Limit cycles and homoclinic orbits and their bifurcation of Bogdanov-Takens system

A quantitative analysis of limit cycles and homoclinic orbits, and the bifurcation curve for the Bogdanov-Takens system are discussed. The parameter incremental method for approximate analytical-expressions of these problems is given. These analytical-expressions of the limit cycle and homoclinic orbit are shown as the generalized harmonic functions by employing a time transformation. Curves of the parameters and the stability characteristic exponent of the limit cycle versus amplitude are drawn. Some of the limit cycles and homoclinic orbits phase portraits are plotted. The relationship curves of parameters μ and A with amplitude a and the bifurcation diagrams about the parameter are also given. The numerical accuracy of the calculation results is good.

LIMIT CYCLE PROBLEM OF QUADRATIC SYSTEM OF TYPE （ Ⅲ ）m=0, （ Ⅲ ）

To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.

On the Number of Limit Cycles in Small Perturbations of a Piecewise Linear Hamiltonian System with a Heteroclinic Loop

In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n(n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [(n+1)/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.

Singular Homoclinic Orbits and Limit Cycles in Positive Second Order Slow-Fast Systems

In the paper we consider a wide class of slow-fast second order systems and give sufficient conditions for the existence of a singular limit cycle related to a homoclinic orbit.

BIFURCATIONS OF LIMIT CYCLES FROM A HETEROCLINIC CYCLE OF HAMILTONIAN SYSTEMS

This paper is concerned with the bifurcations of limit cycles from a heteroclinic cycle of planar Hamiltonian systems under perturbations. The author obtains a simple condition which guarantees the existence of at most two limit cycles near the heteroclinic cycle.

On the Number of Limit Cycles of a Z_4-equivariant Quintic Near-Hamiltonian System

In this paper, we study the number of limit cycles of a near-Hamiltonian system having Za- equivariant quintic perturbations. Using the methods of Hopf and heteroclinic bifurcation theory, we find that the perturbed system can have 28 limit cycles, and its location is also given. The main result can be used to improve the lower bound of the maximal number of limit cycles for some polynomial systems in a previous work, which is the main motivation of the present paper.

On Canard Homoclinic of a Liénard Perturbation System

The classification on the orbits of some Liénard perturbation system with several parameters, which is relation to the example in [1] or [2], is discussed. The conditions for the parameters in order that the system has a unique limit cycle, homoclinic orbits, canards or the unique equilibrium point is globally asymptotic stable are given. The methods in our previous papers are used for the proofs.

Global bifurcation of a cubic system perturbed by degree four

Using the method of multi-parameter perturbation theory and qualitative analysis,a cubic system perturbed by degree four are investigated in this paper. After systematic analysis,it is found that the studied system can have nine limit cycles with their distributions are obtained.

LIMIT CYCLES OF THE GENERALIZED POLYNOMIAL LINARD DIFFERENTIAL SYSTEMS

Using the averaging theory of first and second order we study the maximum number of limit cycles of generalized Linard differential systems{x = y ＋ εhl1（x） ＋ ε2hl2（x）,y=-x- ε（fn1（x）y（2p＋1） ＋ gm1（x）） ＋ ∈2（fn2（x）y（2p＋1） ＋ gm2（x））,which bifurcate from the periodic orbits of the linear center x = y,y=-x,where ε is a small parameter.The polynomials hl1 and hl2 have degree l;fn1and fn2 have degree n;and gm1,gm2 have degree m.p ∈ N and[·]denotes the integer part function.

Homoclinic, heteroclinic and periodic orbits of singularly perturbed systems

The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them compared with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging methods.We also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system.

Exponential Dichotomies and Homoclinic Orbits from Heteroclinic Cycles

In this paper, we investigate the homoclinic bifurcations from a heteroclinic cycle by using exponential dichotomies. We give a Melnikov—type condition assuring the existence of homoclinic orbits form heteroclinic cycle. We improve some important results.

THE POINCAR BIFURCATION IN CUBIC HAMILTONIAN SYSTEMS WITH HETEROCLINIC LOOP

In this paper, we investigate the Poincar bifurcation in cubic Hamiltonian systems with heteroclinic loop, under small general cubic perturbations. We prove that the system has at most two limit cycles and has at least two limit cycles, respectively.

On the number of limit cycles in double homoclinic bifurcations

Let L be a double homoclinic loop of a Hamiltonian system on the plane. We obtain a condition under which L generates at most two large limit cycles by perturbations. We also give conditions for the existence of at most five or six limit cycles which appear near L under perturbations.

Uniqueness of limit cycles in codimension two bifurcations

Consider a three-dimensional autonomous system of the form of =f(u), u∈R^3, (0.1) where f(0)=0, and Df(0)=. By transforming eq. (0.1) into a normal form equation and then unfolding the truncated equation, one can obtain a plane system of the form

含参非线性扰动系统的闭轨分叉分析

研究了一类含参非线性系统的闭轨分叉问题,找到并确定了系统在平衡点附近的极限环及其稳定性.基于后继函数法,引入曲线坐标变换找到系统的后继函数,进而判断该闭轨为二重极限环.得到该系统极限环随参数变化从无到有,再到分裂为多个极限环的闭轨分叉现象.通过数值模拟,验证了系统随参数变化出现极限环的动力学特性.

BIFURCATION OF CUBIC INTEGRABLE SYSTEM UNDER CUBIC PERTURBATION

In this paper, we consider the bifurcation for a class of cubic integrable system under cubic perturbation. Using bifurcation theory and qualitative analysis, we obtain a complete bifurcation diagram of the system in a neighbourhood of the origin for parameter plane.

LIMIT CIRCLES BIFURCATED FROM A SOFT SPRINGDUFFING EQUATION UNDER PERTURBATION

In this paper, the Melnikov function method has been used to analyse the distance between stable manifold and unstable manifold of the soft spring Duffing equation([1]) after its heteroclinic orbits rupture as the result of a small perturbation. The conditions that limit circles are bifurcated are given, and then their stability and location is determined.

低轨星座构型保持研究现状与分析

稳定的构型是星座发挥其正常功能的基础。首先结合卫星在轨运行动力学环境,回顾了地球重力场模型、日月引力模型、大气模型与太阳光压模型的发展,在此基础上介绍了大气模型和太阳光压模型的在轨修正技术。其次,阐述了星座构型保持的必要性与可行性,总结归纳了摄动补偿法、预先设计法、全局优化法、阻力差分法与极限环保持法等构型保持方法,对比了绝对保持与相对保持的控制基准与方法特点。最后,展望了针对不同高度的低轨(low Earth orbit,LEO)星座构型保持,旨在为LEO星座的构型保持策略设计提供参考。

一类多分子反应微分方程模型的定性分析

本文研究一类生化反应的微分方程 dx/dt=1-x^py^q,dy/dt=α(x^py^q-y),其中p,q∈z^+,α>0。对p=1,q≥3和p=2,q=2,我们给出一些定性结果。

一类被开发的捕食-食饵系统的定性分析

讨论了一类食饵种群被开发的两种群捕食系统,主要讨论了系统平衡点的行为以及系统的全局稳定性。用P ioncare切性曲线法及Du lac函数法得到了闭轨不存在的充分条件。用Hop f分支理论及张芷芬唯一性定理证明了极限环的存在唯一性。