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 POINCAR BIFURCATION OF CUBIC HAMILTONIAN SYSTEMS WITH DOUBLE CENTERS

In this paper, we discuss the Poincare bifurcation of cubic Hamiltonian systems with double centers and prove that the systems may at least generate two limit cycles and at most generate three limit cycles outside the lemniscate after a small cubic perturbation.     

The Poincaré Bifurcation of a Class of Planar Hamiltonian Systems

In this paper, we discuss the Poincare bifurcation of a class of Hamiltonian systems having a region consisting of periodic cycles bounded by a parabola and a straight line. We prove that the system can generate at most two limit cycles and may generate two limit cycles after a small cubic polynomial perturbation.     

Z 2-equivariant cubic system which yields 13 limit cycles

For the planar Z_2-equivariant cubic systems having two elementary focuses,the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved.The necessary and sufficient conditions for the existence of the bi-center are obtained.On the basis of this work,in this paper,we show that under small Z_2-equivariant cubic perturbations,this cubic system has at least 13 limit cycles with the scheme 16∪6.     

THE MAXIMAL NUMBER OF LIMIT CYCLES FOR QUADRATIC DIFFERENTIAL SYSTEM  = -y + lx~2 + xy,  = x(l + ax + by)

In this paper, we investigate the maximal number of limit cycles surrounding a first order weak focus for the quadratic differential system. And proved such a system has at most two limit cycles under some certain conditions.     

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.     

LIMIT CYCLES OF QUARTIC AND QUINTIC POLYNOMIAL DIFFERENTIAL SYSTEMS VIA AVERAGING THEORY

We study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging theory. More precisely,we prove that the perturbations of the period annulus of the center located at the origin of a cubic polynomial differential system,by arbitrary quartic and quintic polynomial differential systems,there respectively exist at least 8 and 9 limit cycles bifurcating from the periodic orbits of the period annu...     

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.     

Multiple limit cycles in an immune system

The nonlinear oscillatory phenomenon has been observed in the system of immune response, which corresponds to the limit cycles in the mathematical models. We prove that the system simulating an immune response studied by Huang has at least three limit cycles in the system. The conditions for the multiple limit cycles are useful in analyzing the nonlinear oscillation in immune response.     

Bifurcation of limit cycles in small perturbations of a hyper-elliptic Hamiltonian system with two nilpotent saddles

In this paper we study the first-order Melnikov function for a planar near-Hamiltonian system near a heteroclinic loop connecting two nilpotent saddles. The asymptotic expansion of this Melnikov function and formulas for the first seven coefficients are given. Next, we consider the bifurcation of limit cycles in a class of hyper-elliptic Hamiltonian systems which has a heteroclinic loop connecting two nilpotent saddles. It is shown that this system can undergo a degenerate Hopf bifurcation and Poincarè bifurcation, which emerges at most four limit cycles in the plane for sufficiently small positive ε. The number of limit cycles which appear near the heteroclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. Further more we give all possible distribution of limit cycles bifurcated from the period annulus.     

THE POINCAR BIFURCATION OF A CUBIC HAMILTONIAN SYSTEM WITH HOMOCLINIC LOOP

In this paper, we discuss the Poincare bifurcation of a cubic Hamiltonian system with homoclinic loop. We prove that the system can generate at most seven limit cycles after a small perturbation of general cubic polynomials.     

具有一个细焦点和一个粗焦点的二次系统极限环的个数

Abstract. It is proved that the quadratic system with a weak focus and a strong focus has atmost one limit cycle around the strong focus, and as the weak focus is a 2nd -order (or 3rd-order ) weak focus the quadratic system has at most two (one) limit cycles which have (1,1)-distribution ((0,1)-distribution).     

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.     

Bifurcations of limit cycles in a Z 4-equivariant quintic planar vector field

In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert＇s Problem.     

BIFURCATIONS OF LIMIT CYCLES FORMING COMPOUND EYES IN THE CUBIC SYSTEM

Let H(n)be the maximal number of limit cycle of planar real polynomial differentialsystem with the degree n and C_m~k denote the nest of k limit cycles enclosing m singular points.By computing detection functions,tne authors study bifurcation and phase diagrams in theclass of a planar cubic disturbed Hamiltonian system.In particular,the following conclusionis reached:The planar cubic system(E_ε)has 11 limit cycles,which form the pattern ofcompound eyes of C_9~1(?)2[C'~ε(?)(2C_1~2)and have the symmetrical structure;so the Hilbertnumber H(3)≥11.     

Quadratic systems with a weak focus and a strong focus

It is proved that the quadratic system with a weak focus and a strong focus has a unique limit cycle around one of the two foci, if there exists simultaneously limit cycles around each of the two foci for the system.     

一类Z_2-等变扰动的三次哈密顿系统极限环的个数与分布(英文)

This paper is concerned with the number and distributions of limit cycles of a cubic Z2-symmetry Hamiltonian system under quintic perturbation.By using qualitative analysis of differential equation,bifurcation theory of dynamical systems and the method of detection function,we obtain that this system exists at least 14 limit cycles with the distribution C91  [C11 ＋ 2（C32 2C12）].     

A NECESSARY AND SUFFICIENT CONDITION FOR THE COEXISTENCE OF A CLASS OF CUBIC CURVE SEPARATRIX CYCLES AND LIMIT CYCLES TO THE CUBIC SYSTEM

In this paper, we give the necessary and sufficient condition for the coexistence of a class of cubic curve separatrix cycles and limit cycles to the cubic system, and study their topological structures.     

CLASSIFICATION OF THE PHASE PORTRAIT FOR PLANAR Z_8-EQUIVARIANT HAMILTONIAN SYSTEM OF DEGREE 7

The Hilbert's sixteenth problem, that is the problem on the rlurnber anddistribution of limit cycles of planar polynomial system, has not solved fOr acentury. Since the original problem is so difficuIt, in 1977, V.I.Arnold posed"weakened Hilbert's sixteenth problem"-- the possibility of the number anddistribution of limit cycles fOr polynomial HamiItonian system of degrees n -- lunder perturbation of the polynomial of degrees m 1. From l983, Prof.Li Jibin etc. began to study cubic vecto…     

EXISTENCE AND UNIQUENESS OF LIMIT CYCLES ON A CUBIC KOLMOGOROV DIFFERENTIAL SYSTEM IN THE PREDATOR-PREY RELATION

In this paper,we analyse qualitatively a cubic Kolmogorov system:which is one of the mathematical models in ecology describing the interaction between Predator-Prey of two populations;and give the conditions of nonexistence,existence and uniqueness of limit cycles for three different cases.