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.     

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.     

BIFURCATION OF LIMIT CYCLES FROM A DOUBLE HOMOCLINIC LOOP WITH A ROUGH SADDLE

This paper concerns with the bifurcation of limit cycles from a double homoclinic loop under multiple parameter perturbations for general planar systems. The existence conditions of 4 homoclinic bifurcation curves and small and large limit cycles are especially investigated.     

PERTURBATIONS OF A KIND OF DEGENERATE QUADRATIC HAMILTONIAN SYSTEM WITH SADDLE－LOOP

61. IntroductionConsidcr a Hamiltonian system with smaIl perturbations{::leq:jix:<,\:>!>, (1l.)where X(x, y, e) and Y(I, y, ') are polynomia1s of T, y with coefficients depending analyti-cally on the small paramcter e, and the unperturbed s)ystern (1.l)() has at least onc cel1tersurrounded b}' the compact component r}, of algebraic curveH(x, g) = h, deg H(x, y) = m + 1, nlax{deg X(x. y, '). dog y(J. y. ')} = 7n.Tbe number of ]jm jt cyc]es of (1.1 ). whjcb emergp fIon) Th is equa] to tbe l1…     

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 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.     

GLOBAL STRUCTURES OF THE PLANE CUBIC SYSTEM WITH THE STELLAR NODE

ＧＬＯＢＡＬＳＴＲＵＣＴＵＲＥＳＯＦＴＨＥＰＬＡＮＥＣＵＢＩＣＳＹＳＴＥＭＷＩＴＨＴＨＥＳＴＥＬＬＡＲＮＯＤＥＹａｎｇＤｉａｎｗｕ（杨殿武）（ＪｉｎａｎＵｎｉｖｅｒｓｉｔｙ，济南大学，邮编：２５０００２）ＹａｏＷｅｉｈｏｎｇ（姚蔚红）（ＪｉｎａｎＥｄ...     

THE QUADRATIC SYSTEM WITH NONHYPERBOLIC HOMOCLINIC CYCLE

In this paper, we will show some sufficient conditions to assert certain system existing nonhyperbolic homoclinic cycles.     

HOMOCLINIC BIFURCATION WITH CODIMENSION 3

ＨＯＭＯＣＬＩＮＩＣＢＩＦＵＲＣＡＴＩＯＮＷＩＴＨＣＯＤＩＭＥＮＳＩＯＮ３￥ＺＨＵＤＥＭＩＮＧＡｂｓｔｒａｃｔ：ＦｉｒｓｔｉｔｉｓｐｒｏｖｅｄｔｈａｔｂｏｔｈｔｈｅｉｎｔｅｇｒａｌｏｆｔｈｅｄｉｖｅｒｇｅｎｃｅａｎｄｔｈｅＭｅｌｎｉｋｏｖｆｕｎｃｔｉ...     

HOMOCLINIC ORBITS FOR SECOND ORDER HAMILTONIAN SYSTEM WITH QUADRATIC GROWTH

ＨＯＭＯＣＬＩＮＩＣＯＲＢＩＴＳＦＯＲＳＥＣＯＮＤＯＲＤＥＲＨＡＭＩＬＴＯＮＩＡＮＳＹＳＴＥＭＷＩＴＨＱＵＡＤＲＡＴＩＣＧＲＯＷＴＨＷＵＳＨＡＯＰＩＮＧＡＮＤＬＩＵＪＩＡＱＵＡＮＡｂｓｔｒａｃｔ：Ｓｏｍｅｅｘｉｓｔｅｎｃｅａｎｄｍｕｌｔｉｐｌｉｃｉｔ...     

HOMOCLINIC ORBITS FOR A CLASS OF HAMILTONIAN SYSTEMS WITH POTENTIALS CHANGING SIGN

In this paper, we prove the existence of nontrivial homoclinic orbits for a class of Hamiltonian systems with potential changing sign. We use Mountain Pass Lemma.     

INFINITELY MANY HOMOCLINIC ORBITS FOR A CLASS OF HAMILTONIAN SYSTEMS WITH SYMMETRY

§１．ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＭａｉｎＲｅｓｕｌｔｓＴｈｉｓｐａｐｅｒｉｓａｎｅｘｔｅｎｓｉｏｎｏｆｔｈｅｗｏｒｋ［８］．Ｗｅｃｏｎｓｉｄｅｒｔｈｅｅｘｉｓｔｅｎｃｅｏｆｉｎｆｉｎｉｔｅｌｙｍａｎｙｈｏｍｏｃｌｉｎｉｃｏｒｂｉｔｓｆｏｒｔｈｅｆ...     

THE ANALYSIS OF HOPF BIFURCATION FOR THE PREY-PREDATOR SYSTEM WITH DIFFUSION AND DELAY

In this paper the influence of diffusion for stability of the prey-predator system with delay is studied. We analysed the conditions which raise the Hopf bifurcation. The problem is reduced by centralmanifold, and the approximation form of the stability periodic is obtained     

BIFURCATION OF DEGENERATE HOMOCLINIC ORBITS

ＢＩＦＵＲＣＡＴＩＯＮＯＦＤＥＧＥＮＥＲＡＴＥＨＯＭＯＣＬＩＮＩＣＯＲＢＩＴＳ￥ＺｅｎｇＷｅｉｙａｏ（ＨｕｎａｎＬｉｇｈｔｉｎｄｕｓｔｒｙＣｏｌｌｅｇｅ，４１０００７）Ａｂｓｔｒａｃｔ：Ｔｈｅｍａｉｎｐｕｒｐｏｓｅｏｆｔｈｉｓｐａｐｅｒｉｓｔｏｉｎｖ...     

THE PERIODIC SOLUTIONS OF A CLASS OF CUBIC DELAY DIFFERENTIAL SYSTEM

ＴＨＥＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＳＯＦＡＣＬＡＳＳＯＦＣＵＢＩＣＤＥＬＡＹＤＩＦＦＥＲＥＮＴＩＡＬＳＹＳＴＥＭＣＨＥＮＹＯＮＧＳＨＡＯ（陈永劭）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｏｎｔｈＣｈｉｎａＮｏｒｍａｌＵｎｉｖｅｒｓｉｔ...     

AN ESTIMATE OF THE NUMBER OF ZEROS OF ABELIAN INTEGRALS FOR CUBIC VECTOR FIELDS WITH CUSPIDAL LOOP

１ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｉｎｆｉｎｉｔｅｓｉｍａｌＨｉｌｂｅｒｔ－１６ｔｈｐｒｏｂｌｅｍ，ｐｏｓｅｄｂｙＶ．Ｉ．Ａｒｎｏｌｄ［１］，ｉｓｔｏｇｉｖｅａｎｅｓｔｉｍａｔｅＺ（ｍ，ｎ）ｏｆｔｈｅｎｕｍｂｅｒｏｆｚｅｒｏｓｏｆＡｂｅｌｉａｎｉｎｔｅ...     

GLOBAL ATTRACTOR OF A SPATIALLY DISCRETIZED REACTION-DIFFUSION SYSTEM WITH HAMILTONIAN STRUCTURE

１ＨａｍｉｌｔｏｎｉａｎＳｔｒｕｃｔｕｒｅｉｎＲｅａｃｔｉｏｎ－ＤｉｆｕｓｉｏｎｓｙｓｔｅｍＣｏｎｓｉｄｅｒａｓｙｓｔｅｍｏｆｒｅａｃｔｉｏｎ－ｄｉｆｕｓｉｏｎｅｑｕａｔｉｏｎｓｕｔ＝ｕｘｘ＋ｆ（ｕ，ｖ）ｖｔ＝ｖｘｘ＋ｇ（ｕ，ｖ）｛（１．１）ｗｈｅｒ...     

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.     

BIFURCATION OF PERIODIC ORBITS OF A THREE-DIMENSIONAL SYSTEM

§ 1 . IntroductionThere has been a general theory on bifurcation of periodic orbits for two or higher-dimensional systems, and the theory for two-dimensional systems is much richer than thatfor higher-dimensional systems (see [1–11]). We have a relative…     

A CLASS OF CUBIC SYSTEM WITH ONE SEVENTHORDER SINGULARITY (Ⅱ)

ＡＣＬＡＳＳＯＦＣＵＢＩＣＳＹＳＴＥＭＷＩＴＨＯＮＥＳＥＶＥＮＴＨＯＲＤＥＲＳＩＮＧＵＬＡＲＩＴＹ（Ⅱ）ＨａｎＹｕｌｉａｎｇ（韩玉良）（ＣｈｉｎａＣｏａｌＥｃｏｎｏｍｉｃＣｏｌｌｅｇｅ，中国煤炭经济学院，邮编：２６４００５）Ａｂｓｔｒａｃｔ：Ｉｎｔｈ...