On the Chebyshev property of certain Abelian integrals near a polycycle

F. Dumortier and R. Roussarie formulated in (Discrete Contin. Dyn. Syst. 2 (2009) 723-781] a conjecture concerning the Chebyshev property of a collection I₀,I₁,...,In of Abelian integrals arising from singular perturbation problems occurring in planar slow-fast systems. The aim of this note is to show the validity of this conjecture near the polycycle at the boundary of the family of ovals defining the Abelian integrals. As a corollary of this local result we get that the linear span ⟨I₀,I₁,...,In⟩ is Chebyshev with accuracy k = k(n)

Limit Cycles Bifurcated from Some Z

We study the number and distribution of limit cycles of some planar Z4-equivariant quintic near-Hamiltonian systems. By the theories of Hopf and heteroclinic bifurcation, it is proved that the perturbed system can have 24 limit cycles with some new distributions. The configurations of limit cycles obtained in this paper are new

Limit cycles of cubic polynomial differential systems with rational first integrals of degree 2

Agraïments: FEDER-UNAB-10-4E-378, and a CAPES Grant No. 88881. 030454/2013-01 from the program CSF-PVE. The second authors is partially supported by the project CAPES Grant No. 88881.030454/2013-01 from the program CSF-PVE and CNPq grant "Projeto Universal 472796/2013-5". The second author is supported by CAPES/GDU - 7500/13-0. The last author is supported by FAPESP-2010/17956-1.The main goal of this paper is to study the maximum number of limit cycles that bifurcate from the period annulus of the cubic centers that have a rational first integral of degree 2 when they are perturbed inside the class of all cubic polynomial differential systems using the averaging theory. The computations of this work have been made with Mathematica and Mapl