Abstract
In this paper, we consider nonautonomous differential systems of arbitrary dimension and first find expressions for their inverse Jacobi multipliers and first integrals in some nonautonomous invariant set in terms of the solutions of the differential system. Given an inverse Jacobi multiplier V, we find a relation between the Poincaré translation map Π at time T that extends to arbitrary dimensions the fundamental relation for scalar equations, \({V(T, \Pi(x)) = V(0,x)\Pi'(x)}\), found in García et al. (Trans Am Math Soc 362:3591-3612, 2010). The main result guarantees the existence of continua of T-periodic solutions for T-periodic systems in the presence of T-periodic first integrals and inverse Jacobi multipliers.
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References
Amann H.: Ordinary Differential Equations. An Introduction to Nonlinear Analysis, de Gruyter Studies in Mathematics, vol. 13. Walter de Gruyter & Co., Berline (1990)
Berrone L.R., Giacomini H.: Inverse Jacobi multipliers. Rend. Circ. Mat. Palermo 52(2), 77–130 (2003)
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)
Buică A., García I.A., Maza S.: Multiple Hopf bifurcation in \({\mathbb{R}^3}\) and inverse Jacobi multipliers. J. Differ. Equa. 256, 310–325 (2014)
Chicone C.: Ordinary Differential Equations with Applications. 2nd edn. Springer, New York (2006)
Enciso A., Peralta-Salas D.: Existence and vanishing set of inverse integrating factors for analytic vector fields. Bull. Lond. Math. Soc. 41, 1112–1124 (2009)
Flores-Espinoza R.: Periodic first integrals for Hamiltonian systems of Lie type. Int. J. Geom. Meth. Mod. Phys. 8, 1169–1177 (2011)
García I.A., Grau M.: A survey on the inverse integrating factor. Qual. Theory Dyn. Syst. 9, 115–166 (2010)
García I.A., Giacomini H., Grau M.: The inverse integrating factor and the Poincaré map. Trans. Am. Math. Soc. 362, 3591–3612 (2010)
García I.A., Giacomini H., Grau M.: Generalized Hopf bifurcation for planar vector fields via the inverse integrating factor. J. Dyn. Differ. Equ. 23, 251–281 (2011)
Hale J.K.: Ordinary Differential Equations. 2nd edn. Robert E. Krieger, Huntington (1980)
Jacobi C.G.J.: Sul principio dellultimo moltiplicatore, e suo uso come nuovo principio generale di meccanica. Giornale Arcadico di Scienze. Lettere ed Arti 99, 129–146 (1844)
Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, vol. 176. American Mathematical Society (2011)
Zhang X.: Inverse Jacobian multipliers and Hopf bifurcation on center manifolds. J. Differ. Equ. 256, 3552–3567 (2014)
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The authors are partially supported by a MCYT/FEDER grant number MTM2008-00694 and by a CIRIT grant number 2014 SGR 1204.
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Buică, A., García, I.A. Inverse Jacobi multipliers and first integrals for nonautonomous differential systems. Z. Angew. Math. Phys. 66, 573–585 (2015). https://doi.org/10.1007/s00033-014-0440-7
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DOI: https://doi.org/10.1007/s00033-014-0440-7