scholarly journals Bifurcation of Limit Cycles and Center Conditions for Two Families of Kukles-Like Systems with Nilpotent Singularities

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Peiluan Li ◽  
Yusen Wu ◽  
Xiaoquan Ding
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Singular Points ◽  
Center Problem ◽  
Bifurcation Of Limit Cycles ◽  
Center Conditions

We solve theoretically the center problem and the cyclicity of the Hopf bifurcation for two families of Kukles-like systems with their origins being nilpotent and monodromic isolated singular points.

Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2-equivariant cubic vector fields

2018 ◽  
Vol 265 (10) ◽  
pp. 4965-4992 ◽  
Author(s):  
Feng Li ◽  
Yirong Liu ◽  
Yuanyuan Liu ◽  
Pei Yu
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Singular Points ◽  
Center Problem ◽  
Bifurcation Of Limit Cycles

A Class of Three-Dimensional Quadratic Systems with Ten Limit Cycles

2016 ◽  
Vol 26 (09) ◽  
pp. 1650149 ◽  
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Wentao Huang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Three Dimensional ◽  
Singular Value ◽  
Singular Points ◽  
Quadratic Systems ◽  
Symmetric Points ◽  
Fifth Order

Our work is concerned with a class of three-dimensional quadratic systems with two symmetric singular points which can yield ten small limit cycles. The method used is singular value method, we obtain the expressions of the first five focal values of the two singular points that the system has. Both singular symmetric points can be fine foci of fifth order at the same time. Moreover, we obtain that each one bifurcates five small limit cycles under a certain coefficient perturbed condition, consequently, at least ten limit cycles can appear by simultaneous Hopf bifurcation.

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.

Hopf bifurcation of limit cycles by perturbing piecewise integrable systems

2020 ◽  
Vol 161 ◽  
pp. 102866
Author(s):  
Maoan Han ◽  
Wen Lu
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycles ◽  
Integrable Systems ◽  
Bifurcation Of Limit Cycles

scholarly journals Center problem and the bifurcation of limit cycles for a cubic polynomial system

2015 ◽  
Vol 39 (17) ◽  
pp. 5200-5215 ◽  
Author(s):  
Chaoxiong Du ◽  
Wentao Huang ◽  
Qi Zhang
Keyword(s):  
Limit Cycles ◽  
Polynomial System ◽  
Center Problem ◽  
Bifurcation Of Limit Cycles ◽  
Cubic Polynomial

scholarly journals Fourteen Limit Cycles in a Seven-Degree Nilpotent System

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Wentao Huang ◽  
Ting Chen ◽  
Tianlong Gu
Keyword(s):  
Critical Point ◽  
Computer Algebra ◽  
Limit Cycles ◽  
Computer Algebra System ◽  
Degree Polynomial ◽  
Bifurcation Of Limit Cycles ◽  
Fine Focus ◽  
Center Conditions ◽  
Nilpotent Critical Point ◽  
Lyapunov Constants

Center conditions and the bifurcation of limit cycles for a seven-degree polynomial differential system in which the origin is a nilpotent critical point are studied. Using the computer algebra system Mathematica, the first 14 quasi-Lyapunov constants of the origin are obtained, and then the conditions for the origin to be a center and the 14th-order fine focus are derived, respectively. Finally, we prove that the system has 14 limit cycles bifurcated from the origin under a small perturbation. As far as we know, this is the first example of a seven-degree system with 14 limit cycles bifurcated from a nilpotent critical point.

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.

scholarly journals Limit Cycles and Integrability in a Class of Systems with High-Order Nilpotent Critical Points

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Feng Li ◽  
Jianlong Qiu
Keyword(s):  
Critical Point ◽  
Critical Points ◽  
Limit Cycles ◽  
High Order ◽  
Bifurcation Of Limit Cycles ◽  
Polynomial Differential Systems ◽  
Sufficient And Necessary Conditions ◽  
Center Conditions ◽  
Nilpotent Critical Point ◽  
Lyapunov Constants

A class of polynomial differential systems with high-order nilpotent critical points are investigated in this paper. Those systems could be changed into systems with an element critical point. The center conditions and bifurcation of limit cycles could be obtained by classical methods. Finally, an example was given; with the help of computer algebra system MATHEMATICA, the first 5 Lyapunov constants are deduced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact that there exist 5 small amplitude limit cycles created from the high-order nilpotent critical point is also proved.

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