An algebraic approach to a study of two-dimensional affine differential system (Q947004)

The autonomous polynomial differential system \[ \begin{aligned} \frac{dx^j}{dt}=\sum\limits_{m_i\in\Gamma}P_{m_i}^j(x),\quad(j=1,2)\\ P_{m_i}^j=\sum\limits_{k=0}^{m_i} \left( \begin{matrix} m_i\\ k \end{matrix} \right) \alpha^{ij}_k (x^1)^{m_i-k}(x^2)^k,\quad (j=1,2; i=1,\dots,l) \end{aligned} \tag{1} \] is considered, where \(\Gamma=\{m_i \}_{i=1}^l\) is some finite set of different non-negative integers, \(P_{m_i}^j\) are homogeneous polynomials. System (1) is denoted by \(s^2(\Gamma)\) and is considered with the group of affine transformations \(\text{Aff}(2,\mathbb{R})\) given by \[ \bar{x}^1=\alpha x^1+\beta x^2+h_1,\; \bar{x}^2=\gamma x^1+\delta x^2+h_2,\; \left( \Delta=\left| \begin{matrix} \alpha & \beta\\ \gamma & \delta \end{matrix} \right| \neq 0 \right) \] where \(\alpha\), \(\beta\), \(\gamma\), \(\delta\) are real parameters. In the reviewed article the problem of classification of \(\text{Aff}(2,\mathbb{R})\) orbits' dimension for system \(s^2(0,1)\) is studied.