×

A generalization of the Grobman-Hartman theorem for plane vector fields, through Newton polyhedra. (English) Zbl 0744.58060

Nonlinear dynamics, Proc. Conf., Bologna/Italy 1988, 320-321 (1989).
[For the entire collection see Zbl 0741.00081.]
Let \(X\) be a smooth vector field on \(\mathbb{R}^ 2\), having a singularity at 0; if the linear part of \(X\) is hyperbolic then the Grobman-Hartman theorem says that \(X\) is topologically equivalent to its linear part in a neighborhood of 0. If the linear part is not hyperbolic we must take into account the nonlinear terms of \(X\) in order to determine the qualitative features of the phase portrait of \(X\) near 0.
We define in the following the Principal Part \(X_ \Delta\) of \(X\), which is a generalization of the linear part, and we show that under suitable non-degeneracy hypotheses this Principal Part is sufficient to determine the qualitative behaviour of \(X\) near 0 (i.e., \(X\) and \(X_ \Delta\) are topologically equivalent in a neighborhood of 0).

MSC:

37D99 Dynamical systems with hyperbolic behavior

Citations:

Zbl 0741.00081
PDFBibTeX XMLCite