一类异宿环分支极限环的唯一性

本文研究平面上一类两点或三点异宿环附近极限环的分支,在一简洁条件下证明了异宿环分支极限环的唯一性,并给出了极限环唯一存在的充要条件.作为对三维余维2分支的应用,解决了所出现的两点异宿环产生唯一极限环的问题.     

再论一类二次系统的无界双中心周期环域的POincare分支

本文再一次讨论了具有双曲线与赤道弧为边界的双中心周期环域的二次系统的Poincare分支，并构造出了此系统出现极限环的(0，3)分布或出现一个三重极限环的具体例子．     

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.     

一类异宿环分支问题中极限环的唯一性

在具余维２奇点的四维系统的两参数开折的研究中出现一类三点异宿环的扰动分支,对此异宿环产生极限环的唯一性一直未得到完整的解决,本文圆满地解决了这一问题,并获得了全局分支中极限环的唯一性。     

Limit cycle bifurcations by perturbing a class of integrable systems with a polycycle

In this paper, we deal with the problem of limit cycle bifurcation near a 2-polycycle or 3-polycycle for a class of integrable systems by using the first order Melnikov function. We first get the formal expansion of the Melnikov function corresponding to the heteroclinic loop and then give some computational formulas for the first coefficients of the expansion. Based on the coefficients, we obtain a lower bound for the maximal number of limit cycles near the polycycle. As an application of our main results, we consider quadratic integrable polynomial systems, obtaining at least two limit cycles.     

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.     

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.     

THE HETEROCLINIC CYCLE IN THEMODEL OF COMPETITION BETWEENn SPECIES AND ITS STABILITY

1.IntroductionConsidertheanalogousGause-Lotka-Volterra(GLV)differentialequationsforncompetingpopulationsMayandLeonard[ll,Hcf'bauerandSigmundl21havestudiedthissystemforthecasen=3separately.Bothofthemnotedthatinsuchsystemprobablythereexistsaheterocliniccycle.Biologically,thestudyoftheheterocliniccycleisinterestingsinceitsstabilityiscloselyrelatedtothepermanenceproblem.Ontheotherhand,inrecelltyearsmathematicianshavestudiedwithinterestthestabilityofaheterocltalccycleandthebifurcationofit.Inth…     

Specific locking in populations dynamics: Symmetry analysis for coupled heteroclinic cycles

Population dynamics on two sites of ecological fields are studied. Each site shows oscillatory dynamics with a heteroclinic cycle or a limit cycle attractor, and populations migrate between two sites diffusively. In this system, frequency locking states with specific ratios between the oscillations of two sites are observed. The selection of the ratios are explained with the symmetry of the phase space. Other properties of the locking states as behaviors intrinsic to heteroclinic cycles are also discussed.     

A polycycle and limit cycles in a non-differentiable predator-prey model

For a non-differentiable predator-prey model, we establish conditions for the existence of a heteroclinic orbit which is part of one contractive polycycle and for some values of the parameters, we prove that the heteroclinic orbit is broken and generates a stable limit cycle. In addition, in the parameter space, we prove that there exists a curve such that the unique singularity in the realistic quadrant of the predator-prey model is a weak focus of order two and by Hopf bifurcations we can have at most two small amplitude limit cycles.     

Limit Cycle Bifurcations Of A Kind Of Lienard System With A Hypobolic Saddle And A Nilpotent Cusp

This paper gives a general theorem on the number of limit cycles of a near Hamiltonian system with a heteroclinic loop passing through a hyperbolic saddle and a nilpotent cusp. Then we study a kind of Lienard systems of type (n,4) for 3<=n<=27 and obtain the lower bound of the maximal number of limit cycles for this kind of system.     

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 Z₄- 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.     

THE EXISTENCE AND STABILITY OF THE HETEROCLINIC CYCLE IN A KIND OF BIOLOGICAL SYSTEM

IIntroductlonConsider the n－spedes biological systemlit\1．＝IJ!厂．＋》*i,工上D、忍＝上,’··。n．ti)Ifffi＝1,it is S S－SpSCllS LOthaka－VoltOOYY SystSS．Iftti＝2,It Is S S－sPeCieSKolmongorov system．As to the n－spedes Gause－Lotb－Volterra system矿ti 乙工．＝T;Ii、y Qiil．I。Ti 7 U,on M U,t6)174 AnnofDiff Eqs．VO18M叫 and Leonard[1],Ho凡aner and Sigmund问 have studied this system forthe case n二 3 respectlvelyand noted that thereprobably exists aheterocllnlccyclefor…     

极限环的表达式和计算

本文提出一种解析法和数值法相结合的方法，用来计算多项式微分系统的极限环，极限环表示为x=∑k≥0(ak cos kφ bk sin kφ),y=∑k≥0(ck cos kφ dk sin kφ)，先用解析法求出小参数时极限环的初始表达式，然后用增量法和迭代法求出任意参数时极限环满足给定精度的表达式，半稳定极限环和分叉值也可以计算。     

Distributional convergence in planar dynamics and singular perturbations

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics.     

空间同宿环和异宿环的稳定性

关于平面同（异）宿环的稳定性已有不少文献讨论过，但关于空间同（异）宿环的稳定性尚没有任何结果．本文在可定义回复映射的条件下给出了同（异）宿环在其部分邻域中是渐近稳定的判据．这些结果在某种意义下是平面系统相应结果的推广，包括并推广了［２］，［３］的结果．本文最后讨论了Ｌｏｒｅｎｚ系统同宿环和三种群竞争系统异宿环的稳定性，所得结果和Ｓｐａｒｒｏｗ与Ｍａｙ等的数值结果相吻合．     

具有临界两点异宿环的二次系统

本文给出二次系统存在临界两点异宿环的充要条件,并证明二次系统的临界两点异宿环必由双曲线的一支和直线或由椭圆和直线构成,其内部的奇点必是中心。推广所研究的这种系统,本文对[1]中提出的一个公开问题也给出了解答。     

推广后继函数法研究第二临界情况下同宿环的稳定性

本文通过灵活选取参照闭曲线,推广了研究闭轨线的后继函数法.通过计算后继函数,本文首先获得了二重极限环的半稳定性判据.在此基础上,运用推广的后继函数法,获得了第二临界情况下同宿环的内稳定性判据,事实上,推广的后继函数法可对以往的结果和本文的结果用统一的方法给予证明,并可向更高临界情况推广.最后本文证明了二重极限环及第二临界情况下的同宿环在一定条件下分支出极限环的唯二性.     

Complex dynamics of a simple 3D autonomous chaotic system with four-wing

The present paper revisits a three dimensional (3D) autonomous chaotic system with four-wing occurring in the known literature [Nonlinear Dyn (2010) 60(3): 443--457] with the entitle ``A new type of four-wing chaotic attractors in 3-D quadratic autonomous systems'' and is devoted to discussing its complex dynamical behaviors, mainly for its non-isolated equilibria, Hopf bifurcation, heteroclinic orbit and singularly degenerate heteroclinic cycles, etc. Firstly, the detailed distribution of its equilibrium points is formulated. Secondly, the local behaviors of its equilibria, especially the Hopf bifurcation, are studied. Thirdly, its such singular orbits as the heteroclinic orbits and singularly degenerate heteroclinic cycles are exploited. In particular, numerical simulations demonstrate that this system not only has four heteroclinic orbits to the origin and other four symmetry equilibria, but also two different kinds of infinitely many singularly degenerate heteroclinic cycles with the corresponding two-wing and four-wing chaotic attractors nearby.