On the configuration of limit cycles in planar vector fields

Limit cycles for planar semi-quasi-homogeneous polynomial vector fields

This paper is concerned with the limit cycles for planar semi-quasi-homogeneous polynomial systems. We give some explicit criteria for the nonexistence and existence of periodic orbits. Let $N=N(p,q,m,n)$ be the maximum number of limit cycles of such system. A lower bound is given for $N=N(p,q,m,n)$. The cyclicity and center problem are studied for some subfamilies of semi-quasi-homogeneous polynomial systems. Our results generalize those obtained for polynomial semi-homogeneous systems.

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.

Uniqueness of limit cycles for a class of planar vector fields

In this paper we give sufficient conditions to ensure uniqueness of limit cycles for a class of planar vector fields. We also exhibit a class of examples with exactly one limit cycle.

Limit cycles of planar system defined by the sum of two quasi-homogeneous vecter fields

In this paper we consider the limit cycles of the planar system $$\frac{d}{dt}(x,y)=\mathbf X_n+\mathbf X_m, $$ where $\mathbf X_n$ and $\mathbf X_m$ are quasi-homogeneous vector fields of degree $n$ and $m$ respectively. We prove that under a new hypothesis, the maximal number of limit cycles of the system is $1$. We also show that our result can be applied to some systems when the previous results are invalid. The proof is based on the investigations for the Abel equation and the generalized-polar equation associated with the system, respectively. Usually these two kinds of equations need to…

Number of limit cycles for planar systems with invariant algebraic curves

For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadrat…

具有两个结点的不连续平面分段线性系统至多有两个极限环

The discontinuous planar piecewise linear system with two nodes has at most two limit cycles

针对由直线分隔且两子系统均为结点的平面分段线性系统，论文借助 Poincare 半映射与后继函数分析，证明计入重数后极限环数量至多为两个。