A study on the Upper Number of Zeros for a Liénard System

2021 ◽  
Vol 11 (07) ◽  
pp. 1441-1450
Author(s):  
成群 李
Keyword(s):  
Liénard System ◽  
Number Of Zeros ◽  
Lienard System

scholarly journals Sharp Bound of the Number of Zeros for a Liénard System with a Heteroclinic Loop

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Junning Cai ◽  
Minzhi Wei ◽  
Guoping Pang
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Sharp Bound ◽  
Abelian Integral ◽  
Heteroclinic Loop ◽  
Liénard System ◽  
Algebraic Criterion ◽  
Number Of Zeros ◽  
Lienard System ◽  
Nilpotent Saddle

In the presented paper, the Abelian integral I h of a Liénard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to find the least upper bound of the number of limit cycles bifurcating from periodic annulus.

scholarly journals Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Minzhi Wei ◽  
Junning Cai ◽  
Hongying Zhu
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Abelian Integral ◽  
Chebyshev System ◽  
Homoclinic Loop ◽  
Bifurcation Of Limit Cycles ◽  
Liénard System ◽  
Number Of Zeros ◽  
Lienard System ◽  
Nilpotent Saddle

In present paper, the number of zeros of the Abelian integral is studied, which is for some perturbed Hamiltonian system of degree 6. We prove the generating elements of the Abelian integral from a Chebyshev system of accuracy of 3; therefore there are at most 6 zeros of the Abelian integral.

scholarly journals Global Analysis of a Liénard System with Quadratic Damping

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Feng Guo
Keyword(s):  
Necessary Conditions ◽  
Global Analysis ◽  
Global Bifurcation ◽  
Phase Portraits ◽  
Liénard System ◽  
Closed Orbits ◽  
Lienard System ◽  
Sufficient And Necessary Conditions ◽  
Global Phase Portrait ◽  
Quadratic Damping

In this paper, the global analysis of a Liénard equation with quadratic damping is studied. There are 22 different global phase portraits in the Poincaré disc. Every global phase portrait is given as well as the complete global bifurcation diagram. Firstly, the equilibria at finite and infinite of the Liénard system are discussed. The properties of the equilibria are studied. Then, the sufficient and necessary conditions of the system with closed orbits are obtained. The degenerate Bogdanov-Takens bifurcation is studied and the bifurcation diagrams of the system are given.

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