitru, MATEI, Angela. Center conditions for a cubic differential system having an integrating factor. In: Conference on Applied and Industrial Mathematics: CAIM 2022, Ed. 29, 25-27 august 2022, Chişinău. Chișinău, Republica Moldova: Casa Editorial-Poligrafică „Bons Offices”, 2022, Ediţia a 29, pp. 59-61. ISBN 978-9975-81-074-6. |
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| Conference on Applied and Industrial Mathematics Ediţia a 29, 2022 |
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Conferința "Conference on Applied and Industrial Mathematics" 29, Chişinău, Moldova, 25-27 august 2022 | ||||||
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We consider the cubic system of differential equations x˙ = y + p2(x, y) + p3(x, y), y˙ = −x + q2(x, y) + q3(x, y), (1)where pj(x, y), qj (x, y) ∈ R[x, y] are homogeneous polinomials of degree j. The origin O(0, 0) is a singular point for (1) with purely imaginary eigenvalues, i.e. a focus or a center. The problem of distinguishing between a center and a focus (the problem of the center) is open for general cubic systems. In [1] the problem of the center was solved for cubic system (1) with: four invariant straight lines; three invariant straight lines; two invariant straight lines and one irreducible invariant conic. The center conditions for a cubic differential system (1) with two invariant straight lines and one irreducible invariant cubic curve Φ ≡ a30x3 + a21x2y + a12xy2 + a03y3 + x2 + y2 = 0 were found in [2] and for cubic system (1) having an integrating factor μ−1 = Φh were found in [3], where a30,a21,a12,a03 and h are real parameters. In this talk we give the conditions under which the cubic system (1) has an integrating factor of the form μ−1 = Ψh, (1) where Ψ ≡ a20x2+a11xy+a02y2+a10x+a01y+1 = 0 is an irreducible invariant conic and a20,a11,a02,a10,a01 and h are real parameters. According to [2] the cubic differential systems (1) which have integrating factors of the form (1) have a center at the singular point O(0, 0). |
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