Limit cycles bifurcating from the periodic annulus of the weight-homogeneous polynomial centers of weight-degree 2

Agraïments: FEDER-UNAB-10-4E-378, and a CAPES grant number 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.We obtain an explicit polynomial whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of cubic polynomial differential centers when it is perturbed inside the class of all cubic polynomial differential systems. The family considered is the unique family of weight-homogeneous polynomial differential systems of weight-degree 2 with a center. The computations has been done with the help of the algebraic manipulator Mathematica

On the cyclicity of weight-homogeneous centers

Let W be a weight-homogeneous planar polynomial differential system with a center. We find an upper bound of the number of limit cycles which bifurcate from the period annulus of W under a generic polynomial perturbation. We apply this result to a particular family of planar polynomial systems having a nilpotent center without meromorphic first integral.Comment: 13 pages, no figure

Limit cycles bifurcating from the periodic orbits of the weight-homogeneous polynomial centers of weight-degree 3

In this paper we obtain two explicit polynomials, whose simple positive real roots provide the limit cycles which bifurcate from the periodic orbits of a family of polynomial differential centers of order 5, when this family is perturbed inside the class of all polynomial differential systems of order 5, whose average function of first order is not zero. Then the maximum number of limit cycles that bifurcate from these periodic orbits is 6 and it is reached. The family of centers studied completes the study about the limit cycles which can bifurcate from the periodic orbits of all centers of the weight--homogeneous polynomial differential systems of weight--degree 3, when we perturb them inside the class of all polynomial differential systems having the same degree, and whose average function of first order is not zero

Centers of discontinuous piecewise smooth quasi-homogeneous polynomial differential systems

In this paper we investigate the center problem for the discontinuous piecewise smooth quasi-homogeneous but non-homogeneous polynomial differential systems. First, we provide sufficient and necessary conditions for the existence of a center in the discontinuous piecewise smooth quasi-homogeneous polynomial differential systems. Moreover, these centers are global, and the period function of their periodic orbits is monotonic. Second, we characterize the centers of the discontinuous piecewise smooth quasi-homogeneous cubic and quartic polynomial differential systems

Limit cycles bifurcating from planar polynomial quasi-homogeneous centers

In this paper we find an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi-homogeneous center, which can be obtained using first order averaging method. This result improves the upper bounds given in [7]

Centers of weight-homogeneous polynomial vector fields on the plane

Agraïments: The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013.We characterize all centers of a planar weight-homogeneous polynomial vector fields. Moreover we classify all centers of a planar weight-homogeneous polynomial vector fields of degrees 6 and 7