ANDRONOV–HOPF BIFURCATION OF HIGHER CODIMENSIONS IN A LIÉNARD SYSTEM

2012 ◽  
Vol 22 (11) ◽  
pp. 1250271 ◽  
Author(s):  
ALEXANDER GRIN ◽  
KLAUS R. SCHNEIDER
Keyword(s):  
Hopf Bifurcation ◽  
Parameter Space ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Control Parameter ◽  
Unique Equilibrium ◽  
Liénard System ◽  
Lienard System

We study a polynomial Liénard system depending on three parameters a, b, c and exhibiting the following properties: (i) The origin is the unique equilibrium for all parameters. (ii) If a crosses zero, then the origin changes its stability, and Andronov–Hopf bifurcation arises. We consider a as control parameter and investigate the dependence of Andronov–Hopf bifurcation on the "unfolding" parameters b and c. We establish and describe analytically the existence of surfaces and curves located near the origin in the parameter space connected with the existence of small-amplitude limit cycles of multiplicity two and three (existence of degenerate Andronov–Hopf bifurcation).

HOPF BIFURCATION OF LIÉNARD SYSTEMS BY PERTURBING A NILPOTENT CENTER

2012 ◽  
Vol 22 (08) ◽  
pp. 1250203 ◽  
Author(s):  
JING SU ◽  
JUNMIN YANG ◽  
MAOAN HAN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Nonlinear Oscillators ◽  
Melnikov Function ◽  
Liénard System ◽  
First Order ◽  
Liénard Systems ◽  
Lienard System ◽  
Nilpotent Center

As we know, Liénard system is an important model of nonlinear oscillators, which has been widely studied. In this paper, we study the Hopf bifurcation of an analytic Liénard system by perturbing a nilpotent center. We develop an efficient method to compute the coefficients bl appearing in the expansion of the first order Melnikov function by finding a set of equivalent quantities B2l+1 which are able to calculate directly and can be used to study the number of small-amplitude limit cycles of the system. As an application, we investigate some polynomial Liénard systems, obtaining a lower bound of the maximal number of limit cycles near a nilpotent center.

scholarly journals Small amplitude limit cycles for the polynomial Liénard system

2008 ◽  
Vol 245 (9) ◽  
pp. 2522-2533 ◽  
Author(s):  
Maciej Borodzik ◽  
Henryk Żołądek
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Liénard System ◽  
Lienard System

On the Distribution of Limit Cycles in a Liénard System with a Nilpotent Center and a Nilpotent Saddle

2016 ◽  
Vol 26 (02) ◽  
pp. 1650025 ◽  
Author(s):  
R. Asheghi ◽  
A. Bakhshalizadeh
Keyword(s):  
Limit Cycles ◽  
Upper Bound ◽  
Integral Formula ◽  
Abelian Integral ◽  
Liénard System ◽  
Algebraic Criterion ◽  
Period Annulus ◽  
Lienard System ◽  
Nilpotent Center ◽  
Nilpotent Saddle

In this work, we study the Abelian integral [Formula: see text] corresponding to the following Liénard system, [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are real bounded parameters. By using the expansion of [Formula: see text] and a new algebraic criterion developed in [Grau et al., 2011], it will be shown that the sharp upper bound of the maximal number of isolated zeros of [Formula: see text] is 4. Hence, the above system can have at most four limit cycles bifurcating from the corresponding period annulus. Moreover, the configuration (distribution) of the limit cycles is also determined. The results obtained are new for this kind of Liénard 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 Existence of Limit Cycles for Liénard System with Application

2021 ◽  
pp. 33-37
Author(s):  
Ali Bakur Barsham ALmurad ◽  
Elamin Mohammed Saeed Ali
Keyword(s):  
Mathematical Method ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Planar System ◽  
Liénard System ◽  
System Form ◽  
Lienard System

This paper is part of a wider study limit cycle problems and planar system; The aims of this is to study the existence of limit cycle for Liénard system. We followed the historical analytical mathematical method to present a proof of a result on the existence of limit cycle for Liénard system form x ̇=y-F(x) ,y ̇=-g(x)

Hopf Bifurcation of Z2-Equivariant Generalized Liénard Systems

2018 ◽  
Vol 28 (06) ◽  
pp. 1850069 ◽  
Author(s):  
Yusen Wu ◽  
Laigang Guo ◽  
Yufu Chen
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Polynomial Function ◽  
Complete Classification ◽  
Normal Form Theory ◽  
Liénard Systems ◽  
Form Theory ◽  
Center Conditions

In this paper, we consider a class of Liénard systems, described by [Formula: see text], with [Formula: see text] symmetry. Particular attention is given to the existence of small-amplitude limit cycles around fine foci when [Formula: see text] is an odd polynomial function and [Formula: see text] is an even function. Using the methods of normal form theory, we found some new and better lower bounds of the maximal number of small-amplitude limit cycles in these systems. Moreover, a complete classification of the center conditions is obtained for such systems.

LIMIT CYCLES IN 3D LOTKA–VOLTERRA SYSTEMS APPEARING AFTER PERTURBATION OF HOPF CENTER

2008 ◽  
Vol 18 (12) ◽  
pp. 3647-3656 ◽  
Author(s):  
Ł. J. GOŁASZEWSKI ◽  
P. SŁAWIŃSKI ◽  
H. ŻOŁADEK
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Invariant Torus ◽  
Small Amplitude ◽  
Parameter Family ◽  
Volterra Systems ◽  
Parameter Perturbations ◽  
Two Parameter

We study the system ẋ = x(y+2z+(15/2η2)u), ẏ = y(x-2z-(7/2η2)u), ż = -z(x+y+(4/η2)u), u = x+y+z-1, and its two-parameter perturbations. We show that before perturbation there exists a one-parameter family of periodic solutions obtained via a nondegenarate Hopf bifurcation and after perturbation there remains at most one limit cycle of small amplitude and bounded period. Moreover, we found that a secondary Hopf bifurcation to an invariant torus occurs after the perturbation.

Maximum Number of Small Limit Cycles in Some Rational Liénard Systems with Cubic Restoring Terms

2021 ◽  
Vol 31 (12) ◽  
pp. 2150176
Author(s):  
Jiayi Chen ◽  
Yun Tian
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Upper Bound ◽  
Small Amplitude ◽  
Liénard Systems

In this paper, we obtain an upper bound for the number of small-amplitude limit cycles produced by Hopf bifurcation in one particular type of rational Liénard systems in the form of [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are polynomials in [Formula: see text] with degrees [Formula: see text] and [Formula: see text], respectively. Furthermore, we show that the upper bound presented here is sharp in the case of [Formula: see text].

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