# Journal of Nonlinear Mathematical Physics

Volume 25, Issue 1, February 2018, Pages 106 - 121

# Meromorphic and formal first integrals for the Lorenz system

Authors
Kaiyin Huang
School of Mathematics, Jilin University, Changchun 130012, P. R. China,keiyinhuang@gmail.com
Shaoyun Shi
School of Mathematics, Jilin University, Changchun 130012, P. R. China
State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, P. R. China,shisy@jlu.edu.cn
Wenlei Li*
School of Mathematics, Jilin University, Changchun 130012, P. R. China
Beijing Computational Science Research Center, ZPark II, No. 10 Dongbeiwang, West Road, Haidian District, Beijing 100094, China,lwlei@jlu.edu.cn
*Corresponding author.
Corresponding Author
Wenlei Li
Received 17 September 2017, Accepted 13 October 2017, Available Online 6 January 2021.
DOI
10.1080/14029251.2018.1440745How to use a DOI?
Keywords
Meromorphic first integrals; formal first integrals; Lorenz system
Abstract

The Lorenz system

x˙=σ(yx),y˙=rxyxz,z˙=βz+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 σβ ≠ 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(r1)/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).

© 2018 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
25 - 1
Pages
106 - 121
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2018.1440745How to use a DOI?
© 2018 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

TY  - JOUR
AU  - Kaiyin Huang
AU  - Shaoyun Shi
AU  - Wenlei Li
PY  - 2021
DA  - 2021/01/06
TI  - Meromorphic and formal first integrals for the Lorenz system
JO  - Journal of Nonlinear Mathematical Physics
SP  - 106
EP  - 121
VL  - 25
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1440745
DO  - 10.1080/14029251.2018.1440745
ID  - Huang2021
ER  -