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].

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.

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 Hopf Bifurcation of Limit Cycles in Discontinuous Liénard Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Special Form ◽  
Bifurcation Of Limit Cycles ◽  
Liénard Systems

We consider a class of discontinuous Liénard systems and study the number of limit cycles bifurcated from the origin when parameters vary. We establish a method of studying cyclicity of the system at the origin. As an application, we discuss some discontinuous Liénard systems of special form and study the cyclicity near the origin.

BIFURCATIONS OF LIMIT CYCLES FROM A QUINTIC HAMILTONIAN SYSTEM WITH A FIGURE DOUBLE-FISH

2013 ◽  
Vol 23 (07) ◽  
pp. 1350116 ◽  
Author(s):  
MINGHUI QI ◽  
LIQIN ZHAO
Keyword(s):  
Hamiltonian System ◽  
Limit Cycles ◽  
Upper Bound ◽  
Abelian Integrals ◽  
Liénard Systems

In this paper, we consider Liénard systems of the form [Formula: see text] where 0 < ∣∊∣ ≪ 1 and (α, β, γ) ∈ ℝ3. We prove that the least upper bound of the number of isolated zeros of the related Abelian integrals [Formula: see text] is 4 (counting the multiplicity) and this upper bound is a sharp one.

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.

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

Bifurcation of Limit Cycles in Small Perturbation of a Class of Liénard Systems

2014 ◽  
Vol 24 (01) ◽  
pp. 1450004 ◽  
Author(s):  
Xianbo Sun ◽  
Hongjian Xi ◽  
Hamid R. Z. Zangeneh ◽  
Rasool Kazemi
Keyword(s):  
Limit Cycle ◽  
Small Perturbation ◽  
Limit Cycles ◽  
Upper Bound ◽  
Bifurcation Of Limit Cycles ◽  
Heteroclinic Loop ◽  
Algebraic Criterion ◽  
Liénard Systems ◽  
Limit Cycle Bifurcation ◽  
Nilpotent Saddle

In this article, we study the limit cycle bifurcation of a Liénard system of type (5,4) with a heteroclinic loop passing through a hyperbolic saddle and a nilpotent saddle. We study the least upper bound of the number of limit cycles bifurcated from the periodic annulus inside the heteroclinic loop by a new algebraic criterion. We also prove at least three limit cycles will bifurcate and six kinds of different distributions of these limit cycles are given. The methods we use and the results we obtain are new.

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

DEGENERATE HOPF BIFURCATION IN NONSMOOTH PLANAR SYSTEMS

2012 ◽  
Vol 22 (03) ◽  
pp. 1250057 ◽  
Author(s):  
FENG LIANG ◽  
MAOAN HAN
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Weak Focus ◽  
Liénard Systems ◽  
General Systems ◽  
Planar Systems

In this paper, we mainly discuss Hopf bifurcation for planar nonsmooth general systems and Liénard systems with foci of parabolic–parabolic (PP) or focus–parabolic (FP) type. For the bifurcation near a focus, when the focus is kept fixed under perturbations we prove that there are at most k limit cycles which can be produced from an elementary weak focus of order 2k + 2 ( resp. k + 1)(k ≥ 1) if the focus is of PP (resp. FP) type, and we present the conditions to ensure these upper bounds are achievable. For the bifurcation near a center, the Hopf cyclicicy is studied for these systems. Some interesting applications are presented.

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