scholarly journals Limit Cycle Problem for Quadratic System with some Applications of a Class I

2021 ◽  
pp. 38-44
Author(s):  
Ali Bakur Barsham ALmurad ◽  
Elamin Mohammed Saeed Ali
Keyword(s):  
Limit Cycle ◽  
Quadratic System ◽  
Class I ◽  
Planar System

This paper is part of a wider study limit cycle problems of a class and planar system; we study the existence of limit cycle for polynomial planar system. In this paper, we present a proof of a result on the existence of limit cycle of the Quadratic System: 

Limit cycle problem of quadratic system of type (III) m=0, (III)

2005 ◽  
Vol 20 (4) ◽  
pp. 431-440
Author(s):  
Ali Elamin ◽  
M. Saeed ◽  
Luo Dingjun
Keyword(s):  
Limit Cycle ◽  
Quadratic System ◽  
Type Iii

scholarly journals Existence of Limit Cycles for Liénard System with Application

2021 ◽  
pp. 33-37
Author(s):  
Ali Bakur Barsham ALmurad ◽  
Elamin Mohammed Saeed Ali
Keyword(s):  
Mathematical Method ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Planar System ◽  
Liénard System ◽  
System Form ◽  
Lienard System

This paper is part of a wider study limit cycle problems and planar system; The aims of this is to study the existence of limit cycle for Liénard system. We followed the historical analytical mathematical method to present a proof of a result on the existence of limit cycle for Liénard system form x ̇=y-F(x) ,y ̇=-g(x)

scholarly journals Quadratic systems with a degenerate critical point

1988 ◽  
Vol 38 (1) ◽  
pp. 1-10 ◽  
Author(s):  
W. A. Coppel
Keyword(s):  
Critical Point ◽  
Limit Cycle ◽  
Quadratic System ◽  
Quadratic Systems ◽  
Degenerate Critical Point

It is shown that a quadratic system with a degenerate critical point has at most one limit cycle.

scholarly journals Quadratic systems with a weak focus

1991 ◽  
Vol 44 (3) ◽  
pp. 511-526 ◽  
Author(s):  
Zhang Pingguang ◽  
Cai Suilin
Keyword(s):  
Singular Point ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Relative Position ◽  
Quadratic System ◽  
Singular Points ◽  
Finite Plane ◽  
Quadratic Systems ◽  
Weak Focus

In this paper we study the number and the relative position of the limit cycles of a plane quadratic system with a weak focus. In particular, we prove the limit cycles of such a system can never have (2, 2)-distribution, and that there is at most one limit cycle not surrounding this weak focus under any one of the following conditions:(i) the system has at least 2 saddles in the finite plane,(ii) the system has more than 2 finite singular points and more than 1 singular point at infinity,(iii) the system has exactly 2 finite singular points, more than 1 singular point at infinity, and the weak focus is itself surrounded by at least one limit cycle.

On the limit cycle bifurcation of a polynomial system from a global center

2014 ◽  
Vol 12 (03) ◽  
pp. 251-268 ◽  
Author(s):  
Yanqin Xiong ◽  
Maoan Han
Keyword(s):  
Limit Cycle ◽  
Polynomial System ◽  
Melnikov Function ◽  
Quadratic System ◽  
Hopf Bifurcations ◽  
First Order ◽  
Global Center ◽  
Limit Cycle Bifurcation

In this paper, we consider a quadratic system with a global center under polynomial perturbations of degree n(n ≥ 1). By using the first-order Melnikov function, we study Poincaré and Poincaré–Andronov–Hopf bifurcations. We prove that both Poincaré and Hopf cyclicity are n for n ≥ 2 up to the first order in ε.

scholarly journals On the limit cycle distribution over two nests in quadratic systems

1995 ◽  
Vol 52 (3) ◽  
pp. 461-474 ◽  
Author(s):  
Xianhua Huang ◽  
J.W. Reyn
Keyword(s):  
Limit Cycle ◽  
Critical Points ◽  
Limit Cycles ◽  
Affirmative Answer ◽  
Quadratic System ◽  
Quadratic Systems ◽  
Hilbert’S 16Th Problem ◽  
Hilbert's 16Th Problem

As a contribution to the solution of Hilbert's 16th problem the question is considered whether in a quadratic system with two nests of limit cycles at least in one nest there exists precisely one limit cycle. An affirmative answer to this question is given for the case that the sum of the multiplicities of the finite critical points in the system is equal to three.

Bifurcation of Nonhyperbolic Limit Cycles in Piecewise Smooth Planar Systems with Finitely Many Zones

2017 ◽  
Vol 27 (10) ◽  
pp. 1750162 ◽  
Author(s):  
Yurong Li ◽  
Liping Yuan ◽  
Zhengdong Du
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Bifurcation Of Limit Cycles ◽  
Planar System ◽  
Planar Systems ◽  
The Stability ◽  
Straight Lines ◽  
First Time

Like for smooth systems, it is very important to discuss the stability and bifurcation of limit cycles in a piecewise smooth planar system. Most of the previous works focus only on hyperbolic limit cycles. Few works have considered nonhyperbolic limit cycles. In fact, to date, no concrete examples of piecewise smooth planar system with nonhyperbolic limit cycles have been given in literature. In this paper, we consider for the first time the bifurcation of nonhyperbolic limit cycles in piecewise smooth planar systems with discontinuities on finitely many straight lines intersecting at the origin. We present a method of Melnikov type to derive two quantities which can be used to determine the stability and the number of limit cycles that can bifurcate from a nonhyperbolic limit cycle of a piecewise smooth planar system. As applications, we present two examples of piecewise smooth systems with two and three zones respectively whose unperturbed system has a nonhyperbolic limit cycle.

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