On the topological classification of generic singular points of a planar vector field with the zero $(m-1)$-jet

According to the Grobman-Hartman theorem, a dynamical system defined by a finite-dimensional vector field in the neighborhood of a singular point is topologically equivalent (and even topologically conjugate) to the dynamical system defined by the linearized vector field in the generic case when the eigenvalues of the matrix of the linear part of the field at the singular point have nonzero real parts. The topological classification of such singular points is simple: the number of eigenvalues with negative real part is a complete topological invariant. The paper proposes the following generalization of these results. It shows that for a planar vector field with a zero $(m–1)$-jet at a singular point, the $m$-jet ($m > 1$) in the “generic case” determines the topological type of the singular point. It also presents a topological classification of such singular points.