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.

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 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.

INTEGRABILITY AND BIFURCATIONS OF LIMIT CYCLES IN A CUBIC KOLMOGOROV SYSTEM

2013 ◽  
Vol 23 (04) ◽  
pp. 1350061 ◽  
Author(s):  
FENG LI
Keyword(s):  
Critical Point ◽  
Computer Algebra ◽  
Limit Cycles ◽  
Computer Algebra System ◽  
Algebraic Curves ◽  
Necessary Conditions ◽  
Invariant Algebraic Curves ◽  
Kolmogorov System

We investigate the planar cubic Kolmogorov systems with three invariant algebraic curves which have a equilibrium at (1,1). With the help of computer algebra system MATHEMATICA, we prove that five limit cycles can be bifurcated from a critical point in the first quadrant. Moreover, the necessary conditions of center are obtained, by technical transformation, and its sufficiencies are proved.

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].

ANALYTIC CENTER OF NILPOTENT CRITICAL POINTS

2012 ◽  
Vol 22 (08) ◽  
pp. 1250198 ◽  
Author(s):  
TAO LIU ◽  
LIANGGANG WU ◽  
FENG LI
Keyword(s):  
Critical Points ◽  
Necessary Conditions ◽  
Computation Method ◽  
Analytic Center ◽  
Center Problem ◽  
Third Order ◽  
Small Perturbations ◽  
Integrating Factor ◽  
Sufficient And Necessary Conditions ◽  
Lyapunov Constants

For third-order nilpotent critical points of a planar dynamical system, the analytic center problem is completely solved in this article by using the integrating factor method. The associated quasi-Lyapunov constants are defined and their computation method is given. For a class of cubic-order systems under small perturbations, sufficient and necessary conditions for an analytic center are obtained.

scholarly journals Limit Cycles in a Cubic Kolmogorov System with Harvest and Two Positive Equilibrium Points

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Qi-Ming Zhang ◽  
Feng Li ◽  
Yulin Zhao
Keyword(s):  
Computer Algebra ◽  
Critical Points ◽  
Limit Cycles ◽  
Computer Algebra System ◽  
Necessary Conditions ◽  
Equilibrium Points ◽  
Positive Equilibrium ◽  
Kolmogorov System

A class of planar cubic Kolmogorov systems with harvest and two positive equilibrium points is investigated. With the help of computer algebra system MATHEMATICA, we prove that five limit cycles can be bifurcated simultaneously from the two critical points (1, 1) and (2, 2), respectively, in the first quadrant. Moreover, the necessary conditions of centers are obtained.

scholarly journals Oscillation of Second-Order Differential Equations with Multiple and Mixed Delays under a Canonical Operator

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1323
Author(s):  
Shyam Sundar Santra ◽  
Rami Ahmad El-Nabulsi ◽  
Khaled Mohamed Khedher
Keyword(s):  
Differential Equations ◽  
Necessary Conditions ◽  
Second Order ◽  
Multiple Delays ◽  
Mixed Delays ◽  
Neutral Differential Equations ◽  
Canonical Operator ◽  
Second Order Differential Equations ◽  
The Future ◽  
Sufficient And Necessary Conditions

In this work, we obtained new sufficient and necessary conditions for the oscillation of second-order differential equations with mixed and multiple delays under a canonical operator. Our methods could be applicable to find the sufficient and necessary conditions for any neutral differential equations. Furthermore, we proved the validity of the obtained results via particular examples. At the end of the paper, we provide the future scope of this study.

Sign in / Sign up

Export Citation Format

Share Document