Rocky Mountain Journal of Mathematics

Invariant curves and integrability of planar $\mathcal C^r$ differential systems

Antoni Ferragut and Jaume Llibre

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We improve the known expressions of the $\mathcal {C}^r$ differential systems in the plane having a given $\mathcal {C}^{r+1}$ invariant curve, or a given $\mathcal {C}^{r+1}$ first integral. Their application to polynomial differential systems having either an invariant algebraic curve, or a first integral, also improves the known results on such systems.

Article information

Rocky Mountain J. Math., Volume 48, Number 5 (2018), 1537-1550.

First available in Project Euclid: 19 October 2018

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Zentralblatt MATH identifier

Primary: 34A05: Explicit solutions and reductions 34A34: Nonlinear equations and systems, general 34A55: Inverse problems 34C05: Location of integral curves, singular points, limit cycles 37C10: Vector fields, flows, ordinary differential equations

Planar differential system $\mathcal C^r$ function invariant curve exponential factor cofactor Darboux theory of integrability


Ferragut, Antoni; Llibre, Jaume. Invariant curves and integrability of planar $\mathcal C^r$ differential systems. Rocky Mountain J. Math. 48 (2018), no. 5, 1537--1550. doi:10.1216/RMJ-2018-48-5-1537.

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  • C. Christopher and C. Li, Limit cycles of differential equations, Adv. Courses Math., Birkhäuser Verlag, Basel, 2007.
  • C. Christopher, J. Llibre, Ch. Pantazi and S. Walcher, Inverse problems for multiple invariant curves, Proc. Roy. Soc. Edinburgh 137 (2007), 1197–1226.
  • ––––, Inverse problems in Darboux' theory of integrability, Acta Appl. Math. 120 (2012), 101–126.
  • C. Christopher, J. Llibre, Ch. Pantazi and X. Zhang, Darboux integrability and invariant algebraic curves for planar polynomial systems, J. Phys. Math. 35 (2002), 2457–2476.
  • C. Christopher, J. Llibre and J.V. Pereira, Multiplicity of invariant algebraic curves in polynomial vector fields, Pacific J. Math. 229 (2007), 63–117.
  • G. Darboux, Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré $($Mélanges$)$, Bull. Sci. Math. 2 (1878), 60–96, 123–144, 151–200.
  • F. Dumortier, J. Llibre and J.C. Artés, Qualitative theory of planar differential systems, Universitext, Springer, Berlin, 2006.
  • I.A. Garcí a and J. Giné, Generalized cofactors and nonlinear superposition principles, Appl. Math. Lett. 16 (2003), 1137–1141.
  • I.A. Garcí a and M. Grau, A survey on the inverse integrating factor, Qual. Th. Dynam. Syst. 9 (2010), 115–166.
  • H. Giacomini, J. Giné and M. Grau, Integrability of planar polynomial differential systems through linear differential equations, Rocky Mountain J. Math. 36 (2006), 457–485.
  • J. Llibre and R. Ramí rez, Inverse problems in ordinary differential equations and applications, Progr. Math. 313 (2016).
  • J. Llibre and C. Valls, Liouvillian first integrals for generalized Riccati polynomial differential systems, Adv. Nonlin. Stud. 15 (2015), 951–961.
  • D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields. Bifurcations and periodic orbits of vector fields, NATO Adv. Sci. Inst. Math. Phys. Sci. 408 (1993), 429–467.
  • Y. Ye, Theory of limit cycles, Transl. Math. Mono. 66 (1986).