Zhifen Zhang

Linear estimate for the number of zeros of Abelian integrals for quadratic isochronous centres

The main objective of this paper is to provide an explicit and fairly accurate upper bound for the number of zeros of Abelian integrals defined by quadratic isochronous centres when we perturb them inside the class of all polynomial systems of degree n.

On the limit cycles of polynomial differential systems with homogeneous nonlinearities

We study the limit cycles of a class of cubic polynomial differential systems in the plane and their global shape using the averaging theory. More specifically, we analyze the global shape of the limit cycles which bifurcate: first, from a Hopf bifurcation; second, from periodic orbits of the linear center ?̇? = −y, ?̇? = x; and finally from periodic orbits of the cubic centers ?̇? = −yh(x, y), ?̇? = xh(x, y) where h(x, y) = 0 is a conic. The perturbation of these systems is made inside the class of cubic polynomial differential systems having non quadratic terms.

Abelian integrals for quadratic centres having almost all their orbits formed by quartics

In this paper, we give an upper bound on the number of zeros of Abelian integrals for the quadratic centres having almost all their orbits formed by quartics, under polynomial perturbations of arbitrary degree n. The bound is linearly dependent on n.

Abelian integrals of quadratic hamiltonian vector fields with an invariant straight line

We prove that the lowest upper bound for the number of isolated zeros of the Abelian integrals associated to quadratic Hamiltonian vector fields having a center and an invariant straight line after quadratic perturbations is one.