Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
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Bul. Acad. Ştiinţe Repub. Mold. Mat., 2008, номер 1, страницы 27–83 (Mi basm3)  

Эта публикация цитируется в 19 научных статьях (всего в 20 статьях)

Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four

Dana Schlomiuka, Nicolae Vulpeb

a Département de Mathématiques et de Statistique, Université de Montréal
b Institute of Mathematics and Computer Science, Academy of Sciences of Moldova

Аннотация: In this article we consider the class $\mathbf{QSL}_4$ of all real quadratic differential systems $\dfrac{dx}{dt}=p(x,y)$, $\dfrac{dy}{dt}=q(x,y)$ with $\mathrm{gcd}(p,q)=1$, having invariant lines of total multiplicity four and a finite set of singularities at infinity. We first prove that all the systems in this class are integrable having integrating factors which are Darboux functions and we determine their first integrals. We also construct all the phase portraits for the systems belonging to this class. The group of affine transformations and homotheties on the time axis acts on this class. Our Main Theorem gives necessary and sufficient conditions, stated in terms of the twelve coefficients of the systems, for the realization of each one of the total of 69 topologically distinct phase portraits found in this class. We prove that these conditions are invariant under the group action.

Ключевые слова и фразы: Quadratic differential system, Poincaré compactification, algebraic invariant curve, affine invariant polynomial, configuration of invariant lines, phase portrait.

Полный текст: PDF файл (1520 kB)
Список литературы: PDF файл   HTML файл

Реферативные базы данных:
MSC: 34A26, 34C40, 34C14
Поступила в редакцию: 06.11.2007
Язык публикации: английский

Образец цитирования: Dana Schlomiuk, Nicolae Vulpe, “Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2008, no. 1, 27–83

Цитирование в формате AMSBIB
\RBibitem{SchVul08}
\by Dana~Schlomiuk, Nicolae~Vulpe
\paper Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four
\jour Bul. Acad. \c Stiin\c te Repub. Mold. Mat.
\yr 2008
\issue 1
\pages 27--83
\mathnet{http://mi.mathnet.ru/basm3}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2392678}
\zmath{https://zbmath.org/?q=an:1159.34329}


Образцы ссылок на эту страницу:
  • http://mi.mathnet.ru/basm3
  • http://mi.mathnet.ru/rus/basm/y2008/i1/p27

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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    Эта публикация цитируется в следующих статьяx:
    1. Artés J.C., Llibre J., Vulpe N., “When singular points determine quadratic systems”, Electron. J. Differential Equations, 2008, no. 82, 37 pp.  mathscinet  elib
    2. Artés J.C., Llibre J., Vulpe N., “Quadratic systems with a polynomial first integral: a complete classification in the coefficient space $\mathbb R^{12}$”, J. Differential Equations, 246:9 (2009), 3535–3558  crossref  mathscinet  zmath  isi  elib  scopus
    3. N. Gherstega, V. Orlov, N. Vulpe, “A complete classification of quadratic differential systems according to the dimensions of $Aff(2,\mathbb R)$-orbits”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2009, no. 2, 29–54  mathnet  mathscinet  zmath
    4. Schlomiuk D., Vulpe N., “Bifurcation diagrams and moduli spaces of planar quadratic vector fields with invariant lines of total multiplicity four and having exactly three real singularities at infinity”, Qual. Theory Dyn. Syst., 9:1-2 (2010), 251–300  crossref  mathscinet  zmath  elib  scopus
    5. Schlomiuk, Dana; Vulpe, Nicolae, “Global classification of the planar Lotka–Volterra differential systems according to their configurations of invariant straight lines”, J. Fixed Point Theory Appl., 8:1 (2010), 177–245  crossref  mathscinet  zmath  isi  scopus
    6. Artés, Joan C.; Llibre, Jaume; Vulpe, Nicolae, “Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ?12”, Rendiconti del Circolo Matematico di Palermo, 59:3 (2010), 419–449  crossref  mathscinet  zmath  scopus
    7. Vulpe, Nicolae, “Characterization of the finite weak singularities of quadratic systems via invariant theory”, Nonlinear Anal., 74:17 (2011), 6553–6582  crossref  mathscinet  zmath  isi  elib  scopus
    8. Artés, Joan C.; Llibre, Jaume; Vulpe, Nicolae, “Quadratic systems with an integrable saddle: a complete classification in the coefficient space R12”, Nonlinear Anal., 75:14 (2012), 5416–5447  crossref  mathscinet  zmath  isi  elib  scopus
    9. Dana Schlomiuk, Nicolae Vulpe, “Global topological classification of Lotka-Volterra quadratic differential systems”, Electronic Journal of Differential Equations, 2012:64 (2012), 1–69  crossref  mathscinet
    10. Dana Schlomiuk, “New developments based on the mathematical legacy of C. S. Sibirschi”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2013, no. 1, 3–10  mathnet  mathscinet
    11. J. C. Artés, J. Llibre, D. Schlomiuk, N. Vulpe, “Geometric configurations of singularities for quadratic differential systems with total finite multiplicity lower than 2”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2013, no. 1, 72–124  mathnet  mathscinet
    12. Cristina Bujac, “One subfamily of cubic systems with invariant lines of total multiplicity eight and with two distinct real infinite singularities”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2015, no. 1, 48–86  mathnet
    13. Oliveira R.D.S., Rezende A.C., Schlomiuk D., Vulpe N., “Geometric and Algebraic Classification of Quadratic Differential Systems With Invariant Hyperbolas”, Electron. J. Differ. Equ., 2017, 295  mathscinet  zmath  isi
    14. Bujac C. Vulpe N., “Cubic Differential Systems With Invariant Straight Lines of Total Multiplicity Eight Possessing One Infinite Singularity”, Qual. Theor. Dyn. Syst., 16:1 (2017), 1–30  crossref  mathscinet  zmath  isi  scopus
    15. Bujac C. Vulpe N., “First Integrals and Phase Portraits of Planar Polynomial Differential Cubic Systems With Invariant Straight Lines of Total Multiplicity Eight”, Electron. J. Qual. Theory Differ., 2017, no. 85, 1–35  crossref  mathscinet  isi  scopus
    16. Schlomiuk D., Zhang X., “Quadratic Differential Systems With Complex Conjugate Invariant Lines Meeting At a Finite Point”, J. Differ. Equ., 265:8 (2018), 3650–3684  crossref  mathscinet  zmath  isi  scopus
    17. Llibre J. Valls C., “Global Phase Portraits For the Abel Quadratic Polynomial Differential Equations of the Second Kind With Z(2)-Symmetries”, Can. Math. Bul.-Bul. Can. Math., 61:1 (2018), 149–165  crossref  mathscinet  zmath  isi  scopus
    18. Dana Schlomiuk, “Working with professor Nicolae Vulpe”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2019, no. 2, 154–160  mathnet
    19. Dana Schlomiuk, Nicolae Vulpe, “The topological classification of a family of quadratic differential systems in terms of affine invariant polynomials”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2019, no. 2, 41–55  mathnet
    20. Cristina Bujac, “The classification of a family of cubic differential systems in terms of configurations of invariant lines of the type $(3,3)$”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2019, no. 2, 79–98  mathnet
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