Weigu Li

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.

On the differentiability of first integrals of two dimensional flows

By using techniques of differential geometry we answer the following open problem proposed by Chavarriga, Giacomini, Gine, and Llibre in 1999. For a given two dimensional flow, what is the maximal order of differentiability of a first integral on a canonical region in function of the order of differentiability of the flow? Moreover, we prove that for every planar polynomial differential system there exist finitely many invariant curves and singular points γ i , i = 1, 2, , l, such that R 2 \ (∪ l i=1 γ i ) has finitely many connected open components, and that on each of these connected sets the system has an analytic first integral. For a homogeneous polynomial differential system in R 3 , there exist finitely many invariant straight lines and invariant conical surfaces such that their complement in R 3 is the union of finitely many open connected components, and that on each of these connected open components the system has an analytic first integral.

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.

Planar analytic vector fields with generalized rational first integrals

The main purpose of this paper is to characterize a germ of planar holomorphic vector field at an elementary singular point having a generalized rational first integral. Our results generalize a result due to Poincare on a necessary condition of the existence of a rational first integral for planar polynomial systems. As two applications of our main result, we give the necessary and sufficient conditions on the existence of rational first integral for planar quadratic systems having either a weak nondegenerate singular point, or a degenerate elementary singular point.

Poincare theorems for random dynamical systems

In this paper, we establish analytic conjugacy theorems of Poincare type and Poincare–Dulac type for analytic random dynamical systems based on their Lyapunov exponents.

Polynomial and linearized normal forms for almost periodic difference systems

In this paper, we prove the analytic conjugacy theorems of Poincaré type for almost periodic difference systems based on the Lyapunov exponents of corresponding reduced systems.

Normal forms for almost periodic differential systems

In this paper we prove smooth conjugate theorems of Sternberg type for almost periodic differential systems, based on the Lyapunov exponents of the corresponding reduced systems.