Center cyclicity of a family of quartic polynomial differential system

In this paper we study the cyclicity of the centers of the quartic polynomial family written in complex notation as $$= i z z (A z^2 B z C ^2 ),$$ where A,B,C C. We give an upper bound for the cyclicity of any nonlinear center at the origin when we perturb it inside this family. Moreover we prove that this upper bound is sharp

Vanishing set of inverse Jacobi multipliers and attractor/repeller sets

In this paper, we study conditions under which the zero-set of the inverse Jacobi multiplier of a smooth vector field contains its attractor/repeller compact sets. The work generalizes previous results focusing on sink singularities, orbitally asymptotic limit cycles, and monodromic attractor graphics. Taking different flows on the torus and the sphere as canonical examples of attractor/repeller sets with different topologies, several examples are constructed illustrating the results presented

On the periodic orbit bifurcating from a Hopf bifurcation in systems with two slow and one fast variables

The Hopf bifurcation in slow-fast systems with two slow variables and one fast variable has been studied recently, mainly from a numerical point of view. Our goal is to provide an analytic proof of the existence of the zero Hopf bifurcation exhibited for such systems, and to characterize the stability or instability of the periodic orbit which borns in such zero Hopf bifurcation. Our proofs use the averaging theory.The first and third authors are partially supported by a MICINN grant number MTM2011-22877 and by an AGAUR grant number 2009SGR 381. The second author is partially supported by a MICINN/FEDER grant number MTM2008- 03437, by an AGAUR grant number 2009SGR 410 and by ICREA Academia

On the multiple zeros of a real analytic function with applications to the averaging theory of differential equations

In this work we improve the classical averaging theory applied to -families of analytic T-periodic ordinary differential equations in standard form defined on R. First we characterize the set of points z_0 in the phase space and the parameters where T-periodic solutions can be produced when we vary a small parameter . Second we expand the displacement map in powers of the parameter whose coefficients are the averaged functions. The main contribution consists in analyzing the role that have the multiple zeros z_0 R of the first non-zero averaged function. The outcome is that these multiple zeros can be of two different classes depending on whether the points (z_0, ) belong or not to the analytic set defined by the real variety associated to the ideal generated by the averaged functions in the Noetheriang ring of all the real analytic functions at (z_0, ). Next we are able to bound the maximum number of branches of isolated T-periodic solutions that can bifurcate from each multiple zero z_0. Sometimes these bounds depend on the cardinalities of minimal bases of the former ideal. Several examples illustrate our results

The Hopf cyclicity of the centers of a class of quintic polynomial vector fields

Agraïments: FEDER-UNAB10-4E-378We consider families of planar polynomial vector fields having a singularity with purely imaginary eigenvalues for which a basis of its Bautin ideal B is known. We provide an algorithm for computing an upper bound of the Hopf cyclicity less than or equal to the Bautin depth of B. We also present a method for studying the cyclicity problem for the Hamiltonian and the timereversible centers without the necessity of solving previously the Dulac complex center problem associated to the larger complexified family. As application we analyze the Hopf cyclicity of the quintic polynomial family written in complex notation as ˙z = iz+zz¯(Az3+Bz2z¯+Czz¯2+Dz¯3)

Cyclicity of a simple focus via the vanishing multiplicity of inverse integrating factors

First we provide new properties about the vanishing multiplicity of the inverse integrating factor of a planar analytic differential system at a focus. After we use this vanishing multiplicity for studying the cyclicity of some simple foci of several classes of planar analytic differential systems