On Periodic Orbits of Vector Fields in Arbitrary Dimension Via Autonomous and Nonautonomous Inverse Jacobi Multipliers

• Isaac A. Garcia ; Ernest Latorre ^[1] ; Susanna Maza ^[1]
  1. [1] Universitat de Lleida

     Universitat de Lleida

     Lérida, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 2, 2026
• Idioma: inglés
• DOI: 10.1007/s12346-026-01490-4
• Enlaces
  • Texto completo  1    2  
• Resumen

  • We investigate periodic orbits of C1 autonomous vector fields in Rn using inverse Jacobi multipliers that may depend explicitly on time.We establish a localization principle for T -periodic orbits in arbitrary dimension, extending known planar results and deriving nonexistence conditions through the relation between the time-slices V(0, ·) and V(T , ·). We further characterize hyperbolicity and orbital stability, including a decomposition of characteristic multipliers along invariant surfaces associated with autonomous inverse Jacobi multipliers. A test for the algebraicity of periodic orbits in 3-dimensional vector fields is given based on non-autonomous inverse Jacobi multipliers.

    The interplay between normalizers, inverse Jacobi multipliers and invariants is analyzed, with applications to the Lorenz and Rössler systems.

• Referencias bibliográficas
  • 1. Berrone, L.R., Giacomini, H.: Inverse Jacobi multipliers. Rend. Circ. Mat. Palermo(2) 52, 77–130 (2003)
  • 2. Buic˘a, A., García, I.A.: Inverse Jacobi multipliers and first integrals for nonautonomous differential systems. Z. Angew. Math. Phys....
  • 3. Buic˘a, A., García, I.A., Maza, S.: Existence of inverse Jacobi multipliers around Hopf points in R3: emphasis on the center problem. J....
  • 4. Buic˘a, A., García, I.A., Maza, S.: Multiple Hopf bifurcation in R3 and inverse Jacobi multipliers. J. Differential Equations 256, 310–325...
  • 5. Chicone, C.: Ordinary Differential Equations with Applications, 2nd edn. Springer-Verlag, New York (2006)
  • 6. Enciso, A., Peralta-Salas, D.: Existence and vanishing set of inverse integrating factors for analytic vector fields. Bull. London Math....
  • 7. García, I.A., Giné, J., Llibre, J., Maza, S.: Vanishing set of inverse Jacobi multipliers and attractor/ repeller sets. Chaos 31, 013113...
  • 8. García, I.A., Giné, J., Maza, S.: Periodic solutions of second-order differential equations with twodimensional Lie point symmetry algebra....
  • 9. García, I.A., Grau,M.: A survey on the inverse integrating factor. Qual. Theory Dyn. Syst. 9, 115–166 (2010)
  • 10. García, I.A., Maza, S.: Non-autonomous inverse Jacobi multipliers and periodic orbits of planar vector fields. Commun. Nonlinear Sci....
  • 11. García, I.A., Shafer, D.S.: Integral invariants and limit sets of planar vector fields. J. Differential Equations 217, 363–376 (2005)
  • 12. Giacomini, H., Grau, M.: On the stability of limit cycles for planar differential systems. J. Differential Equations 213, 368–388 (2005)
  • 13. Giacomini, H., Llibre, J., Viano, M.: On the nonexistence, existence, and uniqueness of limit cycles. Nonlinearity 9, 501–516 (1996)
  • 14. Hydon, P.: Symmetry Methods for Differential Equations: A beginner’s guide. Cambridge Texts in Applied Mathematics. Cambridge University...
  • 15. Jacobi, C.G.J.: Sul principio dell’ultimo moltiplicatore, e suo uso come nuovo principio generale di meccanica. Giornale Arcadico di Scienze,...
  • 16. Llibre, J., Pantazi, Ch.: Darboux theory of integrability for a class of nonautonomous vector fields. J. Math. Phys. 50, 102705 (2009)
  • 17. Llibre, J., Zhang, X.: Invariant algebraic surfaces of theLorenz system. J.Math. Phys. 43(3), 1622–1645 (2002)
  • 18. Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (2009)
  • 19. Sparrow, C.: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Springer New York, Applied Mathematical Sciences, vol....
  • 20. Wulfman, C.E.: Limit cycles as invariant functions of Lie groups. J. Phys. A 12, 73–75 (1979)