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SM ISO690:2012 BUJAC, Cristina. Cubic systems with one simple and one triple real infinite singularities with invariant lines of total multiplicity eight. In: Tendinţe contemporane ale dezvoltării ştiinţei: viziuni ale tinerilor cercetători, Ed. 3, 10 martie 2014, Chișinău. Chișinău, Republica Moldova: Universitatea Academiei de Ştiinţe a Moldovei, 2014, Editia 3, T, p. 11. ISBN 978-9975-4257-2-8. |
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Tendinţe contemporane ale dezvoltării ştiinţei: viziuni ale tinerilor cercetători Editia 3, T, 2014 |
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Conferința "Tendinţe contemporane ale dezvoltării ştiinţei: viziuni ale tinerilor cercetători" 3, Chișinău, Moldova, 10 martie 2014 | |||||||
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Teza |
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This paper deals with the family of planar cubic polynomial differential systems. Our main goal is to classify the family of cubic systems according to their geometric properties encoded in the configurations of invariant straight lines of total multiplicity eight (including the line at infinity with its own multiplicity), which these systems possess [2]. We restrict our attention to the subfamily of cubic systems with two real distinct infinite singularities, namely 1 simple and 1 of the multiplicity three. We prove that there are total of 3 such configurations which are distinguished, roughly speaking, by the multiplicity of their invariant lines and by the multiplicities of the singularities of the systems located on these lines. In addition to this, we construct for the normal forms given in our Main Theorem the corresponding perturbations, which prove that the respective invariant straight lines have the indicated multiplicity. Moreover using the algebraic method of invariants of differential systems, developed by Sibirskii and his disciples, we construct necessary and sufficient affine invariant conditions for the realization of each one of these configurations. We remark that cubic systems with maximum number of invariant lines are considered in [1]. The author is partially supported by FP7-PEOPLE-2012-IRSES-316338 and by the grant 12.817-.08.05F from SCSTD of ASM. |