Vassili Gelfreich
Zeeman Building, Coventry CV4 7AL, UK
Mathematics Institute, University of Warwick
Publications:
Gelfreich V. G., Lerman L. M.
Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation
2014, vol. 19, no. 6, pp. 635655
Abstract
We study the splitting of a separatrix in a generic unfolding of a degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We assume that the unperturbed fixed point has two purely imaginary eigenvalues and a nonsemisimple double zero one. It is well known that a oneparameter unfolding of the corresponding Hamiltonian can be described by an integrable normal form. The normal form has a normally elliptic invariant manifold of dimension two. On this manifold, the truncated normal form has a separatrix loop. This loop shrinks to a point when the unfolding parameter vanishes. Unlike the normal form, in the original system the stable and unstable separatrices of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies nonexistence of singleround homoclinic orbits and divergence of series in normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with the behavior of analytic continuation of the system in a complex neighborhood of the equilibrium.

Gelfreich V. G., Turaev D. V.
Universal dynamics in a neighborhood of a generic elliptic periodic point
2010, vol. 15, no. 23, pp. 159164
Abstract
We show that a generic areapreserving twodimensional map with an elliptic periodic point is $C^\omega$universal, i.e., its renormalized iterates are dense in the set of all realanalytic symplectic maps of a twodimensional disk. The results naturally extend onto Hamiltonian and volumepreserving flows.

Gelfreich V. G., Gelfreikh N.
Unique normal forms for area preserving maps near a fixed point with neutral multipliers
2010, vol. 15, no. 23, pp. 300318
Abstract
We study normal forms for families of areapreserving maps which have a fixed point with neutral multipliers $\pm 1$ at $\varepsilon = 0$. Our study covers both the orientationpreserving and orientationreversing 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 nondegeneracy 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.

Gelfreich V. G.
Analitical Invariants of Conformal Transformations. A Dynamical System Approach
1998, vol. 3, no. 4, pp. 4048
Abstract
The paper is devoted to the problem of analytical classification of conformal maps of the form $f : z \mapsto z + z^2 +\ldots$ in a neighborhood of the degenerate fixed point $z=0$. It is shown that the analytical invariants, constructed in the works of Voronin and Ecalle, may be considered as a measure of splitting for stable and unstable (semi) invariant foliations associated with the fixed point. This splitting is exponentially small with respect to the distance to the fixed point.
