Vassili Gelfreich

Zeeman Building, Coventry CV4 7AL, UK
Mathematics Institute, University of Warwick


Gelfreich V. G., Lerman L. M.
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 non-semisimple double zero one. It is well known that a one-parameter 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 single-round 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.
Keywords: Hamiltonian bifurcation, homoclinic orbit, separatrix splitting, asymptotics beyond all orders
Citation: Gelfreich V. G., Lerman L. M.,  Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 635-655
Gelfreich V. G., Turaev D. V.
We show that a generic area-preserving two-dimensional map with an elliptic periodic point is $C^\omega$-universal, i.e., its renormalized iterates are dense in the set of all real-analytic symplectic maps of a two-dimensional disk. The results naturally extend onto Hamiltonian and volume-preserving flows.
Keywords: homoclinic tangency, wild hyperbolic set, Newhouse phenomenon, Hamiltonian system, area-preserving map, volume-preserving flow, exponentially small splitting, KAM theory
Citation: Gelfreich V. G., Turaev D. V.,  Universal dynamics in a neighborhood of a generic elliptic periodic point, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 2-3, pp. 159-164
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
Gelfreich V. G.
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.
Citation: Gelfreich V. G.,  Analitical Invariants of Conformal Transformations. A Dynamical System Approach, Regular and Chaotic Dynamics, 1998, vol. 3, no. 4, pp. 40-48

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