Bogdanov-Takens系统的三次齐次扰动

对Bogdanov-Takens向量场的三次齐次扰动系统进行了讨论，得到了当其前二阶Melnikov函数恒为0时，则其后各阶Melnikov函数一定为0，且对于小的扰动参数，此系统为可积的或为Hamilton的；并对M1（h)≠0和M1(h)≡0，M2(h)≠0两种情形该系统的分岔结构给出了较全面的结论。     

On the Bifurcations of a Hamiltonian Having Three Homoclinic Loops under Z3 Invariant Quintic Perturbations

A cubic system having three homoclinic loops perturbed by Z3 invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurcation, homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied. It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.     

QUALITATIVE STUDY FOR A CUBIC SYSTEM

This article analyses a non-Lienard type planar cubic system, and a complete qualitative analysis is given for the system, especially the conclusions for the non-existence, existence and uniqueness of limit cycles are obtained.     

奇摄动下的共振不变环面分支

本文研究高维退化系统在小扰动下的动力学行为,在共振的情况下,利用延拓的方法,讨论了扰动 系统不变环面的保存性,并利用推广的Melnikov函数、横截性理论讨论了同宿于不变环面的横截同宿 轨道存在的条件,推广和改进了一些文献的结果.     

THE HOPF BIFURCATION OF QUADRATIC SYSTEM (I)_(n=0)

δlm is the parameter space of quadratic system (I)n=0. A partition of parameters corresponding to the existence and nonexistence of the limit cycle of the system is given in detail. The Hopf bifurcation surfaces of (I)m=0 are obtained, and the sketch of Hopf bifurcation surfaces of (I)n=0 are drawn.     

GLOBAL BIFURCATIONS OF LIMIT CYCLES FOR A CUBIC SYSTEM

1MainResultsConsiderthecubicsystemwheree>Oissmall,parametersa=(a,,a,)EDCR2withDboundedandcl)cz)bl?bzareconstants.LetWesupposep'(y)2ofory/oorequivalentlyoscf53c2.(1.3)Assumethatg(x)hasonlysimplezeroswith6152/o.Withoutlossofgene-ralitywecansuppose610,5z<0(onecenterandtwosaddles);(3)5l-46z<0,bz>0(twocentersandonesaddle),Inthispaperweonlyconsiderthefirsttwocases.Obviously,(1'1).=ohasafamily0fperiodicorbitswhe…     

一类三次系统的极限环个数与奇点分支

给出二次系统I的一类相伴系统在奇点O(0,0)的焦点量公式,证明了O至多为2阶细焦点,δlmn=0时系统在O外围至多有一个极限环,从而说明了系统在细焦点外围至多有一个极限环。最后给出了各个奇点的分支情况及几何特征。     

一类捕食食饵微分经济系统的稳定性与Hopf分支

本文主要研究了一个带有对捕食者进行捕获的微分代数经济系统的稳定性和Hopf分支问题.利用了动力系统和微分代数系统中的稳定性理论和分支理论的方法,得到了稳定性和Hopf分支稳定性的相关结论.本文对Ratio-Dependent捕食食饵模型进行了一定程度的完善,并且选取经济效益μ为分支参数进行研究,最后利用Matlab进行数值模拟,这样使得到的结论更符合现实意义.     

SAME DISTRIBUTION OF LIMIT CYCLES TO PERTURBED CUBIC HAMILTONIAN SYSTEMS WITH DIFFERENT PERTURBATIONS

Using qualitative analysis, we study perturbed Hamiltonian systems with different n-th order polynomial as perturbation terms. By numerical simulation, we show that these perturbed systems have the same distribution of limit cycles. Our results imply that these perturbed systems are equivalent in the sense of distribution of limit cycles. This is useful for studying limit cycles of perturbed systems.     

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.     

一类三次系统的极限环

本文讨论了一类三次系统 x=-y(1-αx~2)+δx-ιx~3,y=x(1-βx~2)和 x=-y(1-ax)(1-bx)+δx-ιx~3,y=x(1-cx)(1-bx)的极限环问题。     

THE PROBLEM OF LIMIT CYCLES FOR A LIENARD TYPE CUBIC SYSTEM

The purpose of this paper is to study a general Lienard type cubic system with one antisaddle and two saddles. We give some results of the existence and uniqueness of limit cycles as well as the evolution of limit cycles around the antisaddle for system (2) in the following when parameter a1 changes.     

Chemostat系统中Hopf分支的存在性

本文应用Hopf分支理论研究了具有内在代谢形式的Chemostat系统存在Hopf分支的条件,同时得到周期解的存在性及稳定性的结果.     

HOPF BIFURCATION ANALYSIS OF A 3D CHAOTIC SYSTEM

In this paper,a 3D chaotic system with multi-parameters is introduced. The dynamical systems of the original ADVP circuit and the modified ADVP model are regarded as special examples to the system.Some basic dynamical behaviors such as the stability of equilibria,the existence of Hopf bifurcation are investigated.We analyse the Hopf bifurcation of the system comprehensively using the first Lyapunov coefficient by precise symbolic computation.As a result,in fact we have studied the further dynamical behaviors.     

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.     

LIMIT CYCLES BIFURCATION FOR A CLASS OF DEGENERATE SINGULARITY

In this paper, bifurcation of limit cycles for the degenerate equilibrium to a three- dimensional system is investigated. Firstly, we use formal series to calculate the focal values at the high-order critical point on center manifold. Then an example is studied, and the existence of 3 limit cycles on the center manifold is proved. In terms of high- order singularities in high-dimensional systems, our results are new.     

THE LIMIT CYCLES AND HOPF BIFURCATION OF A CLASS OF SIMPLIFIED HOLLING TYPE-IV PREDATOR-PREY SYSTEM WITH LINEAR STATE FEEDBACK

In this paper, a class of simplified Type-IV predator-prey system with linear state feedback is investigated. We prove the boundedness of the positive solutions to this system, and analyze the quality of the equilibria and the existence of limit cycles of the system surrounding the positive equilibra. By Hopf bifurcation theory, the result of having two limit cycles to the system is obtained.     

二维非线性动力系统的多极限环分叉的参数控制

本文研究了一类具有Z2-等变性质的5次扰动平面Hamilton向量场，利用动力系统的分叉理论和判定函数法，通过控制其参数，得到系统在两组不同的参数条件下分别存在20和23个极限环，以及它们之间相对位置的不同构型.这一结果对机械振动中相应的控制问题有理论指导意义。     

Cyclical strategies in two-dimensional optimal control models: Necessary conditions and existence

This paper derives necessary conditions such that cyclical policies may be optimal in concave, two state variable (economic) control problems. These conditions identify four different routes. One major implication is that two of these four conditions may be met by separable models. This possibility has been overlooked so far. Therefore, even separable and structurally very simple models may be characterized by optimal cyclical policies. Indeed, it will be shown that stable limit cycles exist for concave and separable control problems.