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SM ISO690:2012 VACARAŞ, Olga. Cubic differential systems with two invariant straight lines of multiplicity m﴾2;5). In: Tendinţe contemporane ale dezvoltării ştiinţei: viziuni ale tinerilor cercetători, Ed. 4, 10 martie 2015, Chișinău. Chișinău, Republica Moldova: Universitatea Academiei de Ştiinţe a Moldovei, 2015, Ediția 4, p. 27. |
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Tendinţe contemporane ale dezvoltării ştiinţei: viziuni ale tinerilor cercetători Ediția 4, 2015 |
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Conferința "Tendinţe contemporane ale dezvoltării ştiinţei: viziuni ale tinerilor cercetători" 4, Chișinău, Moldova, 10 martie 2015 | |||||||
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We consider the real cubic system of differential equations ()()(),1,;,/,,/===QPGCDyxQdtdyyxPdtdx (1) ()(){}.3deg,degmax=QP A straight line ,0=++γβαyx ,,,C∈γβα )0,0(),(≠βα is invariant for (1) if there exists a polynomial ()yxK, such that the identity ()()()()yxKyxyxQyxP,,,γβαβα++≡+ holds. In this paper we show that if the cubic system admits an affine straight line of multiplicity two, then the line at infinity cannot have multiplicity greater than five. Theorem. Any cubic system having two invariant straight lines (including the line at infinity) of the multiplicity )5;2(m via affine transformation and time rescaling can be brought to the form .0,,23≠+++==bybxaxxyxx&& (2) For (2) 0=x is invariant straight line of multiplicity 21=m and the line at infinity has multiplicity 52=m. The perturbed cubic system 9/))4129)(3()69)(23()23)(((,9/)23)(23(22222222232εεεεεεεεεεεεεεεεεababayxyxbaaaxybyayxbxaxybxabxaxx−+−++++−++++++++++=++−+=&& (3) has six distinct invariant straight lines: ,23,21εεεbxalyxl−+=+= yyxababaxaxbaaallxlbxal)2)(4129()23()223)(23)(1(,,2322222226543εεεεεεεεεεεεεε+−+−+++++++++=⋅=++=2 which converge to 0=x when .0 |