Every period annulus is both reversible and symmetric

The criticality of reversible quadratic centers at the outer boundary of its period annulus

This paper deals with the period function of the reversible quadratic centers \begin{equation*} X_{\np}=-y(1-x)\partial_x+(x+Dx^2+Fy^2)\partial_y, \end{equation*} where $\np=(D,F)\in\R^2.$ Compactifying the vector field to $\Sc^2$, the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle $\out_\np$ at the outer boundary. A critical period is an isolated critical point of the period function.…

关于拟齐次多项式向量场的周期环域的循环性

On the cyclicity of the period annulus of quasi-homogeneous polynomial vector fields

研究在单参数多项式扰动下，具有中心的拟齐次多项式向量场从周期环域分岔出的极限环个数，通过改进的Francoise算法和组合技术建立了k阶Melnikov函数孤立零点数的上界公式。

On the cyclicity of the period annulus of quadratic Hamiltonian triangle vector field

This paper is concerned with the cyclicity of the period annulus of quadratic Hamiltonian triangle vector field under quadratic perturbations. This problem has been studied by Iliev (J. Differential Equations {\bf 128}(1996)), based on the displacement function obtained by Żoładek (J. Differential Equations {\bf 109}(1994)). Recently, P. Mardešić etc. (J. Dynamical and Control Systems {\bf 17}(2011)) studied unfoldings of the Hamiltonian triangle within quadratic vector fields. It turned out that the displacement function is not precise of the form given by Żoładek. Using the corrected form of…

Invariant Curves of Almost Periodic Reversible Mappings

In this paper, we prove some invariant curve theorems for the planar almost periodic reversible mappings. As an application, we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the nonlinear oscillator $x"+g(x)x'+\varpi^{2}x+φ(x)=f(t)$ with $f(t)$ almost periodic.

Cyclicity of period annuli and principalization of Bautin ideals

We prove that the maximal number of limit cycles which bifurcate from an open period annulus under a given multi-parameter analytic deformation of a given analytic vector field is the same as in an appropriate one-parameter analytic deformation of the field, provided that this cyclicity is finite. Along the same lines we give also a bound of the cyclicity of homoclinic saddle loops.

The period functions' higher order derivatives

We prove a formula for the $n$-th derivative of the period function $T$ in a period annulus of a planar differential system. For $n = 1$, we obtain Freire, Gasull and Guillamon formula for the period's first derivative \cite{FGG}. We apply such a result to hamiltonian systems with separable variables and other systems. We give some sufficient conditions for the period function of conservative second order O.D.E.'s to be convex.