C1 integrability via periodic orbits
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LLIBRE, Jaume. C1 integrability via periodic orbits. In: Mathematics and IT: Research and Education. 1-3 iulie 2021, Chişinău. Chișinău, Republica Moldova: 2021, pp. 51-52.
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Mathematics and IT: Research and Education 2021
Conferința "Mathematics and IT: Research and Education "
Chişinău, Moldova, 1-3 iulie 2021

C1 integrability via periodic orbits

Pag. 51-52

Llibre Jaume
Universitat Autònoma de Barcelona
Disponibil în IBN: 30 iunie 2021


These last years the Ziglin’s and the Morales-Ramis’ theories has been used for studying the non-meromorphic integrability of an autonomous differential system. 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 nonequilibrium solutions as function of the complex time using the monodromy group of their variational equations. Ziglin’s theory was extended to the socalled 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 [6] 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 Poincar´e (see Arnold [1]), 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 Poincar´e’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 Poincar´e was forgotten in the mathematical community until that modern Russian mathematicians (specially Kozlov) have recently publish on it (see [1, 4]). We shall apply Poincar´e’s results for studying the C1 integrability of the Lorenz system, the Rossler system, the Michelson system, the H´enon–Heiles Hamiltonian system and the Yang-Mills Hamiltonian system (see [2, 3, 5]).