Non-integrability of a class of Painlevé IV equations as Hamiltonian systems

Shi, Shaoyun; Li, Wenlei

doi:10.1063/1.4826601

Cited by 1 publication

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“…Roughly speaking, their results show that if Hamiltonian system with n degrees of freedom admits n meromorphic first integrals which are in involution and independent, then the identity component of the differential Galois group of the normal variational equations along a non-equilibrium solution must be commutative. Many scholars applied Morales-Ramis theory to large numbers of practical models, see for instance [8,21,26]. For general dynamical systems, the authors in [18,19] presented some results which reveal the relation between the meromorphic first integrals and the differential Galois group of normal variational equations, which will be used to prove Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

Meromorphic and formal first integrals for the Lorenz system

Huang¹,

Shi²,

Li³

2021

JNMP

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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).

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