Classification of \(\mathrm{Aff}(2,\mathbb R)\)-orbit’s dimensions for quadratic differential system. (English) Zbl 1158.34324

Gherstega, N.; Orlov, V.

Bul. Acad. Ştiinţe Repub. Mold., Mat. 2008, No. 2(57), 122-126 (2008).

Let \(\text{Aff} (2,{\mathbb R})\) be the group of non-degenerated affine transformations of the plane. The authors calculate the dimension of \(\text{Aff} (2,{\mathbb R})\)-orbit for the system
\[ \frac{x^j}{dt} = a^j + \sum_{\alpha} a^j_{\alpha} x^{\alpha} + \sum_{\alpha, \beta} a^j_{\alpha \beta} x^{\alpha}x^{\beta}, \quad j, \alpha, \beta \in \{1,2\}. \tag{1} \]
The representation operators of the group \(\text{Aff} (2,{\mathbb R})\) form a 6-dimensional Lie algebra. The authors prove that the dimension of \(\text{Aff} (2,{\mathbb R})\)-orbit for the generic system (1) is equal to 6, and obtain explicit necessary and sufficient conditions of it.

Reviewer: Alexey Remizov (Porto)

MSC:

34C14

Symmetries, invariants of ordinary differential equations

Keywords:

linear-quadratic differential system; affine transformations; invariants; orbits