Structural stability of planar quasi-homogeneous vector fields
Let \({\mathbf t}=(t_1,t_2)\), where \(t_1\) and \(t_2\) are relatively prime nonnegative integers. Denote by \({\mathcal P}^{\mathbf t}_k\) the set of polynomials \(f\) of two variables such that \[ f(\varepsilon ^{t_1}x,\varepsilon ^{t_2}y)=\varepsilon ^{k}f(x,y). \] The authors consider the space \({\mathcal Q}^{\mathbf t}_{k}\) of planar vector fields \(F=(P,Q)\), where \[ P\in {\mathcal P}^{\mathbf t}_{k+t_1},\quad Q\in {\mathcal P}^{\mathbf t}_{k+t_2}. \] They obtain conditions on a vector field \(F\in {\mathcal Q}^{\mathbf t}_{k}\) under which \(F\) is topologically equivalent to the perturbed fields \(H=F+\varepsilon G\), where \(\varepsilon\) is a small parameter and \(G\in {\mathcal Q}^{\mathbf t}_{k}\).
scientific article