scholarly journals Simultaneous Bifurcation of Limit Cycles and Critical Periods

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Regilene D. S. Oliveira ◽  
Iván Sánchez-Sánchez ◽  
Joan Torregrosa
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Critical Periods ◽  
Bifurcation Of Limit Cycles ◽  
Period Function ◽  
Polynomial Differential Equations ◽  
Return Map ◽  
Lie Bracket ◽  
Liénard Systems ◽  
Quadratic Family

AbstractThe present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.

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.

The number of limit cycles of certain polynomial differential equations

1984 ◽  
Vol 98 (3-4) ◽  
pp. 215-239 ◽  
Author(s):  
T. R. Blows ◽  
N. G. Lloyd
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Equations ◽  
Image Position ◽  
Cubic Systems

SynopsisTwo-dimensional differential systemsare considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

Bifurcation of Limit Cycles for Some Liénard Systems with a Nilpotent Singular Point

2015 ◽  
Vol 25 (05) ◽  
pp. 1550066 ◽  
Author(s):  
Junmin Yang ◽  
Xianbo Sun
Keyword(s):  
Singular Point ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Bifurcation Of Limit Cycles ◽  
Liénard Systems ◽  
Nilpotent Singular Point ◽  
Nilpotent Saddle

In this paper, we first present some general theorems on bifurcation of limit cycles in near-Hamiltonian systems with a nilpotent saddle or a nilpotent cusp. Then we apply the theorems to study the number of limit cycles for some polynomial Liénard systems with a nilpotent saddle or a nilpotent cusp, and obtain some new estimations on the number of limit cycles of these systems.

scholarly journals On the polynomial limit cycles of polynomial differential equations

2011 ◽  
Vol 181 (1) ◽  
pp. 461-475 ◽  
Author(s):  
Jaume Giné ◽  
Maite Grau ◽  
Jaume Llibre
Keyword(s):  
Differential Equations ◽  
Limit Cycles ◽  
Polynomial Differential Equations

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.

scholarly journals On the critical periods of Liénard systems with cubic restoring forces

2004 ◽  
Vol 2004 (61) ◽  
pp. 3259-3274 ◽  
Author(s):  
Zhengdong Du
Keyword(s):  
Taylor Series ◽  
Type I ◽  
Type Ii ◽  
Critical Periods ◽  
Period Function ◽  
Weak Center ◽  
Local Bifurcations ◽  
Liénard Systems ◽  
Lienard System ◽  
Restoring Forces

We study local bifurcations of critical periods in the neighborhood of a nondegenerate center of a Liénard system of the formx˙=−y+F(x),y˙=g(x), whereF(x)andg(x)are polynomials such thatdeg(g(x))≤3,g(0)=0, andg′(0)=1,F(0)=F′(0)=0and the system always has a center at(0,0). The set of coefficients ofF(x)andg(x)is split into two strata denoted bySIandSIIand(0,0)is called weak center of type I and type II, respectively. By using a similar method implemented in previous works which is based on the analysis of the coefficients of the Taylor series of the period function, we show that for a weak center of type I, at most[(1/2)deg(F(x))]−1local critical periods can bifurcate and the maximum number can be reached. For a weak center of type II, the maximum number of local critical periods that can bifurcate is at least[(1/4)deg(F(x))].

scholarly journals On the limit cycles of polynomial differential systems with homogeneous nonlinearities

2000 ◽  
Vol 43 (3) ◽  
pp. 529-543 ◽  
Author(s):  
Chengzhi Li ◽  
Weigu Li ◽  
Jaume Llibre ◽  
Zhifen Zhang
Keyword(s):  
Differential Equations ◽  
Periodic Orbits ◽  
Limit Cycles ◽  
Homogeneous Polynomials ◽  
Differential Systems ◽  
Polynomial Differential Equations ◽  
Polynomial Differential Systems

AbstractWe consider three classes of polynomial differential equations of the form ẋ = y + establish Pn (x, y), ẏ = x + Qn (x, y), where establish Pn and Qn are homogeneous polynomials of degree n, having a non-Hamiltonian centre at the origin. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centres when we perturb them inside the class of all polynomial differential systems of the above form. A more detailed study is made for the particular cases of degree n = 2 and n = 3.

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