On the non-integrability of a generalized Darboux Halphen system
type: Item
instance of: scholarly article
title: On the non-integrability of a generalized Darboux Halphen system
zbMATH Open document ID: 1115.34003
author: Claudia Valls
published in: Journal of Geometry and Physics
publication date: 2006-12-07T00:00:00Z
review text: The author consider a generalizes Darboux-Halphen system \[ \dot x_i=\prod_{j=1,j\neq i}^3x_j- x_i \sum_{j=1,j\neq i}^3x_j- \sum_{k=1}^3\alpha_k^2\prod_{j=1,j\neq k}^3(x_k-x_j), \quad i=1,2,3, \] where \(x_1,\) \(x_2,\) \(x_3\) are real unknown functions, \(\alpha_1,\) \(\alpha_2,\) \(\alpha_3\) are real constants. When \(\alpha_1=\alpha_2=\alpha_3=0\), the system becomes the classical Halphen system which first appeared in Darboux's work in 1878. It is proved that for every \((\alpha_1,\alpha_2,\alpha_3) \in \mathbb R^3 \backslash \{(0,0,0)\} \) the system does admit neither nonconstant formal first integrals (Theorem 1) nor nonconstant rational or Darboux global first integral (Theorem 4, 5). The author asserts that these results represent a first step toward proving that the system is chaotic.
Mathematics Subject Classification ID: 34A05
zbMATH DE Number: 5078786
zbMATH Keywords: generalized Halphen system
zbMATH Keywords: analytic and rational first integrals
reviewed by: Vladimir L. Makarov
MaRDI profile type: Publication
full work available at URL: j.geomphys.2006.02.004
OpenAlex ID: W2027129807
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DOI: 10.1016/J.GEOMPHYS.2006.02.004
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description: scientific article; zbMATH DE number 5078786