scholarly journals Chapter 9. Center-Focus Problem and Bifurcations of Limit Cycles for a Z2-Equivariant Cubic System

2014 ◽  
pp. 272-307
Keyword(s):  
Limit Cycles ◽  
Cubic System

A Cubic System with Eight Small-Amplitude Limit Cycles

1991 ◽  
Vol 47 (2) ◽  
pp. 163-171 ◽  
Author(s):  
E. M. JAMES ◽  
N. G. LLOYD
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Cubic System

Bifurcations of limit cycles in a cubic system with cubic perturbations

2006 ◽  
Vol 176 (1) ◽  
pp. 341-358 ◽  
Author(s):  
Hong Zang ◽  
Tonghua Zhang ◽  
Maoan Han
Keyword(s):  
Limit Cycles ◽  
Cubic System

LIMIT CYCLE BIFURCATIONS IN NEAR-HAMILTONIAN SYSTEMS BY PERTURBING A NILPOTENT CENTER

2008 ◽  
Vol 18 (10) ◽  
pp. 3013-3027 ◽  
Author(s):  
MAOAN HAN ◽  
JIAO JIANG ◽  
HUAIPING ZHU
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Unperturbed System ◽  
Bifurcation Theorem ◽  
First Order ◽  
Cubic System ◽  
Limit Cycle Bifurcation ◽  
Almost All ◽  
Nilpotent Center ◽  
Computing Method

As we know, Hopf bifurcation is an important part of bifurcation theory of dynamical systems. Almost all known works are concerned with the bifurcation and number of limit cycles near a nondegenerate focus or center. In the present paper, we study a general near-Hamiltonian system on the plane whose unperturbed system has a nilpotent center. We obtain an expansion for the first order Melnikov function near the center together with a computing method for the first coefficients. Using these coefficients, we obtain a new bifurcation theorem concerning the limit cycle bifurcation near the nilpotent center. An interesting application example & a cubic system having five limit cycles & is also presented.

scholarly journals Five limit cycles for a simple cubic system

1997 ◽  
Vol 41 ◽  
pp. 199-208 ◽  
Author(s):  
N. G. Lloyd ◽  
J. M. Pearson
Keyword(s):  
Limit Cycles ◽  
Simple Cubic ◽  
Cubic System

scholarly journals Uniqueness of Limit Cycles for a Class of Cubic Systems with Two Invariant Straight Lines

2010 ◽  
Vol 2010 ◽  
pp. 1-17
Author(s):  
Xiangdong Xie ◽  
Fengde Chen ◽  
Qingyi Zhan
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Uniqueness Of Limit Cycles ◽  
Cubic System ◽  
Straight Lines ◽  
Cubic Systems

A class of cubic systems with two invariant straight linesdx/dt=y(1-x2),  dy/dt=-x+δy+nx2+mxy+ly2+bxy2.is studied. It is obtained that the focal quantities ofO(0,0)are,W0=δ; ifW0=0, thenW1=m(n+l); ifW0=W1=0, thenW2=−nm(b+1); ifW0=W1=W2=0, thenOis a center, and it has been proved that the above mentioned cubic system has at most one limit cycle surrounding weak focalO(0,0). This paper also aims to solve the remaining issues in the work of Zheng and Xie (2009).

ON THE NUMBER AND DISTRIBUTION OF LIMIT CYCLES IN A CUBIC SYSTEM

2004 ◽  
Vol 14 (12) ◽  
pp. 4285-4292 ◽  
Author(s):  
MAOAN HAN ◽  
TONGHUA ZHANG ◽  
HONG ZANG
Keyword(s):  
Qualitative Analysis ◽  
Limit Cycles ◽  
Bifurcation Theory ◽  
Cubic System

This paper concerns the number of limit cycles in a cubic system. Eleven limit cycles are found and two different distributions are given by using the methods of bifurcation theory and qualitative analysis.

Bifurcations in a Cubic System with a Degenerate Saddle Point

2014 ◽  
Vol 24 (11) ◽  
pp. 1450144
Author(s):  
Desheng Shang ◽  
Yaoming Zhang
Keyword(s):  
Perturbation Theory ◽  
Qualitative Analysis ◽  
Saddle Point ◽  
Limit Cycles ◽  
Blow Up ◽  
Cubic System ◽  
Cubic Systems

Bifurcations in a cubic system with a degenerate saddle point are investigated using the technique of blow-up, the method of planar perturbation theory and qualitative analysis. It has been found that after appropriate perturbations, at least 12 limit cycles can bifurcate from a degenerate saddle point in a type of cubic systems.

BIFURCATIONS OF LIMIT CYCLES CREATED BY A MULTIPLE NILPOTENT CRITICAL POINT OF PLANAR DYNAMICAL SYSTEMS

2011 ◽  
Vol 21 (02) ◽  
pp. 497-504 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Dynamical Systems ◽  
Critical Point ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Hopf Bifurcations ◽  
Bifurcation Phenomena ◽  
Cubic System ◽  
The Stability ◽  
Nilpotent Critical Point ◽  
Planar Dynamical Systems

Bifurcations of limit cycles created from a multiple critical point of planar dynamical systems are studied. It is different from the usual Hopf bifurcations of limit cycles created from an elementary critical point. This bifurcation phenomena depends on the stability of the multiple critical point and the multiple number of the critical point. As an example, a cubic system which can created four small amplitude limit cycles from the origin (a multiple critical point) is given.

BIFURCATIONS OF LIMIT CYCLES FOR A PERTURBED CUBIC SYSTEM WITH DOUBLE FIGURE EIGHT LOOP

2013 ◽  
Vol 23 (04) ◽  
pp. 1350067
Author(s):  
TONGHUA ZHANG ◽  
HONG ZANG ◽  
MOSE O. TADE
Keyword(s):  
Hamiltonian System ◽  
Qualitative Analysis ◽  
Limit Cycles ◽  
Bifurcation Theory ◽  
Cubic System

This paper is concerned with the distribution and number of limit cycles for a cubic Hamiltonian system under cubic perturbation. The fact that there exist seven limit cycles is proved. The different distributions of limit cycles are given by using the methods of bifurcation theory and qualitative analysis, and the distributions of seven limit cycles are newly established.

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