C^1 integrability via periodic orbits [ Back ]

UAB - Dept. Matemàtiques (C1/-128)
Jaume Llibre
Universitat Autònoma de Barcelona


These last years the Ziglin's and the Morales-Ramis' theories has been used for studying the non-meromorphic integrability of a autonomous differential systems. In some sense the Ziglin's theory is a continuation of Kovalevskaya's ideas used for studying the integrability of the rigid body because it relates the non integrability of the considered system with the behavior of some of its non-equilibrium solutions as function of the complex time using the monodromy group of their variational equations. Ziglin's theory was extended to the so-called Morales-Ramis' theory which replace the study of the monodromy group of the variational equations by the study of their Galois differential group, which is easier to analyze (see [JR] for more details and the references therein). But as Ziglin's theory the Morales-Ramis' theory only can study the non-existence of meromorphic first integrals. Kovalevskaya's idea and consequently Ziglin's and Morales-Ramis' theory go back to Poincaré (see Arnold [Ar]), who used the multipliers of the monodromy group of the variational equations associated to periodic orbits for studying the non integrability of autonomous differential systems. The main difficulty for applying Poincaré's non integrability method to a given autonomous differential system is to find for such an equation periodic orbits having multipliers different from 1.

It seems that this result of Poincaré was forgotten in the mathematical community until that modern Russian mathematicians (specially Kozlov) have recently publish on it, see [Ar, K]. We shall apply Poincaré's results for studying the $C^1$ integrability of the Lorenz system, the Rossler system, the Michelson system, the Hénon-Heiles Hamiltonian system and the Yang-Mills Hamiltonian system, see [LV, JL1, JL2].


[Ar] V. I. Arnold, Forgotten and neglected theories of Poincaré, Russian Math. Surveys 61 No.1 (2006), 1-18.

[JL1] L. Jiménez and J. Llibre, Periodic orbits and non integrability of Henon-Heiles systems, J. of Physics A: Math. Gen. 44 (2011), 205103-14pp.

[JL2] L. Jiménez and J. Llibre, Periodic orbits and non-integrability of generalized classical Yang-Mills Hamiltonian systems, J. of Math. Phys. 52 (2011), 032901.

[K] V. V. Kozlov, Integrability and non-integrability in Hamiltonian mechanics, Russian Math. Surveys 38 No.1 (1983), 1-76.

[LV] J. Llibre and C. Valls, On the C^1 non-integrability of differential systems via periodic orbits, European J. of Applied Mathematics 22 (2011), 381-391.

[JR] J. J. Morales-Ruiz, Differential Galois Theory and non-integrability of Hamiltonian systems, Progress in Math. Vol. 178, Birkhauser, Verlag, Basel, 1999.