Limit Cycles Coming from Some Uniform Isochronous Centers

© 2016 by De Gruyter. This article is about the weak 16th Hilbert problem, i.e. we analyze how many limit cycles can bifurcate from the periodic orbits of a given polynomial differential center when it is perturbed inside a class of polynomial differential systems. More precisely, we consider the uniform isochronous centers x =-y+x 2 y(x 2 +y 2 ) n ,y =x+xy 2 (x 2 +y 2 ) n ,$ {x}= -y + x2 y (x2 + y2)n, \quad \dot{y}= x + x y2 (x2 + y2)n, $ of degree 2n+3${2n+3}$ and we perturb them inside the class of all polynomial differential systems of degree 2n+3${2n+3}$ . For n=0,1${n=0,1}$ we provide the maximum number of limit cycles, 3 and 8 respectively, that can bifurcate from the periodic orbits of these centers using averaging theory of first order, or equivalently Abelian integrals. For n = 2 we show that at least 12 limit cycles can bifurcate from the periodic orbits of the center.