一类非光滑对称三次系统的全局分支

该文对一类对称三次Hamilton系统在非光滑对称摄动下产生的极限环数目进行研究.通过多参数摄动理论和定性分析方法,得到这类在非光滑摄动下的三次系统可以存在至少19个极限环.     

一类奇摄动系统奇异极限环的不变环面分支

对一类奇异摄动系统中由奇异极限环产生的不变环面分支进行了研究并利用不变环面的分支理论,讨论了由快系统的二重极限环和三重环分支出的不变环面的存在性.     

具有十个大振幅极限环的五次多项式系统

本文研究一类五次平面多项式系统赤道极限环分支问题.运用奇点量方法,首次证明了五次多项式系统可在赤道分支出十个极限环.     

软弹簧型Duffing方程在摄动下分支出的极限环

在这篇文章中,作者用Melnikov函数方法分析了软弹簧型Duffing方程[1]在摄动下异宿轨道破裂后稳定流形与不稳定流形的相对位置,给出了方程在不同摄动下分支出极限环的条件与极限环的位置.     

三种单节点摄动对混合值逻辑网络极限集的影响

研究了三种单节点摄动对混合值逻辑网络不动点和极限环的影响.受布尔网络单节点摄动的启发,文章首先在结构矩阵的基础上提出了混合值逻辑网络中单节点摄动的定义,并利用矩阵半张量积的方法得到了单节点摄动下混合值逻辑网络的代数形式.然后,研究了结构矩阵与状态转移矩阵间的关系,并提出了三种特殊类型的单节点摄动.最后,得到了三种单节点摄动对混合值逻辑网络的不动点及极限环影响的充分必要条件.     

中心对称三次系统的一类双纽线有界周期环域的poincare分支

证明中心对称三次系统的一类双纽线有界周期环域的 poincare分支至少可以出现作对称 (3,3)分布的六个极限环 .     

一类奇摄动系统奇异极限环的不变环面分支

对一类奇异摄动系统中由奇异极限环产生的不变环面分支进行了研究并利用不变环面的分支理论,讨论了由快系统的二重极限环和三重环分支出的不变环面的存在性．     

一类五次系统的全局分支

本文利用多参数扰动法并进行定性分析,对一类三次哈密尔顿系统进行五次扰动,得到了五个极限环.     

一类五次多项式系统的奇点量与极限环分支

该文研究一类五次多项式微分系统在高次奇点与无穷远点的极限环分支问题. 该系统的原点是高次奇点, 赤道环上没有实奇点. 首先推导出计算高次奇点与无穷远点奇点量的代数递推公式,并用之计算系统原点、无穷远点的奇点量，然后分别讨论了系统原点、无穷远点中心判据. 给出了多项式系统在高次奇点分支出5个极限环同时在无穷远点分支出2个极限环的实例. 这是首次在同步扰动的条件下讨论高次奇点与无穷远点分支出极限环的问题.     

一类五次系统赤道环的稳定性与极限环分枝

本文解决了一类五次系统赤道环的稳定性与极限环分枝问题,所得的结论与二次系统的若干结论形成有趣的对比.     

Dynamics and bifurcation study on an extended Lorenz system

In this paper, we study dynamics and bifurcation of limit cycles in a recently developed new chaotic system, called extended Lorenz system. A complete analysis is provided for the existence of limit cycles bifurcating from Hopf critical points. The system has three equilibrium solutions: a zero one at the origin and two non-zero ones at two symmetric points. It is shown that the system can either have one limit cycle around the origin, or three limit cycles enclosing each of the two symmetric equilibria, giving a total six limit cycles. It is not possible for the system to have limit cycles simultaneously bifurcating from all the three equilibria. Simulations are given to verify the analytical predictions.     

具有三次曲线解的中心对称三次系统的极限环的存在性问题

本文证明了具有三次曲线解y＝αx3的中心对称三次系统可以存在极限环，从而纠正了文[1]认为具有三次曲线解的中心对称三次系统不可能存在极限环的错误结论     

LIMIT CYCLES OF A QUADRATIC HAMILTONIAN SYSTEM BY CUBIC PERTURBATIONS

In this paper,we are concerned with a cubic near-Hamiltonian system,whose unperturbed system is quadratic and has a symmetric homoclinic loop.By using the method developed in [12],we find that the system can have 4 limit cycles with 3 of them being near the homoclinic loop.Further,we give a condition under which there exist 4 limit cycles.     

The Hopf Bifurcations in the Permanent Magnet Synchronous Motors

Based on the focus quantities and other techniques, the stability properties of equilibria and the limit cycles arising from Hopf bifurcations are investigated for two models of permanent magnet synchronous motors. The first model is of surface-magnet type and can have at most two unstable small limit cycles, which are symmetric with respect to $x$-axis. The other model is of interior-magnet type and can have at most four small limit cycles in two symmetric nests.     

一类具有两个抛物线解的中心对称三次系统存在极限环的条件

本文给出了具有两个抛线解的中心对称三次系统存在在极限环的条件，它可能也是充要条件。     

Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems

In this paper, we study the bifurcation of limit cycles in piecewise smooth systems by perturbing a piecewise Hamiltonian system with a generalized homoclinic or generalized double homoclinic loop. We first obtain the form of the expansion of the first Melnikov function. Then by using the first coefficients in the expansion, we give some new results on the number of limit cycles bifurcated from a periodic annulus near the generalized (double) homoclinic loop. As applications, we study the number of limit cycles of a piecewise near-Hamiltonian systems with a generalized homoclinic loop and a central symmetric piecewise smooth system with a generalized double homoclinic loop.     

Hopf bifurcation at infinity in 3D symmetric piecewise linear systems. Application to a Bonhoeffer–van der Pol oscillator

In this work, a Hopf bifurcation at infinity in three-dimensional symmetric continuous piecewise linear systems with three zones is analyzed. By adapting the so-called closing equations method, which constitutes a suitable technique to detect limit cycles bifurcation in piecewise linear systems, we give for the first time a complete characterization of the existence and stability of the limit cycle of large amplitude that bifurcates from the point at infinity. Analytical expressions for the period and amplitude of the bifurcating limit cycles are obtained. As an application of these results, we study the appearance of a large amplitude limit cycle in a Bonhoeffer–van der Pol oscillator.     

Small limit cycles bifurcating from fine focus points in quartic order Z3-equivariant vector fields

This paper is concerned with the number of limit cycles for a quartic polynomial Z₃-equivariant vector fields. The system under consideration has a fine focus point at the origin, and three fine focus points which are symmetric about the origin. By the computation of the singular point values, sixteen limit cycles are found and their distributions are studied by using the new methods of bifurcation theory and qualitative analysis. This is a new result in the study of the second part of the 16th Hilbert problem. It gives rise to the conclusion: H(4)?16, where H(n) is the Hilbert number for the second part of Hilbert's 16th problem. The process of the proof is algebraic and symbolic. As far as know, the technique employed in this work is different from more usual ones, the calculation can be readily done with using computer symbol operation system such as Mathematica.     

Cubic Lienard Equations with Quadratic Damping (Ⅱ)

Abstract Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienardequations with quadratic damping have at most three limit cycles. This implies that the guess in which thesystem has at most two limit cycles is false. We give the sufficient conditions for the system has at most threelimit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by usingnumerical simulation.     

The dynamics ofn weakly coupled identical oscillators

Summary We present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of “submaximal” limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.