The aim of this paper is to investigate a generalized Rikitake system from the integrability point of view. For the integrable case, we derive a family of integrable deformations of the generalized Rikitake system by altering its constants of motion, and give two classes of Hamilton–Poisson structures which implies these integrable deformations, including the generalized Rikitake system, are bi-Hamiltonian and have infinitely many Hamilton–Poisson realizations. By analyzing properties of the differential Galois groups of normal variational equations (NVEs) along certain particular solution, we show that the generalized Rikitake system is not rationally integrable in an extended Liouville sense for almost all parameter values, which is in accord with the fact that this system admits chaotic behaviors for a large range of its parameters. The non-existence of analytic first integrals are also discussed.
The Lorenz systemẋ = σ (y − x),ẏ = rx − y − xz,ż = −β z + xy, is completely integrable with two functional independent first integrals when σ = 0 and β , r arbitrary. In this paper, we study the integrability of the Lorenz system when σ , β , r take the remaining values. For the case of σ β = 0, we consider the non-existence of meromorphic first integrals for the Lorenz system, and show that it is not completely integrable with meromorphic first integrals, and furthermore, if 2 (σ + 1) 2 + 4σ (r − 1)/β is not an odd number, then it also dose not admit any meromorphic first integrals and is not integrable in the sense of Bogoyavlensky. For the case of σ = 0, β = 0, we study the existence of formal first integrals and present a necessary condition of the Lorenz system processing a time-dependent formal first integral in the form of Φ(x, y, z) exp(λt).
We investigate the analytic, rational and C 1 first integrals of the Maxwell-Bloch systeṁwhere κ, γ ⊥ , g , γ , 0 are real parameters. In addition, we prove this system is rationally non-integrable in the sense of Bogoyavlenskij for almost all parameter values.
Résumé. Nous étudions les premières intégrales analytiques, rationnelles et C 1 du système de Maxwell-où κ, γ ⊥ , g , γ , 0 sont des paramètres réels. En outre, nous prouvons que ce système est non intégrable rationnel dans le sens de Bogoyavlenskij pour presque toutes les valeurs de paramètres.
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