平面上含鞍结点与中心型鞍点的余维3环的三参数开折

本文应用有限光滑正规形理论研究了含有一个重数为2的鞍结点与一个中心型双曲鞍点（双曲比率为1）的余维3的平面环（Polycycle）的一般三参数开折与环性,证明了这类环至多分支出三条极限环,并且在一般性条件下环性为3,给出了分支图和相应的相图．作为应用,证明了文［4］提出的图（I192）环性为1的猜想．     

PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem

This paper is part of the program launched in (J. Differential Equations 110(1) (1994) 86) to prove the finiteness part of Hilbert's 16th problem for quadratic system, which consists in proving that 121 graphics have finite cyclicity among quadratic systems. We show that any pp-graphic through a multiplicity 3 nilpotent singularity of elliptic type which does not surround a center has finite cyclicity. Such graphics may have additional saddles and/or saddle-nodes. Altogether we show the finite cyclicity of 15 graphics of (J. Differential Equations 110(1) (1994) 86). In particular we prove the finite cyclicity of a pp-graphic with an elliptic nilpotent singular point together with a hyperbolic saddle with hyperbolicity ≠1 which appears in generic 3-parameter families of vector fields and hence belongs to the zoo of Kotova and Stanzo (Concerning the Hilbert 16th problem, American Mathematical Society Translation Series 2, Vol. 165, American Mathematical Society, Providence, RI, 1995, pp. 155-201).     

The Hopf cyclicity of Lienard systems

We study the Hopf cyclicity for general systems on the plane and establish an algebraic method to compute the Hopf cyclicity for a general Lienard system.     

Absolute cyclicity, Lyapunov quantities and center conditions

In this paper we consider analytic vector fields X₀ having a non-degenerate center point e. We estimate the maximum number of small amplitude limit cycles, i.e., limit cycles that arise after small perturbations of X₀ from e. When the perturbation (Xλ) is fixed, this number is referred to as the cyclicity of Xλ at e for λ near 0. In this paper, we study the so-called absolute cyclicity; i.e., an upper bound for the cyclicity of any perturbation Xλ for which the set defined by the center conditions is a fixed linear variety. It is known that the zero-set of the Lyapunov quantities correspond to the center conditions (Caubergh and Dumortier (2004) [6]). If the ideal generated by the Lyapunov quantities is regular, then the absolute cyclicity is the dimension of this so-called Lyapunov ideal minus 1. Here we study the absolute cyclicity in case that the Lyapunov ideal is not regular.     

Finite Cyclicity of Graphics with a Nilpotent Singularity of Saddle or Elliptic Type

In this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside C^∞ families of planar vector fields. In some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3. By blowing up the family of vector fields, we obtain all the limit periodic sets. We calculate two different types of Dulac maps in the blown-up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point. The finite cyclicity theorems are derived by a generalized derivation-division method introduced by Roussarie.     

Cyclicity of planar polycycles and ensembles with codimension 3 degeneration through a saddle-node and two hyperbolic saddles

The cyclicity of four classes of codimension 3 plnnar polycycles and ensembles containing a saddle-node and two hyperbdic saddles is dealt with. The exnct cyclicity or cyclicity bound of them is obtained by finitely-smooth normal form theory.     

Cyclicity of homoclinic loops and degenerate cubic Hamiltonians

New conditions for a planar homoclinic loop to have cyclicity two under multiple parameter perturbations have been obtained. As an application it is proved that a homoclinic loop of a nongeneric cubic Hamiltonian has cyclicity two under arbitrary quadratic perturbations.     

Cyclicity of graphs

The cyclicity of a graph is the largest integer n for which the graph is contractible to the cycle on n vertices. By analyzing the cycle space of a graph, we establish upper and lower bounds on cyclicity. These bounds facilitate the computation of cyclicity for several classes of graphs, including chordal graphs, complete n-partite graphs, n-cubes, products of trees and cycles, and planar graphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 160–170, 1999     

Cyclicity of homoclinic loops and degenerate cubic Hamiltonians

New conditions for a planar homoclinic loop to have cyclicity two under multiple parameter perturbations have been obtained. As an application it is proved that a homoclinic loop of a nongeneric cubic Hamiltonian has cyclicity two under arbitrary quadratic perturbations. Project supported by the National Natural Science Foundation of China (Grant Nos. 19531070 and 19771037).     

Quadratic perturbations of a quadratic reversible center of genus one

In this paper, we study a reversible and non-Hamitonian system with a period annulus bounded by a hemicycle in the Poincaré disk. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. This verifies some results of the conjecture given by Gautier et al.     

Bifurcation of 2-periodic orbits from non-hyperbolic fixed points

We introduce the concept of 2-cyclicity for families of one-dimensional maps with a non-hyperbolic fixed point by analogy to the cyclicity for families of planar vector fields with a weak focus. This new concept is useful in order to study the number of 2-periodic orbits that can bifurcate from the fixed point. As an application we study the 2-cyclicity of some natural families of polynomial maps.     

Cyclicity of planar homoclinic loops and quadratic integrable systems

A general method for a homoclinic loop of planar Hamiltonian systems to bifurcate two or three limit cycles under perturbations is established. Certain conditions are given under which the cyclicity of a homoclinic loop equals 1 or 2. As an application to quadratic systems, it is proved that the cyclicity of homoclinic loops of quadratic integrable and non-Hamiltonian systems equals 2 except for one case. Project supported by the National Natural Science Foundation of China.     

添加捷径对环状DEDS的影响分析

研究具有环性拓扑的离散事件动态系统添加捷径后的时序性能变化问题.分析了添加捷径对周期长度和周期时间的影响.发现添加少量捷径即可使周期长度为1的概率有显著提高.     

Hopf Cyclicity of a Class of Li\'{e}nard-Type Systems

The Hopf cyclicities of some smooth polynomial, rational polynomial and piecewise smooth Li\''{e}nard systems are studied. For two Li\''{e}nard systems with the same damping term and different restoring (or potential) terms, we provide sufficient conditions that the two systems have the same Hopf cyclicity. Then, some examples are given to illustrate the efficiency and applicability of our results.     

Hypercyclic Sequences of Differential and Antidifferential Operators

In this paper, we provide some extensions of earlier results about hypercyclicity of some operators on the Fréchet space of entire functions of several complex variables. Specifically, we generalize in several directions a theorem about hyper- cyclicity of certain infinite order linear differential operators with constant coefficients and study the corresponding property for certain kinds of “antidifferential” operators which are introduced in the paper. In addition, the existence of hypercyclic functions for certain sequences of differential operators with additional properties, for instance, boundedness or with some nonvanishing derivatives, is established.     

On Hopf bifurcations of piecewise planar Hamiltonian systems

In this paper we study the number of limit cycles appearing in Hopf bifurcations of piecewise planar Hamiltonian systems. For the case that the Hamiltonian function is a piecewise polynomials of a general form we obtain lower and upper bounds of the number of limit cycles near the origin respectively. For some systems of special form we obtain the Hopf cyclicity.     

On L-functions of cyclotomic function fields

We study two criterions of cyclicity for divisor class groups of function fields, the first one involves Artin L-functions and the second one involves “affine” class groups. We show that, in general, these two criterions are not linked.     

Cyclicity of a simple focus via the vanishing multiplicity of inverse integrating factors

First we provide new properties about the vanishing multiplicity of the inverse integrating factor of a planar analytic differential system at a focus. After we use this vanishing multiplicity for studying the cyclicity of foci with pure imaginary eigenvalues and with homogeneous nonlinearities of arbitrary degree having either its radial or angular speed independent of the angle variable in polar coordinates. After we study the cyclicity of a class of nilpotent foci in their analytic normal form.     

Limit cycle bifurcation by perturbing a cuspidal loop of order 2 in a Hamiltonian system

This paper deals with the analytical property of the first Melnikov function for general Hamiltonian systems possessing a cuspidal loop of order 2 and its expansion at the Hamiltonian value corresponding to the loop. The explicit formulas for the first coefficients of the expansion have been given. We prove that at least 13 limit cycles can bifurcate from the cuspidal loop of order 2 under certain conditions. Then we consider the cyclicity of a cuspidal loop in some Liénard and Hamiltonian systems, and determine the number of limit cycles that can bifurcate from the perturbed system.     

Cyclicity of a kind of degenerate polycycles through three singular points

This paper deals with the cyclicity of a kind of degenerate planar polycycles through a saddle-node P0 and two hyperbolic saddles P1 and P2, where the hyperbolicity ratio of the saddle P1 (which connects the saddle-node with hh-connection) is equal to 1 and that of the other saddle P2 is irrational. It is assumed that the connections between P0 to P2 and P0 to P1 keep unbroken. Then the cyclicity of this kind of polycycle is no more than m 3 if the saddle P1 is of order m and the hyperbolicity ratio of P2 is bigger than m. Furthermore, the cyclicity of this polycycle is no more than 7 if the saddle P1 is of order 2 and the hyperbolicity ratio of P2 is located in the interval (1,2).