Inverse Jacobi Multipliers: Recent Applications in Dynamical Systems

  • Adriana BuicăEmail author
  • Isaac A. García
  • Susanna Maza
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 54)


In this paper we show novel applications of the inverse Jacobi multiplier focusing on questions of bifurcations and existence of periodic solutions admitted by both autonomous and non-autonomous systems of ordinary differential equations. In the autonomous case we focus on dimension n ≥ 3 whereas in the non-autonomous we study the cases with n ≥ 2. We summarize results already published and additionally we state some recent results to appear. The principal object of this research is two fold: first to prove the existence and smoothness of inverse Jacobi multiplier V in the region of interest in the phase space and second to show that the invariant set under the flow given by the zero-set of an inverse Jacobi multiplier contains under some assumptions orbits which are relevant in its phase portrait such as periodic orbits, limit cycles, stable, unstable and center manifolds and so on. In the non-autonomous T-periodic case we show some relationships between T-periodic orbits and T-periodic inverse Jacobi multipliers.


Vector Field Periodic Orbit Center Manifold Infinitesimal Generator Linear Partial Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Belitskii, G.R.: Smooth equivalence of germs of vector fields with a single zero eigenvalue or a pair of purely imaginary eigenvalues. Funct. Anal. Appl. 20, 253–259 (1986)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berrone, L.R., Giacomini, H.: Inverse Jacobi multipliers. Rend. Circ. Mat. Palermo (2) 52, 77–130 (2003)Google Scholar
  3. 3.
    Bibikov, Y.N.: Local Theory of Nonlinear Analytic Ordinary Differential Equations. Lecture Notes in Mathematics, vol. 702. Springer, New York (1979)Google Scholar
  4. 4.
    Buică, A., García, I.A., Maza, S.: Existence of inverse Jacobi multipliers around Hopf points in \({\mathbb{R}}^{3}\): emphasis on the center problem. J. Differ. Equ. 252, 6324–6336 (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Buică, A., García, I.A.: Inverse Jacobi multipliers for non-autonomous differential systems (2013, preprint)Google Scholar
  6. 6.
    Buică, A., García, I.A., Maza, S.: Multiple Hopf bifurcation in \({\mathbb{R}}^{3}\) and inverse Jacobi multipliers (2013, preprint)Google Scholar
  7. 7.
    Buică, A., García, I.A., Maza, S.: Some remarks on inverse Jacobi multipliers around Hopf points (2013, preprint)Google Scholar
  8. 8.
    Darboux, G.: De l’emploi des solutions particulières algébriques dans l’intégration des systèmes d’équations différentielles algébriques. Acad. Sci. Paris C. R. 86, 1012 (1878)zbMATHGoogle Scholar
  9. 9.
    García, I.A., Grau, M.: A survey on the inverse integrating factor. Qual. Theory Dyn. Syst. 9, 115–166 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    García, I.A., Giacomini, H., Grau, M.: The inverse integrating factor and the Poincaré map. Trans. Am. Math. Soc. 362, 3591–3612 (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    García, I.A., Giné, J., Maza, S.: Periodic solutions of second–order differential equations with two–dimensional Lie point symmetry algebra. Nonlinear Anal. Real World Appl. 11, 4128–4140 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    García, I.A., Maza, S., Shafer, D.S.: Properties of monodromic points on center manifolds in \({\mathbb{R}}^{3}\) via Lie symmetries (2013, preprint)Google Scholar
  13. 13.
    Giacomini, H., Llibre, J., Viano, M.: On the nonexistence, existence, and uniqueness of limit cycles. Nonlinearity 9, 501–516 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jacobi, C.G.J.: Sul principio dell’ultimo moltiplicatore, e suo uso come nuovo principio generale di meccanica. Giornale Arcadico di Scienze Lettere ed Arti 99, 129–146 (1844)Google Scholar
  15. 15.
    Lie, S.: Verallgemeinerung und neue Verwertung der Jacobischen Multiplikatortheorie. Forhandlinger Christiania 1874, 255–274 (1875). Reprinted in Abhandlungen 3, 188–205. [November 1874.]Google Scholar
  16. 16.
    Llibre, J., Pantazi, C.: Darboux theory of integrability for a class of nonautonomous vector fields. J. Math. Phys. 50, 102705, 19 pp (2009)Google Scholar
  17. 17.
    Llibre, J., Peralta-Salas, D.: A Note on the first integrals of vector fields with integrating factors and normalizers. SIGMA 8, 1–9 (2012)MathSciNetGoogle Scholar
  18. 18.
    Mahdi, A., Pessoa, C., Shafer, D.S.: Centers on center manifolds in the Lü system. Phys. Lett. A 375, 3509–3511 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mello, L.F., Coelho, S.F.: Degenerate Hopf bifurcations in the Lü system. Phys. Lett. A 373, 1116–1120 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nucci, M.C., Leach, P.G.L.: Jacobi’s last multiplier and symmetries for the Kepler problem plus a lineal story. J. Phys. A 37, 7743–7753 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nucci, M.C., Leach, P.G.L.: Jacobi’s last multiplier and the complete symmetry group of the Ermakov-Pinney equation. J. Nonlinear Math. Phys. 12, 305–320 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Adriana Buică
    • 1
    Email author
  • Isaac A. García
    • 2
  • Susanna Maza
    • 2
  1. 1.Department of Applied MathematicsBabeş-Bolyai UniversityCluj-NapocaRomania
  2. 2.Departament de MatemàticaUniversitat de LleidaLleidaSpain

Personalised recommendations