Analytic classification of resonant saddles and foci. (English) Zbl 0596.34021

Martinet, Jean; Ramis, Jean-Pierre

Singularities and dynamical systems, Proc. Int. Conf., Heraklion/Greece 1983, North-Holland Math. Stud. 103, 109-135 (1985).

[For the entire collection see Zbl 0547.00033.]
The authors classify, up to local analytic transformation of \({\mathbb{R}}^ 2\), the phase portraits near zero of the differential systems: (1) Saddles: \(\dot x=qx+...,\quad \dot y=-py+... ;\) (2) Foci: \(\dot x=y+...,\quad \dot y=-x+...,\) where p, q are relatively prime positive integers and, in both cases, the dots denote convergent series of order at least two. Some results from formal classification of resonant differential forms are recalled. The basic admission to the classification problem in question consists in the complexification of the real resonant system.

Reviewer: A.Klič

MSC:

34C20	Transformation and reduction of ordinary differential equations and systems, normal forms
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37C85	Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
34M99	Ordinary differential equations in the complex domain

Keywords:

resonant differential form; foliation; fibration; formally; equivalent systems; first order differential equation; phase portraits; Saddles; Foci

Citations:

Zbl 0547.00033