scholarly journals POLYNOMIAL VECTOR FIELDS IN ℝ3 WITH INFINITELY MANY LIMIT CYCLES

2013 ◽  
Vol 23 (02) ◽  
pp. 1350029 ◽  
Author(s):  
ANTONI FERRAGUT ◽  
JAUME LLIBRE ◽  
CHARA PANTAZI
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Hamiltonian Structure ◽  
The Other ◽  
Constructive Method ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Period Annulus ◽  
Melnikov Integral

We provide a constructive method to obtain polynomial vector fields in ℝ3 having infinitely many limit cycles starting from polynomial vector fields in ℝ2 with a period annulus. We present two examples of polynomial vector fields in ℝ3 having infinitely many limit cycles, one of them of degree 2 and the other one of degree 12. The main tools of our method are the Melnikov integral and the Hamiltonian structure.

scholarly journals A number of limit cycle of sextic polynomial differential systems via the averaging theory

2021 ◽  
Vol 39 (4) ◽  
pp. 181-197
Author(s):  
Amour Menaceur ◽  
Salah Boulaaras
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Averaging Theory ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Period Annulus ◽  
Cubic System ◽  
Cubic Polynomial

The main purpose of this paper is to study the number of limit cycles of sextic polynomial differential systems (SPDS) via the averaging theory which is an extension to the study of cubic polynomial vector fields in (Nonlinear Analysis 66 (2007), 1707--1721), where we provide an accurate upper bound of the maximum number of limit cycles that SPDS can have bifurcating from the period annulus surrounding the origin of a class of cubic system.

Bifurcation of Limit Cycles from the Center of a Family of Cubic Polynomial Vector Fields

2018 ◽  
Vol 28 (05) ◽  
pp. 1850063 ◽  
Author(s):  
Shiyou Sui ◽  
Liqin Zhao
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Abelian Integral ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Number Of Zeros ◽  
Period Annulus ◽  
Cubic Polynomial

In this paper, we consider the number of zeros of Abelian integral for the system [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are arbitrary polynomials of degree [Formula: see text]. We obtain that [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the maximum number of limit cycles bifurcating from the period annulus up to the first order in [Formula: see text]. So, the bounds for [Formula: see text] or [Formula: see text], [Formula: see text], [Formula: see text] are exact.

Limit cycles of three-dimensional polynomial vector fields

Nonlinearity ◽  
2004 ◽  
Vol 18 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Marcin Bobie ski ◽  
Henryk o a dek
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

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.

scholarly journals Limit cycles of planar polynomial vector fields

Scholarpedia ◽  
2010 ◽  
Vol 5 (8) ◽  
pp. 9648 ◽  
Author(s):  
Maoan Han ◽  
Jibin Li ◽  
Chengzhi Li
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Polynomial Vector ◽  
Polynomial Vector Fields
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