Lubomir Gavrilov

The infinitesimal 16th Hilbert problem in the quadratic case

Abstract.Let H(x,y) be a real cubic polynomial with four distinct critical values (in a complex domain) and letn

Petrov modules and zeros of Abelian integrals

{X_H} = {H_y}frac{partial }{{partial x}} - {H_x}frac{partial }{{partial y}}

THE DISPLACEMENT MAP ASSOCIATED TO POLYNOMIAL UNFOLDINGS OF PLANAR HAMILTONIAN VECTOR FIELDS

be the corresponding Hamiltonian vector field. We show that there is a neighborhood ? of XH in the space of all quadratic plane vector fields, such that any X∈? has at most two limit cycles.

Isochronicity of plane polynomial Hamiltonian systems

Abstract We prove that the Petrov module P f associated to an arbitrary semiweighted homogeneous polynomial f ϵ C[ x , y ] is free and finitely generated. We compute its generators and use this to obtain a lower bound for the maximal number of zeros of complete Abelian integrals.

Complete hyperelliptic integrals of the first kind and their non-oscillation

<abstract abstract-type=TeX><p>We study the displacement map associated to small one-parameter polynomial unfoldings of polynomial Hamiltonian vector fields on the plane. Its leading term, the generating function <i>M</i>(<i>t</i>), has an analytic continuation in the complex plane and the real zeroes of <i>M</i>(<i>t</i>) correspond to the limit cycles bifurcating from the periodic orbits of the Hamiltonian flow. We give a geometric description of the monodromy group of <i>M</i>(<i>t</i>) and use it to formulate sufficient conditions for <i>M</i>(<i>t</i>) to satisfy a differential equation of Fuchs or Picard-Fuchs type. As examples, we consider in more detail the Hamiltonian vector fields [inline-graphic xmlns:xlink=http://www.w3.org/1999/xlink xlink:href=01i /] and [inline-graphic xmlns:xlink=http://www.w3.org/1999/xlink xlink:href=02i /], possessing a rotational symmetry of order two and three, respectively. In both cases <i>M</i>(<i>t</i>) satisfies a Fuchs-type equation but in the first example <i>M</i>(<i>t</i>) is always an Abelian integral (that is to say, the corresponding equation is of Picard-Fuchs type) while in the second one this is not necessarily true. We derive an explicit formula of <i>M</i>(<i>t</i>) and estimate the number of its real zeroes.

Generalized Jacobians of spectral curves and completely integrable systems

We study isochronous centres of plane polynomial Hamiltonian systems, and more generally, isochronous Morse critical points of complex polynomial Hamiltonian functions. Our first result is that if the Hamiltonian function H is a non-degenerate semi-weighted homogeneous polynomial, then it cannot have an isochronous Morse critical point, unless the associate Hamiltonian system is linear, that is to say H is of degree two. Our second result gives a topological obstruction for isochronicity. Namely, let be a continuous family of one-cycles contained in the complex level set , and vanishing at an isochronous Morse critical point of H, as . We prove that if H is a good polynomial with only simple isolated critical points and the level set contains a single critical point, then represents a zero homology cycle on the Riemann surface of the algebraic curve . We give several examples of `non-trivial complex Hamiltonians with isochronous Morse critical points and explain how their study is related to the famous Jacobian conjecture.

Quadratic perturbations of quadratic codimension-four centers

Let P(x) be a real polynomial of degree 2g + 1, H = y 2 + P(x) and δ(h) be an oval contained in the level set {H = h}. We study complete Abelian integrals of the form I(h) = ∫ δ(h) (α 0 + α 1 x +... + α g-1 x g-1 /y)dx y, h ∈ Σ, where α i are real and E ⊂ R is a maximal open interval on which a continuous family of ovals {δ(h)} exists. We show that the g-dimensional real vector space of these integrals is not Chebyshev in general: for any g > 1, there are hyperelliptic Hamiltonians H and continuous families of ovals δ(h) ⊂ {H = h}, h E Σ, such that the Abelian integral I(h) can have at least [3/2g] - 1 zeros in Σ. Our main result is Theorem 1 in which we show that when g = 2, exceptional families of ovals {δ(h)} exist, such that the corresponding vector space is still Chebyshev.

Two-dimensional Fuchsian systems and the Chebyshev property ☆

Abstract. Consider an ordinary differential equation which has a Lax pair representation n

Cyclicity of period annuli and principalization of Bautin ideals

dot{A}(x)= [A(x),B(x)]

On the explicit solutions of the elliptic Calogero system

, where A(x) is a matrix polynomial with a fixed regular leading coefficient and the matrix B(x) depends only on A(x). Such an equation can be considered as a completely integrable complex Hamiltonian system. We show that the generic complex invariant manifold n [ { A(x): {rm det}(A(x)-y I)= P(x,y) } ] of this Lax pair is an affine part of a non-compact commutative algebraic group – the generalized Jacobian of the spectral curve n