Hostname: page-component-669899f699-2mbcq Total loading time: 0 Render date: 2025-04-25T13:46:16.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrability of special quadratic systems with invariant hyperbolas

Part of: Smooth dynamical systems: general theory Qualitative theory

Published online by Cambridge University Press:  02 December 2024

Y Paulina Mancilla-Martínez Open the ORCID record for Y Paulina Mancilla-Martínez [Opens in a new window]  and
Claudia Valls Open the ORCID record for Claudia Valls [Opens in a new window]
Y Paulina Mancilla-Martínez
Affiliation:
Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, VIII-Región, Concepción, Chile ([email protected]) (corresponding author)
Claudia Valls
Affiliation:
Center for Mathematical Analysis, Geometry and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ. 45(2021), 1–90, the authors investigate about the integrability of the family QSH (the whole class of non-degenerate planar quadratic systems possessing at least one invariant hyperbola). However, some very difficult cases are left open in Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ. 45(2021), 1–90, and the main aim of this article is to study the Liouvillian integrability some of the systems that were left behind in Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ. 45(2021), 1–90.

Keywords

Darboux polynomialsinvariant algebraic curvesLiouvillian first integralspolynomial vector fieldsPuiseux series

MSC classification

Primary: 34C05: Location of integral curves, singular points, limit cycles
Secondary: 37C10: Vector fields, flows, ordinary differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 27
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bruno, A.D.. Power geometry in algebraic and differential equations. Elsevier Science (North-Holland), 2000.Google Scholar
Bruno, A.D.. Asymptotic behaviour and expansions of solutions of an ordinary differential equation. Russ. Math. Surv. 59 (3) (2004), 429–481.CrossRefGoogle Scholar
Chandrasekhar, S.. An introduction to the study of stellar structure, University of Chicago Press, 1939, Chicago.Google Scholar
Christopher, C.J. and Llibre, J.. Algebraic aspects of integrability for polynomial systems. Qualitative Theor. Planar Differential Equations 1 (1999), 71–95.Google Scholar
Christopher, C.J., Llibre, J. and Pereira, J.V.. Multiplicity of invariant algebraic curves in polynomial vector fields. Pacific J. of Math. 229 (1) (2007), 63–117.CrossRefGoogle Scholar
Darboux, G.. Mémoire sur les équations différentialles algébriques du premier ordre et du premier degré (Melanges). Bull. Sci. Math. (1978), .Google Scholar
Demina, M.V.. Invariant algebraic curves for Liénard dynamical systems. Appl. Math. Lett. 84 (2018), .CrossRefGoogle Scholar
Demina, M.V.. Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems. Phys. Lett. A 382 (20) (2018), 1353–1360.CrossRefGoogle Scholar
Demina, M.V. and Kudryashov, N.A. Meromorphic solutions in the FitzHugh-Nagumo model. Appl. Math. Lett. 82 (2018), 18–23.CrossRefGoogle Scholar
Dumortier, F., Llibre, J. and Artes, J.C.. Qualitative theory of planar differential systems, University Text, 2006, Springer.Google Scholar
Gine, J. and Llibre, J.. A note on Liouvillian integrability, J. Math. Anal. Appl. 387 (2012), 1044–1049.CrossRefGoogle Scholar
Lotka, A.J.. Analytical note on certain rhythmic relations in organic systems. Proc. Natl. Acad. Sci. 6 (1920), 410–415.CrossRefGoogle ScholarPubMed
Oliveira, R. D. S., Rezende, A. C. and Vulpe, N.. Family of quadratic dfferential systems with invariant hyperbolas: a complete classification in the space $\mathbb{R}^{12}$. Electron. J. Differ. Equ. 162 (2016), 1–50.Google Scholar
Oliveira, R. D. S., Schlomiuk, D., and Travaglini, A. M.. Geometry and integrability of quadratic systems with invariant hyperbolas. Electron. J. Qual. Theory Differ. Equ. 6 (2021), 1–56.Google Scholar
Oliveira, R.D.S, Rezende, A.C., Schlomiuk, D. and Vulpe, N.. Geometric and algebraic classification of quadratic differential systems with invariant hyperbolas. Electron. J. Differ. Equ. 295 (201), 1–112.Google Scholar
Oliveira, R.D.S, Schlomiuk, D., Travaglini, A.M., and Valls, C.. Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability. Electron. J. Qual. Theory Differ. Equ. 45(2021), 1–90. doi:10.14232/ejqtde.2021.1.45.Google Scholar
Poincaré, H.. Mémoire sur les courbes définies par les équations différentielles, J. Math. Pures Appl. 4, 1, 165–244. Ouvres de Henri Poincaré, Vol. 1, Gauthier–Villard, Paris, 1951, .Google Scholar
Poincaré, H.. Sur l’intégration algébrique des équations différentielles. C. R. Acad. Sci. Paris, 112 (1891), 761–764.Google Scholar
Poincaré, H.. Sur l’intégration algébrique des équations différentielles du premier ordre et du premier degré I, Rend. Circ. Mat. Palermo 5 (1891), 169–191.Google Scholar
Roth, J.R.. Periodic small-amplitude solutions to Volterra’s problem of two conflicting population and their application to the plasma continuity equations. J. Math. Phys. 10 (1969), 1–43.CrossRefGoogle Scholar
Singer, M.F.. Liouvillian first integrals of differential equations. Trans. Amer. Math. Soc. 333 (1992), 673–688.CrossRefGoogle Scholar
Volterra, V.. Leçons sur la théorie mathématique de la lutte pour la vie Gauthier Villars, 1931, Paris.Google Scholar