Natalia Gelfreikh

7/9 Universitetskaya Emb. St Petersburg 199034 Rus
Saint-Petersburg State University


Gelfreikh N. G.
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.
Keywords: area-preserving maps, resonant fixed point, normal form, bifurcation
Citation: Gelfreikh N. G.,  Normal Forms for Three-parameter Families of Area-preserving Maps near an Elliptic Fixed Point, Regular and Chaotic Dynamics, 2018, vol. 23, no. 3, pp. 273-290
Gelfreich V. G., Gelfreikh N. G.
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.
Keywords: area-preserving map, unique normal form, parabolic fixed point
Citation: Gelfreich V. G., Gelfreikh N. G.,  Unique normal forms for area preserving maps near a fixed point with neutral multipliers, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 300-318

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