# Some Studies on Limit Cycle Bifurcations for Near-Hamiltonian Systems

## Dr. Maoan Han

### Department of Mathematics, Shanghai Normal University, China

September 7, 2007

3:00 - 4:00 (Refreshments at 2:30 in 201 Madison Hall)

## Abstract

As we know, the second part of the 16th Hilbert problem is to ask the maximal number of limit cycles and their distributions of polynomial vector fields of degree n on the plane. Many works have been done related to the problem although it is still open for any n > 1. A closely related problem posed by Arnold is to study the maximal number of zeros of an Abelian integral, which is called the weak 16th Hilbert problem. The study of this problem helps give a lower or upper bound for the number of limit cycles of some near-Hamiltonian systems in many cases. In this talk we outline our methods to find limit cycles for general near-Hamiltonian systems by perturbating a homoclinic or double-homoclinic loop or a center point by using the coefficients of expansions of the Abelian integral or changing stability. Using this method we can give a lower bound for the number of limit cycles for some near-Hamiltonian systems. We also mention the new development of the method to some cases with a nilpotent singular point.