Bifurcation of Limit Cycles and Their Relations in Three Perturbed Integrable Systems

2019 ◽  
pp. 557-568
Author(s):  
Lijun Hong ◽  
Wenyue Fu ◽  
Xiaochun Hong
Keyword(s):  
Limit Cycles ◽  
Integrable Systems ◽  
Bifurcation Of Limit Cycles

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.

BIFURCATION OF LIMIT CYCLES AND ISOCHRONOUS CENTERS FOR A QUARTIC SYSTEM

2013 ◽  
Vol 23 (10) ◽  
pp. 1350172 ◽  
Author(s):  
WENTAO HUANG ◽  
AIYONG CHEN ◽  
QIUJIN XU
Keyword(s):  
Limit Cycles ◽  
Necessary Conditions ◽  
Polynomial System ◽  
Isochronous Center ◽  
Eighth Order ◽  
Bifurcation Of Limit Cycles ◽  
Quartic Polynomial ◽  
Fine Focus ◽  
Sufficient And Necessary Conditions ◽  
Isochronous Centers

For a quartic polynomial system we investigate bifurcations of limit cycles and obtain conditions for the origin to be a center. Computing the singular point values we find also the conditions for the origin to be the eighth order fine focus. It is proven that the system can have eight small amplitude limit cycles in a neighborhood of the origin. To the best of our knowledge, this is the first example of a quartic system with eight limit cycles bifurcated from a fine focus. We also give the sufficient and necessary conditions for the origin to be an isochronous 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.

scholarly journals FOUR LIMIT CYCLES IN QUADRATIC NEAR-INTEGRABLE SYSTEMS

2011 ◽  
Vol 1 (2) ◽  
pp. 291-298 ◽  
Author(s):  
Pei Yu ◽  
◽  
Maoan Han ◽  
Keyword(s):  
Limit Cycles ◽  
Integrable Systems

QUADRATIC PERTURBATIONS OF A CLASS OF QUADRATIC REVERSIBLE LOTKA–VOLTERRA SYSTEMS

2013 ◽  
Vol 23 (08) ◽  
pp. 1350137
Author(s):  
YI SHAO ◽  
A. CHUNXIANG
Keyword(s):  
Limit Cycles ◽  
Positive Answer ◽  
Abelian Integrals ◽  
Volterra System ◽  
Bifurcation Of Limit Cycles ◽  
Number Of Zeros ◽  
Period Annulus ◽  
Volterra Systems

This paper is concerned with the bifurcation of limit cycles of a class of quadratic reversible Lotka–Volterra system [Formula: see text] with b = -1/3. By using the Chebyshev criterion to study the number of zeros of Abelian integrals, we prove that this system has at most two limit cycles produced from the period annulus around the center under quadratic perturbations, which provide a positive answer for a case of the conjecture proposed by S. Gautier et al.

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