On the limit cycles of the polynomial differential systems with a linear node and homogeneous nonlinearities

Agraïments: The second author is partially supported by NNSF of China grant number 10971133. The third author is partially supported by NNSF of China grant number 11271252 and RFDP of Higher Education of China grant 20110073110054. The first and third authors are also supported by FP7-PEOPLE-2012-IRSES-316338 of Europe.We consider the class of polynomial differential equations ˙x = λx + Pn(x, y), y˙ = µy + Qn(x, y) in R2 where Pn(x, y) and Qn(x, y) are homogeneous polynomials of degree n > 1 and λ 6= µ, i.e. the class of polynomial differential systems with a linear node with different eigenvalues and homogeneous nonlinearities. For this class of polynomial differential equations we study the existence and non-existence of limit cycles surrounding the node localized at the origin of coordinates

Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory

In this paper we classify the phase portraits in the Poincar\'e disc of the centers of the generalized class of Kukles systems $$=-y,=x ax^3y bxy^3,$$ symmetric with respect to the y-axis, and we study, using the averaging theory up to sixth order, the limit cycles which bifurcate from the periodic solutions of these centers when we perturb them inside the class of all polynomial differential systems of degree 4

Global phase portraits of Kukles differential systems with homogenous polynomial nonlinearities of degree 5 having a center and their small limit cycles

We provide the nine topological global phase portraits in the Poincaré disk of the family of the centers of Kukles polynomial differential systems of the form x = -y, y= x ax^5y bx^3y^3 cxy^5, where x,y\R and a,b,c are real parameters satisfying a^2 b^2 c^2 0. Using averaging theory up to sixth order we determine the number of limit cycles which bifurcate from the origin when we perturb this system first inside the class of all homogeneous polynomial differential systems of degree 6, and second inside the class of all polynomial differential systems of degree 6