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.

SEVEN LARGE-AMPLITUDE LIMIT CYCLES IN A CUBIC POLYNOMIAL SYSTEM

2006 ◽  
Vol 16 (02) ◽  
pp. 473-485 ◽  
Author(s):  
YIRONG LIU ◽  
WENTAO HUANG
Keyword(s):  
Singular Point ◽  
Limit Cycles ◽  
Large Amplitude ◽  
Polynomial System ◽  
Isochronous Center ◽  
Rational System ◽  
Cubic Polynomial

In this paper, the problem of limit cycles bifurcated from the equator for a cubic polynomial system is investigated. The best result so far in the literature for this problem is six limit cycles. By using the method of singular point value, we prove that a cubic polynomial system can bifurcate seven limit cycles from the equator. We also find that a rational system has an isochronous center at the equator.

scholarly journals Limit Cycles and Analytic Centers for a Family of4n-1Degree Systems with Generalized Nilpotent Singularities

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Yusen Wu ◽  
Cui Zhang ◽  
Changjin Xu
Keyword(s):  
Limit Cycles ◽  
Computer Algebra System ◽  
Necessary Conditions ◽  
Rigorous Proof ◽  
Analytic Centers ◽  
Interesting Phenomenon ◽  
Parameter Perturbations ◽  
Sufficient And Necessary Conditions ◽  
Lyapunov Constants ◽  
Point Type

With the aid of computer algebra systemMathematica8.0 and by the integral factor method, for a family of generalized nilpotent systems, we first compute the first several quasi-Lyapunov constants, by vanishing them and rigorous proof, and then we get sufficient and necessary conditions under which the systems admit analytic centers at the origin. In addition, we present that seven amplitude limit cycles can be created from the origin. As an example, we give a concrete system with seven limit cycles via parameter perturbations to illustrate our conclusion. An interesting phenomenon is that the exponent parameterncontrols the singular point type of the studied system. The main results generalize and improve the previously known results in Pan.

DERIVATION OF ISOCHRONICITY CONDITIONS FOR QUASI-CUBIC HOMOGENEOUS ANALYTIC SYSTEMS

2013 ◽  
Vol 23 (09) ◽  
pp. 1350149
Author(s):  
YUSEN WU ◽  
FENG LI ◽  
PEILUAN LI
Keyword(s):  
Computer Algebra ◽  
Polynomial System ◽  
Isochronous Center ◽  
Interesting Phenomenon ◽  
Cubic Polynomial ◽  
Center Conditions ◽  
Isochronous Centers ◽  
Computer Algebra Software ◽  
Special Case

In this article, we deal with the problems of characterizing isochronous centers for real planar quasi-cubic homogeneous analytic system. The technique is based on reducing the quasi-cubic analytic system into an analytic system. With the help of common computer algebra software-MATHEMATICA, we compute the period constants of the origin and obtain the necessary isochronous center conditions for the transformed system. Finally, we give a proof of the sufficiency by various methods. Similar results are less so far. Our work is new in terms of research about quasi-cubic analytic system and consists of the existing results related to cubic polynomial system as a special case. What is worth pointing out is that we offer a kind of interesting phenomenon that the exponent parameter λ controls the nonanalyticity of the studied system (5).

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.

Limit Cycles in a Family of Planar Piecewise Linear Differential Systems with a Nonregular Separation Line

2019 ◽  
Vol 29 (08) ◽  
pp. 1950109
Author(s):  
Song-Mei Huan ◽  
Xiao-Song Yang
Keyword(s):  
Limit Cycles ◽  
Necessary Conditions ◽  
Piecewise Linear ◽  
Differential Systems ◽  
System Parameters ◽  
Separation Line ◽  
Linear Differential Systems ◽  
Inflection Points ◽  
Sufficient And Necessary Conditions ◽  
Linear Differential

In this paper, we investigate the number of crossing limit cycles in a family of planar piecewise linear differential systems with two zones separated by a nonregular line formed by two rays starting at the origin. By studying the dynamics of each subsystem, a thorough study including the parameteric expressions and main properties of some section maps is performed. Especially, different to the case with a straight separation line, it is proved that each section map can be piecewise with two different pieces and can have at most one inflection point. In addition to this, for the existence of these inflection points, some sufficient and necessary conditions satisfied by the system parameters are obtained. Based on these results, the importance played by these inflection points in increasing the maximum number of limit cycles in such systems is verified by providing a concrete example having five nested limit cycles with two crossing one separation ray and the other three crossing both separation rays. So, the five limit cycles obtained here are different from that obtained in the existing literature, where all the limit cycles cross both separation rays.

Global Dynamics of a Planar Filippov System with Symmetry

2020 ◽  
Vol 30 (07) ◽  
pp. 2050105
Author(s):  
Hongjie Pan ◽  
Xiaofeng Chen ◽  
Jiao Pu ◽  
Xiaoxing Chen
Keyword(s):  
Limit Cycles ◽  
Dynamic Property ◽  
Upper Bound ◽  
Necessary Conditions ◽  
Phase Portraits ◽  
Global Dynamics ◽  
Heteroclinic Loop ◽  
Global Dynamic ◽  
Sufficient And Necessary Conditions ◽  
Unique Limit

Chen [2016a, 2016b] studied global dynamics of the Filippov systems [Formula: see text], respectively. To study the global dynamics of [Formula: see text] completely, since the dynamics of [Formula: see text] is very simple, we are only interested in the global dynamics of [Formula: see text] in this paper. Firstly, we use Briot–Bouquet transformations and normal sector methods to discuss these degenerate equilibria at infinity. Secondly, we discuss the number of limit cycles completely. Then, the sufficient and necessary conditions of existence of the heteroclinic loop are found. To estimate the upper bound of the heteroclinic loop bifurcation function on parameter space, a result on the amplitude of a unique limit cycle of a discontinuous Liénard system is given. Finally, the complete bifurcation diagram and all global phase portraits are presented. The global dynamic property of system [Formula: see text] is totally different from systems [Formula: see text].

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