Principal part of multi-parameter displacement functions

AbstractIn this paper we investigate planar polynomial multi-parameter deformations of Hamiltonian vector fields. We study first all coefficients in the development of the displacement function on a transversal to the period annulus. We show that they can be expressed through iterated integrals, whose length is bounded by the degree of the monomials.A second result expresses the principal terms in the division of the displacement function in the Bautin ideal. More precisely, the principal terms in its division in a reduced basis of the Bautin ideal are given by iterated integrals. Our approach is algorithmic and generalizes Françoise algorithm for one-parameter families

Multiplicity of fixed points and growth of epsilon-neighbourhoods of orbits

We study the relationship between the multiplicity of a fixed point of a function g, and the dependence on epsilon of the length of epsilon-neighborhood of any orbit of g, tending to the fixed point. The relationship between these two notions was discovered before (Elezovic, Zubrinic, Zupanovic) in the differentiable case, and related to the box dimension of the orbit. Here, we generalize these results to non-differentiable cases introducing a new notion of critical Minkowski order. We study the space of functions having a development in a Chebyshev scale and use multiplicity with respect to this space of functions. With the new definition, we recover the relationship between multiplicity of fixed points and the dependence on epsilon of the length of epsilon-neighborhoods of orbits in non-differentiable cases. Applications include in particular Poincare maps near homoclinic loops and hyperbolic 2-cycles, and Abelian integrals. This is a new approach to estimate the cyclicity, by computing the length of the epsilon-neighborhood of one orbit of the Poincare map (for example numerically), and by comparing it to the appropriate scale.Comment: 29 pages, 2 figures, to appear in Journal of Differential Equation

Realization of analytic moduli for parabolic Dulac germs

International audienceIn a previous paper [P. Mardešić and M. Resman. Analytic moduli for parabolic Dulac germs. Russian Math. Surveys , to appear, 2021, arXiv:1910.06129v2.] we determined analytic invariants, that is, moduli of analytic classification, for parabolic generalized Dulac germs. This class contains parabolic Dulac (almost regular) germs, which appear as first-return maps of hyperbolic polycycles. Here we solve the problem of realization of these moduli

Analytic moduli for parabolic Dulac germs

International audienceThis paper gives moduli of analytic classification for parabolic Dulac germs (that is, almost regular germs). Dulac germs appear as first return maps of hyperbolic polycycles. Their moduli are given by a sequence of ecalle-Voronin-type germs of analytic diffeomorphisms. The result is stated in a broader class of parabolic generalized Dulac germs having power- logarithmic asymptotic expansions