Search results for "34C08"
showing 8 items of 8 documents
Darboux integrable system with a triple point and pseudo-abelian integrals
2016
We study pseudo-abelian integrals associated with polynomial perturbations of Dar-boux integrable system with a triple point. Under some assumptions we prove the local boundedness of the number of their zeros. Assuming that this is the only non-genericity, we prove that the number of zeros of the corresponding pseudo-abelian integrals is bounded uniformly for nearby Darboux integrable foliations.
Principal Poincar\'e Pontryagin Function associated to some families of Morse real polynomials
2014
It is known that the Principal Poincar\'e Pontryagin Function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is an Abelian integral also in non generic cases. In non generic cases it is an iterated integral. Uribe [17, 18] gives in a special case a precise description of the Principal Poincar\'e Pontryagin Function, an iterated integral of length at most 2, involving logarithmic functions with only one ramification at a point at infinity. We extend this result to some non isodromic families of real Morse polynomials.
The structure of the moduli spaces of toric dynamical systems
2023
We consider complex-balanced mass-action systems, or toric dynamical systems. They are remarkably stable polynomial dynamical systems arising from reaction networks seen as Euclidean embedded graphs. We study the moduli spaces of toric dynamical systems, called the toric locus: given a reaction network, we are interested in the topological structure of the set of parameters giving rise to toric dynamical systems. First we show that the complex-balanced equilibria depend continuously on the parameter values. Using this result, we prove that the toric locus of any toric dynamical system is connected. In particular, we emphasize its product structure: it is homeomorphic to the product of the s…
Redundant Picard–Fuchs System for Abelian Integrals
2001
We derive an explicit system of Picard-Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit majorants, appears only in dimension approximately two times greater than the standard Picard-Fuchs system. The result is used to obtain a partial solution to the tangential Hilbert 16th problem. We establish upper bounds for the number of zeros of arbitrary Abelian integrals on a positive distance from the critical locus. Under the additional assumption that the critical values of the Hamiltonian are distant from each other (after a proper normalization), we were…
Infinite orbit depth and length of Melnikov functions
2019
Abstract In this paper we study polynomial Hamiltonian systems d F = 0 in the plane and their small perturbations: d F + ϵ ω = 0 . The first nonzero Melnikov function M μ = M μ ( F , γ , ω ) of the Poincare map along a loop γ of d F = 0 is given by an iterated integral [3] . In [7] , we bounded the length of the iterated integral M μ by a geometric number k = k ( F , γ ) which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system F and its orbit γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations d F + ϵ ω with arbitrary high length first nonzero Melnikov function M μ along…
Pseudo-abelian integrals: Unfolding generic exponential case
2009
The search for bounds on the number of zeroes of Abelian integrals is motivated, for instance, by a weak version of Hilbert's 16th problem (second part). In that case one considers planar polynomial Hamiltonian perturbations of a suitable polynomial Hamiltonian system, having a closed separatrix bounding an area filled by closed orbits and an equilibrium. Abelian integrals arise as the first derivative of the displacement function with respect to the energy level. The existence of a bound on the number of zeroes of these integrals has been obtained by A. N. Varchenko [Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 14–25 ; and A. G. Khovanskii [Funktsional. Anal. i Prilozhen. 18 (1984), n…
Vanishing Abelian integrals on zero-dimensional cycles
2011
In this paper we study conditions for the vanishing of Abelian integrals on families of zero-dimensional cycles. That is, for any rational function $f(z)$, characterize all rational functions $g(z)$ and zero-sum integers $\{n_i\}$ such that the function $t\mapsto\sum n_ig(z_i(t))$ vanishes identically. Here $z_i(t)$ are continuously depending roots of $f(z)-t$. We introduce a notion of (un)balanced cycles. Our main result is an inductive solution of the problem of vanishing of Abelian integrals when $f,g$ are polynomials on a family of zero-dimensional cycles under the assumption that the family of cycles we consider is unbalanced as well as all the cycles encountered in the inductive proce…
Darboux systems with a cusp point and pseudo-abelian integrals
2018
International audience; We study pseudo-abelian integrals associated with polynomial deformations of Darboux systems having a cuspidal singularity. Under some genericity hypothesis we provide locally uniform boundedness of on the number of their zeros.