scholarly journals Weak center problem and bifurcation of critical periods for a Z 4 -equivariant cubic system

2013 ◽  
Vol 2013 (1) ◽  
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Canhui Liu
Keyword(s):  
Center Problem ◽  
Critical Periods ◽  
Weak Center ◽  
Cubic System ◽  
Bifurcation Of Critical Periods

scholarly journals Weak Bi-Center and Critical Period Bifurcations of a Z2-Equivariant Quintic System

2021 ◽  
Vol 31 (08) ◽  
pp. 2150117
Author(s):  
Yusen Wu
Keyword(s):  
Numerical Analysis ◽  
Computer Algebra ◽  
Critical Period ◽  
Computer Algebra System ◽  
Center Problem ◽  
Critical Periods ◽  
Symbolic Calculation ◽  
Weak Center

With the help of computer algebra system-Mathematica, this paper considers the weak center problem and local critical periods for bi-center of a [Formula: see text]-equivariant quintic system with eight parameters. The order of weak bi-center is identified and the exact maximum bifurcation number of critical periods generated from the bi-center is given via the combination of symbolic calculation and numerical analysis.

Weak Center and Bifurcation of Critical Periods in a Cubic Z2-Equivariant Hamiltonian Vector Field

2015 ◽  
Vol 25 (11) ◽  
pp. 1550143 ◽  
Author(s):  
Yusen Wu ◽  
Wentao Huang ◽  
Yongqiang Suo
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Local Bifurcation ◽  
Hamiltonian Vector Field ◽  
Critical Periods ◽  
Weak Center ◽  
Hamiltonian Vector Fields ◽  
Center Conditions ◽  
Bifurcation Of Critical Periods

This paper focuses on the problems of weak center and local bifurcation of critical periods for a class of cubic Z2-equivariant planar Hamiltonian vector fields. By computing the period constants carefully, one can see that there are three weak centers: (±1, 0) and the origin. The corresponding weak center conditions are also derived. Meanwhile, we address the problem of the coexistence of bifurcation of critical periods that occurred from (±1, 0) and the origin.

scholarly journals Bifurcation of critical periods of polynomial systems

2015 ◽  
Vol 259 (8) ◽  
pp. 3825-3853 ◽  
Author(s):  
Brigita Ferčec ◽  
Viktor Levandovskyy ◽  
Valery G. Romanovski ◽  
Douglas S. Shafer
Keyword(s):  
Polynomial Systems ◽  
Critical Periods ◽  
Bifurcation Of Critical Periods

Bifurcations of Critical Periods for a Class of Quintic Liénard Equation

2020 ◽  
Vol 30 (14) ◽  
pp. 2050201
Author(s):  
Zhiheng Yu ◽  
Lingling Liu
Keyword(s):  
Phase Portraits ◽  
Liénard Equation ◽  
Local Bifurcation ◽  
Exact Order ◽  
Critical Periods ◽  
Isochronous Centers ◽  
Bifurcation Of Critical Periods ◽  
Lienard Equation

In this paper, we investigate a quintic Liénard equation which has a center at the origin. We give the conditions for the parameters for the isochronous centers and weak centers of exact order. Then, we present the global phase portraits for the system having isochronous centers. Moreover, we prove that at most four critical periods can bifurcate and show with appropriate perturbations that local bifurcation of critical periods occur from the centers.

scholarly journals Critical Periods of Perturbations of Reversible Rigidly Isochronous Centers

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jiamei Zhou ◽  
Na Li ◽  
Maoan Han
Keyword(s):  
Polynomial System ◽  
Polynomial Systems ◽  
New Method ◽  
Critical Periods ◽  
Period Function ◽  
Number Of Zeros ◽  
Isochronous Centers ◽  
Bifurcation Of Critical Periods

We study the problem of bifurcation of critical periods of a time-reversible polynomial system of degreen. We first present a new method to find the number of zeros of the period function. Then applying our results, we study the number of critical periods for some polynomial systems and obtain new results.

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))].

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