scholarly journals A Note on Small Amplitude Limit Cycles of Liénard Equations Theory

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Yassine Bouattia ◽  
Djalil Boudjehem ◽  
Ammar Makhlouf ◽  
Sulima Ahmed Zubair ◽  
Sahar Ahmed Idris
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Extra Condition ◽  
Liénard Equations ◽  
Liénard Systems

In this paper, we demonstrate using a counterexample for a theorem of the small amplitude limit cycles in some Liénard systems and show that that there will be no solutions unless we add an extra condition. A new condition is derived for some specific Liénard systems where a violation of the small amplitude limit cycles theorem takes place.

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.

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].

Small-amplitude limit cycles of certain Liénard systems

1988 ◽  
Vol 418 (1854) ◽  
pp. 199-208 ◽  
Keyword(s):  
Critical Point ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Liénard Systems

The paper is concerned with the number of limit cycles of systems of the form ẋ = y – F ( x ), ẏ = –g( x ), where F and g are polynomials. For several classes of such systems, the maximum number of limit cycles that can bifurcate out of a critical point under perturbation of the coefficients in F and g is obtained (in terms of the degree of F and g ).

Small-amplitude limit cycles of Liénard systems

CALCOLO ◽  
1990 ◽  
Vol 27 (1-2) ◽  
pp. 1-32 ◽  
Author(s):  
S. Lynch
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Liénard Systems

Small-amplitude limit cycles of polynomial Liénard systems

2013 ◽  
Vol 56 (8) ◽  
pp. 1543-1556 ◽  
Author(s):  
MaoAn Han ◽  
Yun Tian ◽  
Pei Yu
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Liénard Systems

scholarly journals Small-Amplitude Limit Cycles in Liénard Systems via Multiplicity

1999 ◽  
Vol 159 (1) ◽  
pp. 186-211 ◽  
Author(s):  
Armengol Gasull ◽  
Joan Torregrosa
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Liénard Systems

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.

Limit Cycles of Liénard Equations With Non Linear Damping

1993 ◽  
Vol 36 (2) ◽  
pp. 251-256 ◽  
Author(s):  
A.M. Urbina ◽  
G. León De La Barra ◽  
M. León De La Barra ◽  
M. Cañas
Keyword(s):  
Limit Cycles ◽  
Small Amplitude ◽  
Polar Coordinates ◽  
Liénard Equations ◽  
Non Linear ◽  
Linear Damping

AbstractWe consider the Liénard's equation with f(x) polynomial. By using the generalized polar coordinates we establish the maximum possible number of small amplitude limit cycles of such equation in terms of the degree of f(x).

The number of small-amplitude limit cycles of Liénard equations

1984 ◽  
Vol 95 (2) ◽  
pp. 359-366 ◽  
Author(s):  
T. R. Blows ◽  
N. G. Lloyd
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Small Amplitude ◽  
Extensive Literature ◽  
Second Order Differential Equations ◽  
Liénard Equations ◽  
Comprehensive Survey ◽  
Image Position

We consider second order differential equations of Liénard type:Such equations have been very widely studied and arise frequently in applications. There is an extensive literature relating to the existence and uniqueness of periodic solutions: the paper of Staude[6] is a comprehensive survey. Our interest is in the number of periodic solutions of such equations.

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