Limit cycle bifurcations from a nilpotent focus or center of planar systems

In this paper, we study the analytical property of the Poincare return map and the generalized focal values of an analytical planar system with a nilpotent focus or center. Then we use the focal values and the map to study the number of limit cycles of this kind of systems with parameters, and obtain some new results on the lower and upper bounds of the maximal number of limit cycles near the nilpotent focus or center.Comment: This paper was submitted to Journal of Mathematical Analysis and Application

Existence and Uniqueness of the Periodic Orbits of a Class of Cylinder Equations

AbstractIn this paper, we investigate the global behaviors of a class of cylinder equations and obtain certain sufficient conditions for the existence and uniqueness of the periodic orbits

The number of limit cycles for a family of polynomial systems

AbstractIn this paper, the number of limit cycles in a family of polynomial systems was studied by the bifurcation methods. With the help of a computer algebra system (e.g., Maple 7.0), we obtain that the least upper bound for the number of limit cycles appearing in a global bifurcation of systems (2.1) and (2.2) is 5n + 5 + (1 − (−1)n)/2 for c ≠ 0 and n for c ≡ 0

Bifurcations of limit cycles from quintic Hamiltonian systems with a double figure eight loop☆☆The work was supported in part by Australia Research Counsil under the Discovery Projects scheme (grant ID: DP0559111).

AbstractThis paper deals with Liénard equations of the form x˙=y, y˙=P(x)+yQ(x,y), with P and Q polynomials of degree 5 and 4 respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight loop. The number of limit cycles and their distributions are given by using the methods of bifurcation theory and qualitative analysis

On the zero-Hopf bifurcation of the Lotka-Volterra systems in R3

Here we study the Lotka-Volterra systems in R3, i.e. the differential systems of the form dxi/dt = xi(ri - Σ3j=1 aijxj), i = 1, 2, 3. It is known that some of these differential systems can have at least four periodic orbits bifurcating from one of their equilibrium points. Here we prove that there are some of these differential systems exhibiting at least six periodic orbits bifurcating from one of their equilibrium points. The tool for proving this result is the averaging theory of third order

Hopf Bifurcation of Limit Cycles in Discontinuous Liénard Systems

We consider a class of discontinuous Liénard systems and study the number of limit cycles bifurcated from the origin when parameters vary. We establish a method of studying cyclicity of the system at the origin. As an application, we discuss some discontinuous Liénard systems of special form and study the cyclicity near the origin

The Number of Limit Cycles of a Polynomial System on the Plane

We perturb the vector field x˙=-yC(x,y), y˙=xC(x,y) with a polynomial perturbation of degree n, where C(x,y)=(1-y2)m, and study the number of limit cycles bifurcating from the period annulus surrounding the origin