Quadratic vector fields in \({\mathbb{C}}\mathbb{P}^2\) with two saddle-node type singularities at infinity
The author studies some group properties of the monodromy transformation of a family of vector fields in the projective space \(\mathbb{C}\mathbb{P}^2\). It is supposed that the fields considered have three different singular points at infinity, two of them saddle-nodes and that they leave the line at infinity invariant. Up to the affine change of coordinates these will be fields written as ODE: \[ \frac{dw}{dz}= \frac{p_{10}z+p_{01}w+zw-w^2} {q_{10}z+q_{01}w} \tag{1} \] and we are interested in the group properties of the monodromy transformation of (1) at infinity. Theorem 1 says that, generically, a necessary and sufficient condition for (1) to have commutative monodromy group at infinity is the invariance of the \(w\)-axis with respect to (1). Theorem 2 gives the conditions of embeddability of monodromy germs for a subfamily of (1). The work is closely related to the works of the Russian school and uses nonlinear Stokes phenomena and groups of germs of conformal mappings (Il'yashenko, Elizarov).
scientific article; zbMATH DE number 1431790
scientific article