Abstract
Consider the class QSH of all non-degenerate planar quadratic differential systems possessing an invariant algebraic hyperbola. In this paper we consider a specific two-parametric family of systems in QSH possessing three pairs of infinite singularities, namely family QSH(D). For this family we have generically a presence of one couple of parallel invariant straight lines and one invariant algebraic hyperbola. Our goal is to explore the relationship among the topological bifurcation diagram, geometry of configurations of invariant algebraic curves and its type of integrability. For this study we construct the topological bifurcation diagram of configurations and phase portraits of the family QSH(D) altogether. We also study the integrability, we obtain all the distinct configurations of invariant algebraic curves, and we get all the topologically distinct phase portraits in the Poincaré disc. More precisely, we prove that the family under consideration is Liouvillian integrable, there are 53 distinct configurations of invariant algebraic curves, and there exist 18 topologically distinct phase portraits.
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Artés, J.C., Llibre, J., Schlomiuk, D.: The geometry of quadratic differential systems with a weak focus of second order. Int. J. Bifur. Chaos Appl. Sci. Eng. 16, 3127–3194 (2006)
Artés, J.C., Llibre, J., Schlomiuk, D., Vulpe, N.: Geometric Configurations of Singularities of Planar Polynomial Differential Systems-A Global Classification in the Quadratic Case. Birkhäuser, Cham (2021)
Artés, J.C., Mota, M.C., Rezende, A.C.: Quadratic differential systems with a finite saddle-node and an infinite saddle-node \((1,1)SN\) - (A). Int. J. Bifur. Chaos Appl. Sci. Eng. 31, 1–24 (2021)
Artés, J.C., Mota, M.C., Rezende, A.C.: Quadratic differential systems with a finite saddle-node and an infinite saddle-node \((1,1)SN\) - (B). Int. J. Bifur. Chaos Appl. Sci. Eng. 31, 1–110 (2021)
Artés, J.C., Rezende, A.C., Oliveira, R.D.S.: Global phase portraits of quadratic polynomial differential systems with a semi-elemental triple node. Int. J. Bifur. Chaos Appl. Sci. Eng. 23, 1–21 (2013)
Artés, J.C., Rezende, A.C., Oliveira, R.D.S.: The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node \((A, B)\). Int. J. Bifur. Chaos Appl. Sci. Eng. 24, 1–30 (2014)
Artés, J.C., Rezende, A.C., Oliveira, R.D.S.: The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node \(C\). Int. J. Bifur. Chaos Appl. Sci. Eng. 25, 1–111 (2015)
Bautin, N.N.: On periodic solutions of a system of differential equations. Prikl. Mat. Meh. 18, 1–128 (1954)
Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: Darboux integrability and the inverse integrating factor. J. Differ. Equ. 194, 116–139 (2013)
Christopher, C. (1989). Quadratic systems having a parabola as an integral curve. Proc. R. Soc. Edinb. Sect. A Math 112(1-2), 113–134
Christopher, C.J.: Invariant algebraic curves and conditions for a centre. Proc. R. Soc. Edinb. Sect. A Math. 124, 1209–1229 (1994)
Christopher, C., Kooij, R.E.: Algebraic invariant curves and the integrability of polynomial systems. Appl. Math. Lett. 6(4), 51–53 (1993)
Christopher, C., Llibre, J.: Integrability via invariant algebraic curves for planar polynomial differential systems. Ann. Differ. Equ. 14, 5–19 (2000)
Christopher, C., Llibre, J., Pantazi, C., Walcher, S.: On planar polynomial vector fields with elementary first integrals. J. Differ. Equ. 267, 4572–4588 (2019)
Christopher, C., Llibre, J., Pereira, J.V.: Multiplicity of invariant algebraic curves in polynomial vector fields. Pacific J. Math. 229(1), 63–117 (2007)
Darboux, G.: Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges). Bull. Sci. Math. 60–96(123–144), 151–200 (1878)
Goriely, A.: Integrability and Nonintegrability of Dynamical Systems. World Scientific Publishing Co. Pte. Ltd., Singapore (2001)
Jouanolou, J.P.: Equations de Pfaff algebriques. Lectures Notes in Mathematics 708, Springer, New York (1979)
Llibre, J., Zhang, X.: Darboux theory of integrability in \({\mathbb{R} }^n\) taking into account the multiplicity at infinity. J. Differ. Equ. 133, 765–778 (2009)
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. 295, 1–112 (2017)
Oliveira, R.D.S., Rezende, A.C., Vulpe, N.: Family of quadratic differential systems with invariant hyperbolas: a complete classification in the space \(\mathbb{R}^{12}\). Electron. J. Differ. Equ. 162, 1–50 (2016)
Oliveira, R.D.S., Schlomiuk, D., Travaglini, A.M.: Geometry and integrability of quadratic systems with invariant hyperbolas. Electron. J. Qual. Theory Differ. Equ. 6, 1–56 (2021)
Oliveira, R.D.S., Schlomiuk, D., Travaglini, A.M., Valls, C.: Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of the Darboux theory of integrability. Electron J. Qual. Theory Differ Equ. 45, 1–90 (2021)
Pereira, J.V.: Vector fields, invariant varieties and linear systems. Ann. Inst. Fourier (Grenoble) 51, 1385–1405 (2001)
Prelle, M.J., Singer, M.F.: Elementary first integrals of differential equations. Trans. Am. Math. Soc. 279, 215–229 (1983)
Schlomiuk, D., Vulpe, N.: Planar quadratic vector fields with invariant lines of total multiplicity at least five. Qual. Theory Dyn. Syst. 5, 135–194 (2004)
Schlomiuk, D., Vulpe, N.: Integrals and phase portraits of planar quadratic systems with invariant lines of at least five total multiplicity. Rocky Mountain J. Math. 38(6), 2015–2075 (2008)
Schlomiuk, D., Vulpe, N.: Integrals and phase portraits of planar quadratic systems with invariant lines of total multiplicity four. Bul. Acad. Stiin. Repub. Mold. Mat. 56, 27–83 (2008)
Schlomiuk, D., Vulpe, N.: The full study of planar quadratic differential systems possessing a line of singularities at infinity. J. Dyn. Differ. Equ. 20, 737–775 (2008)
Schlomiuk, D., Vulpe, N.: Planar quadratic differential systems with invariant straight lines of the total multiplicity four. Nonlinear Anal. Theory Methods Appl. 68, 681–715 (2008)
Singer, M.F.: Liouvillian first integrals of differential equations. Trans. Am. Math. Soc. 333, 673–688 (1992)
Travaglini, A.M.: Integrability and geometry of quadratic differential systems with invariant hyperbolas. Tese de Doutorado do Programa de Pós-Graduação em Matemática (PPG-Mat), Universidade de São Paulo, pp. 1–398 (2021)
Zhang, X.: Integrability of Dynamical Systems: Algebra and Analysis. Developments in Mathematics. Springer Nature, Singapore (2017)
Acknowledgements
The first author is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Grant Number 166449/2020-2. The second author is partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Grant “Projeto Temático” 2019/21181-0 and CNPq Grant Number 304766/2019-4. This paper was developed during the Postdoctoral Program of the first and third authors at ICMC-USP.
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Mota, M.C., Oliveira, R. & Travaglini, A.M. The interplay among the topological bifurcation diagram, integrability and geometry for the family QSH(D). Geom Dedicata 217, 95 (2023). https://doi.org/10.1007/s10711-023-00827-6
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DOI: https://doi.org/10.1007/s10711-023-00827-6
Keywords
- Quadratic differential system
- Invariant algebraic curve
- Darboux and Liouvillian integrability
- Configuration of invariant hyperbolas and lines
- Bifurcation diagram and phase portrait