scholarly journals LIMIT CYCLE BIFURCATION FOR A NILPOTENT SYSTEM IN Z3-EQUIVARIANT VECTOR FIELD

2017 ◽  
Vol 7 (4) ◽  
pp. 1463-1477
Author(s):  
Chaoxiong Du ◽  
◽  
Qinlong Wang ◽  
Yirong Liu ◽  
Qi Zhang ◽  
...  
Keyword(s):  
Vector Field ◽  
Limit Cycle ◽  
Limit Cycle Bifurcation ◽  
Equivariant Vector Field

AN IMPROVED LOWER BOUND ON THE NUMBER OF LIMIT CYCLES BIFURCATING FROM A QUINTIC HAMILTONIAN PLANAR VECTOR FIELD UNDER QUINTIC PERTURBATION

2010 ◽  
Vol 20 (01) ◽  
pp. 63-70 ◽  
Author(s):  
TOMAS JOHNSON ◽  
WARWICK TUCKER
Keyword(s):  
Vector Field ◽  
Hamiltonian System ◽  
Lower Bound ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Hamiltonian Vector Field ◽  
Planar Vector ◽  
Limit Cycle Bifurcation ◽  
The Given ◽  
Planar Hamiltonian System

The limit cycle bifurcation of a Z2 equivariant quintic planar Hamiltonian vector field under Z2 equivariant quintic perturbation is studied. We prove that the given system can have at least 27 limit cycles. This is an improved lower bound on the possible number of limit cycles that can bifurcate from a quintic planar Hamiltonian system under quintic perturbation.

Limit Cycle Bifurcation of Infinity and Degenerate Singular Point in Three-Dimensional Vector Field

2016 ◽  
Vol 26 (09) ◽  
pp. 1650152
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Qi Zhang
Keyword(s):  
Vector Field ◽  
Singular Point ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Three Dimensional ◽  
Nonlinear Phenomenon ◽  
Dimensional Vector ◽  
Degenerate Singular Point ◽  
Limit Cycle Bifurcation ◽  
Three Dimensional Vector Field

Our work focuses on investigating limit cycle bifurcation for infinity and a degenerate singular point of a fifth degree system in three-dimensional vector field. By using singular value method to compute focal values carefully, we give the expressions of the focal values (Lyapunov constants) at the origin and at infinity. Moreover, we obtain that four limit cycles at most can bifurcate from the origin and three limit cycles can bifurcate from infinity. At the same time, we show the structure of limit cycles from the origin and the infinity. It is interesting for this kind of nonlinear phenomenon that a string of large limit cycles encircle a string of small limit cycles by simultaneous Hopf bifurcation, which is hardly seen for similar published results in three-dimensional vector field, our result is new.

scholarly journals Melnikov function and limit cycle bifurcation from a nilpotent center

2008 ◽  
Vol 132 (3) ◽  
pp. 182-193 ◽  
Author(s):  
Jiao Jiang ◽  
Maoan Han
Keyword(s):  
Limit Cycle ◽  
Melnikov Function ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Center

The index of an equivariant vector field and addition theorems for Pontryagin classes

2000 ◽  
Vol 64 (2) ◽  
pp. 223-247
Author(s):  
V M Bukhshtaber ◽  
K E Feldman
Keyword(s):  
Vector Field ◽  
Addition Theorems ◽  
Equivariant Vector Field

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.

ASYMPTOTIC EXPANSIONS OF MELNIKOV FUNCTIONS AND LIMIT CYCLE BIFURCATIONS

2012 ◽  
Vol 22 (12) ◽  
pp. 1250296 ◽  
Author(s):  
MAOAN HAN
Keyword(s):  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Asymptotic Expansions ◽  
Melnikov Function ◽  
Heteroclinic Loop ◽  
First Order ◽  
Melnikov Functions ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Center

In the study of the perturbation of Hamiltonian systems, the first order Melnikov functions play an important role. By finding its zeros, we can find limit cycles. By analyzing its analytical property, we can find its zeros. The main purpose of this article is to summarize some methods to find its zeros near a Hamiltonian value corresponding to an elementary center, nilpotent center or a homoclinic or heteroclinic loop with hyperbolic saddles or nilpotent critical points through the asymptotic expansions of the Melnikov function at these values. We present a series of results on the limit cycle bifurcation by using the first coefficients of the asymptotic expansions.

Limit Cycles in a Class of Quartic Kolmogorov Model with Three Positive Equilibrium Points

2015 ◽  
Vol 25 (06) ◽  
pp. 1550080 ◽  
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Qi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Positive Equilibrium ◽  
Bifurcation Problem ◽  
Kolmogorov Model ◽  
Limit Cycle Bifurcation ◽  
Perturbed Model

Limit cycle bifurcation problem of Kolmogorov model is interesting and significant both in theory and applications. In this paper, we will focus on investigating limit cycles for a class of quartic Kolmogorov model with three positive equilibrium points. Perturbed model can bifurcate three small limit cycles near (1, 2) or (2, 1) under a certain condition and can bifurcate one limit cycle near (1, 1). In addition, we have given some examples of simultaneous Hopf bifurcation and the structure of limit cycles bifurcated from three positive equilibrium points. The limit cycle bifurcation problem for Kolmogorov model with several positive equilibrium points are less seen in published references. Our result is good and interesting.

BIFURCATIONS OF LIMIT CYCLES IN A Z2-EQUIVARIANT PLANAR POLYNOMIAL VECTOR FIELD OF DEGREE 7

2006 ◽  
Vol 16 (04) ◽  
pp. 925-943 ◽  
Author(s):  
JIBIN LI ◽  
MINGJI ZHANG ◽  
SHUMIN LI
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Limit Cycles ◽  
Bifurcation Theory ◽  
Vector Fields ◽  
Compound Eyes ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Planar Dynamical Systems ◽  
Equivariant Vector Field

By using the bifurcation theory of planar dynamical systems and the method of detection functions, the bifurcations of limit cycles in a Z2-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 7 are studied. An example of a special Z2-equivariant vector field having 50 limit cycles with a configuration of compound eyes are given.

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