On the non-integrability of a generalized Darboux Halphen system
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.
scientific article; zbMATH DE number 5078786
scientific article