# Moscow Mathematical Journal

Volume 21, Issue 2, April–June 2021 pp. 413–426.

Smooth Local Normal Forms of Hyperbolic Roussarie Vector Fields

**Authors**:
N.G. Pavlova (1) and A.O. Remizov (2)

**Author institution:**(1) Moscow Institute of Physics and Technology,
Institutskii per. 9, 141700 Dolgoprudny, Russia;

Institute of Control Sciences (RAS),
Profsoyuznaya str. 65, 117997 Moscow, Russia;

Peoples' Friendship University of Russia, Mikluho-Maklaya str. 6, 117198 Moscow, Russia

(2) Moscow Institute of Physics and Technology, Institutskii per. 9, 141700 Dolgoprudny, Russia

**Summary: **

In 1975, Roussarie studied a special class of vector fields, whose singular points fill a submanifold of codimension two and the ratio between two non-zero eigenvalues $\lambda_1:\lambda_2=1:-1$. He established a smooth orbital normal form for such fields at points where $\lambda_{1,2}$ are real and the quadratic part of the field satisfied a certain genericity condition. In this paper, we establish smooth orbital normal forms for such fields at points where this condition fails. Moreover, we prove similar results for vector fields, whose singular points fill a submanifold of codimension two and the ratio between two non-zero eigenvalues $\lambda_1:\lambda_2=p:-q$ with arbitrary integers $p,q \ge 1$.

2020 Math. Subj. Class. Primary: 34C20; Secondary: 34C05

**Keywords:**Vector field, singular point, resonance, normal form.

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