The Estimation of the Number of Zeros of the Abelian Integrals for a Class of Hamiltonian Systems

In this paper, we discuss the estimation of the number of zeros of the Abelian integral for the quadratic system which has a periodic region with a parabola and a straight line as its boundary when we perturb the system inside the class of all polynomial systems of degree n. The main result is that the upper bound for the number of zeros of the Abelian integral associated to this system is 3n-1.

AN ESTIMATE OF THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR CUBIC VECTOR FIELDS WITH CUSPIDAL LOOP

An upper bound B(n)≤ 12{7n+12((-1) n-1)} is derived for the number of zeros of Abelian integralsI(h)=∮ Γ h g(x,y)dy-f(x,y)dx on the open interval (-112,0)∪ (0,+∞), where Γ h is the compact component of the algebraic curve H(x,y)=12y 2+13x 3+14x 4=h,f(x,y) and g(x,y) are polynomials of x and y,n= max s{ deg f(x,y), deg g(x,y)} .

Global Existence and Uniqueness of Periodic Waves for a Perturbed Combined Double-Dispersive Equation

In this paper, we study the periodic wave propagation phenomenon in elastic waveguides modeled by a combined double-dispersive partial differential equation(PDE).The traveling wave ansazt transforms the PDE model into a perturbed integrable ordinary differential equation(ODE). The global bifurcation theory is applied for the perturbed ODE model to establish the existence and uniqueness of the limit cycle, which corresponds the periodic traveling wave for the PDE model. The main tool is the Abelian integral taken from Poincaré bifurcation theory. Simulation is carried out to verify the theoretical result.

Smooth Periodic Solutions with Equal Period for KP-MEW (2,2) Equation

In this paper, the KP-MEW(2,2) equation is considered under a certain parametric condition. We prove that the equation has two isochronous centers under certain parametric conditions, and there exist two families of periodic solutions with equal period.

On the Period Function of a Class of Reversible Quadratic Centers

Abstract This paper is devoted to the study of the period function for a class of reversible quadratic system x=-2xy,y=k-1-2kx＋（k＋1）x^2-1/2y^2.We determine the monotonicity of the period function for each k ∈ R. It is proved that the period function has at most one critical point.

Limit Cycles Bifurcating from a Quadratic Reversible Lotka-Volterra System with a Center and Three Saddles

This paper is concerned with limit cycles which bifurcate from a period annulus of a quadratic reversible Lotka-Volterra system with sextic orbits.The authors apply the property of an extended complete Chebyshev system and prove that the cyclicity of the period annulus under quadratic perturbations is equal to two.

PERTURBATIONS OF A KIND OF DEGENERATE QUADRATIC HAMILTONIAN SYSTEM WITH SADDLE-LOOP

The authors investigate a kind of degenerate quadratic Hamiltonian systems with saddle-loop. Under quadratic perturbations, it is proved that the perturbed system has at most two limit cycles in the finite plane. The proof relies on a careful analysis of a related Abelian integral.

Perturbations from a kind of quartic Hamiltonians under general cubic polynomials

In this paper we investigate the perturbations from a kind of quartic Hamiltonians under general cubic polynomials. It is proved that the number of isolated zeros of the related abelian integrals around only one center is not more than 12 except the case of global center. It is also proved that there exists a cubic polynomial such that the disturbed vector field has at least 3 limit cycles while the corresponding vector field without perturbations belongs to the saddle loop case.

Quadratic perturbations of a class of quadratic reversible center of genus one

This paper is concerned with the quadratic perturbations of a one-parameter family of quadratic reversible system, having a center of genus one. The exact upper bound of the number of limit cycles emerging from the period annulus surrounding the center of the unperturbed system is given.

Bifurcations of Limit Cycles from a Quintic Hamiltonian System with a Heteroclinic Cycle

In this paper,we consider Li′enard systems of the form dx/dt=y,dy/dt=x＋bx3-x5＋ε（α＋βx2＋γx4）y,where b∈R,0〈｜ε｜〈〈1,（α,β,γ）∈D∈R3 and D is bounded.We prove that for ｜b｜〉〉1（b〈0） the least upper bound of the number of isolated zeros of the related Abelian integrals I（h）=∮Γh（α＋βx2＋γx4）ydx is 2（counting the multiplicity） and this upper bound is a sharp one.

Bifurcation of Limit Cycles for a Perturbed Piecewise Quadratic Diferential Systems

In this paper,the bifurcation of limit cycles for planar piecewise smooth systems is studied which is separated by a straight line.We give a new form of Abelian integrals for piecewise smooth systems which is simpler than before.In application,for piecewise quadratic system the existence of 10 limit cycles and 12 small-amplitude limit cycles is proved respectively.

LIMIT CYCLE BIFURCATIONS OF NON-SMOOTH NEAR-HAMILTONIAN SYSTEM

In this paper,using the Abelian integral,we investigate the limit cycle bifurcation of two classes of non-smooth near-Hamiltonian systems,and obtain the maximum number of limit cycles of the system.

A SPECIAL COMPLETE HYPER-ELLIPTIC INTEGRAL OF THE FIRST KIND

In this paper, we study a class of complete Abelian integral. We give an exact number and the upper bound of the number of zero points of the integral under some conditions.