7/9 Universitetskaya Emb. St Petersburg 199034 Rus
Saint-Petersburg State University
Gelfreikh N. G.
Normal Forms for Three-parameter Families of Area-preserving Maps near an Elliptic Fixed Point
2018, vol. 23, no. 3, pp. 273-290
We study dynamics of area-preserving maps in a neighborhood of an elliptic fixed point. We describe simplified normal forms for a fixed point of codimension 3. We also construct normal forms for a generic three-parameter family which contains such degeneracy and use normal form theory to describe generic bifurcations of periodic orbits in these families.
Gelfreich V. G., Gelfreikh N. G.
Unique normal forms for area preserving maps near a fixed point with neutral multipliers
2010, vol. 15, no. 2-3, pp. 300-318
We study normal forms for families of area-preserving maps which have a fixed point with neutral multipliers $\pm 1$ at $\varepsilon = 0$. Our study covers both the orientation-preserving and orientation-reversing cases. In these cases Birkhoff normal forms do not provide a substantial simplification of the system. In the paper we prove that the Takens normal form vector field can be substantially simplified. We also show that if certain non-degeneracy conditions are satisfied no further simplification is generically possible since the constructed normal forms are unique. In particular, we provide a full system of formal invariants with respect to formal coordinate changes.