scholarly journals A KIND OF BIFURCATION OF LIMIT CYCLES FROM A NILPOTENT CRITICAL POINT

2018 ◽  
Vol 8 (1) ◽  
pp. 10-18
Author(s):  
Tao Liu ◽  
◽  
Yirong Liu ◽  
Feng Li ◽  
◽  
...  
Keyword(s):  
Critical Point ◽  
Limit Cycles ◽  
Bifurcation Of Limit Cycles ◽  
Nilpotent Critical Point

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.

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.

scholarly journals BIFURCATION OF LIMIT CYCLES AT A NILPOTENT CRITICAL POINT IN A SEPTIC LYAPUNOV SYSTEM

2020 ◽  
Vol 10 (6) ◽  
pp. 2575-2591
Author(s):  
Yusen Wu ◽  
◽  
Ming Zhang ◽  
Jinxiu Mao ◽  
◽  
...  
Keyword(s):  
Critical Point ◽  
Limit Cycles ◽  
Bifurcation Of Limit Cycles ◽  
Lyapunov System ◽  
Nilpotent Critical Point

BIFURCATIONS OF LIMIT CYCLES CREATED BY A MULTIPLE NILPOTENT CRITICAL POINT OF PLANAR DYNAMICAL SYSTEMS

2011 ◽  
Vol 21 (02) ◽  
pp. 497-504 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Dynamical Systems ◽  
Critical Point ◽  
Limit Cycles ◽  
Small Amplitude ◽  
Hopf Bifurcations ◽  
Bifurcation Phenomena ◽  
Cubic System ◽  
The Stability ◽  
Nilpotent Critical Point ◽  
Planar Dynamical Systems

Bifurcations of limit cycles created from a multiple critical point of planar dynamical systems are studied. It is different from the usual Hopf bifurcations of limit cycles created from an elementary critical point. This bifurcation phenomena depends on the stability of the multiple critical point and the multiple number of the critical point. As an example, a cubic system which can created four small amplitude limit cycles from the origin (a multiple critical point) is given.

CYCLICITY OF PERIOD ANNULUS OF A QUADRATIC REVERSIBLE SYSTEM WITH DEGENERATE CRITICAL POINT

2013 ◽  
Vol 23 (06) ◽  
pp. 1350106 ◽  
Author(s):  
HAIHUA LIANG ◽  
JIANFENG HUANG
Keyword(s):  
Critical Point ◽  
Limit Cycles ◽  
Upper Bound ◽  
Outer Boundary ◽  
Reversible System ◽  
Bifurcation Of Limit Cycles ◽  
Period Annulus ◽  
Degenerate Critical Point

This paper is concerned with the bifurcation of limit cycles from the period annulus of a quadratic reversible system. The outer boundary of the period annulus contains a degenerate critical point. The exact upper bound of the number of limit cycles is given. Our result shows that the conjecture on the cyclicity of (r4) system is correct.

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