scholarly journals On the number of limit cycles of a quartic polynomial system

2018 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Min Li ◽  
◽  
Maoan Han ◽  
Keyword(s):  
Limit Cycles ◽  
Polynomial System ◽  
Quartic Polynomial

BIFURCATION OF LIMIT CYCLES AND ISOCHRONOUS CENTERS FOR A QUARTIC SYSTEM

2013 ◽  
Vol 23 (10) ◽  
pp. 1350172 ◽  
Author(s):  
WENTAO HUANG ◽  
AIYONG CHEN ◽  
QIUJIN XU
Keyword(s):  
Limit Cycles ◽  
Necessary Conditions ◽  
Polynomial System ◽  
Isochronous Center ◽  
Eighth Order ◽  
Bifurcation Of Limit Cycles ◽  
Quartic Polynomial ◽  
Fine Focus ◽  
Sufficient And Necessary Conditions ◽  
Isochronous Centers

For a quartic polynomial system we investigate bifurcations of limit cycles and obtain conditions for the origin to be a center. Computing the singular point values we find also the conditions for the origin to be the eighth order fine focus. It is proven that the system can have eight small amplitude limit cycles in a neighborhood of the origin. To the best of our knowledge, this is the first example of a quartic system with eight limit cycles bifurcated from a fine focus. We also give the sufficient and necessary conditions for the origin to be an isochronous center.

SEVEN LARGE-AMPLITUDE LIMIT CYCLES IN A CUBIC POLYNOMIAL SYSTEM

2006 ◽  
Vol 16 (02) ◽  
pp. 473-485 ◽  
Author(s):  
YIRONG LIU ◽  
WENTAO HUANG
Keyword(s):  
Singular Point ◽  
Limit Cycles ◽  
Large Amplitude ◽  
Polynomial System ◽  
Isochronous Center ◽  
Rational System ◽  
Cubic Polynomial

In this paper, the problem of limit cycles bifurcated from the equator for a cubic polynomial system is investigated. The best result so far in the literature for this problem is six limit cycles. By using the method of singular point value, we prove that a cubic polynomial system can bifurcate seven limit cycles from the equator. We also find that a rational system has an isochronous center at the equator.

ON THE NUMBER AND DISTRIBUTIONS OF LIMIT CYCLES IN A QUINTIC PLANAR VECTOR FIELD

2008 ◽  
Vol 18 (07) ◽  
pp. 1939-1955 ◽  
Author(s):  
YUHAI WU ◽  
YONGXI GAO ◽  
MAOAN HAN
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Hilbert Problem ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Analysis Method ◽  
Quintic Polynomial ◽  
Hilbert Number ◽  
Planar Vector ◽  
16Th Hilbert Problem

This paper is concerned with the number and distributions of limit cycles in a Z2-equivariant quintic planar vector field. By applying qualitative analysis method of differential equation, we find that 28 limit cycles with four different configurations appear in this special planar polynomial system. It is concluded that H(5) ≥ 28 = 52+ 3, where H(5) is the Hilbert number for quintic polynomial systems. The results obtained are useful to the study of the second part of 16th Hilbert problem.

On the Limit Cycles of a Perturbed Z3-Equivariant Planar Quintic Vector Field

2015 ◽  
Vol 25 (05) ◽  
pp. 1550073
Author(s):  
Yunlei Ma ◽  
Yuhai Wu
Keyword(s):  
Vector Field ◽  
Limit Cycles ◽  
Hilbert Problem ◽  
Polynomial System ◽  
Heteroclinic Loop ◽  
Planar Vector ◽  
16Th Hilbert Problem

In this paper, the number and distributions of limit cycles in a Z3-equivariant quintic planar polynomial system are studied. 24 limit cycles with two different configurations are shown in this quintic planar vector field by combining the methods of double homoclinic loops bifurcation, heteroclinic loop bifurcation and Poincaré–Bendixson Theorem. The results obtained are useful to the study of weakened 16th Hilbert problem.

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