Uniqueness of Limit Cycle for the Quadratic Systems with Weak Saddle and Focus

2004 ◽  
Vol 20 (4) ◽  
pp. 647-652 ◽  
Author(s):  
Shen Qi Zhao ◽  
Ping Guang Zhang
Keyword(s):  
Limit Cycle ◽  
Quadratic Systems ◽  
Weak Saddle

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.

Quadratic systems having a parabola as an integral curve

1989 ◽  
Vol 112 (1-2) ◽  
pp. 113-134 ◽  
Author(s):  
Colin Christopher
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Integral Curve ◽  
Canonical Forms ◽  
Quadratic Systems ◽  
Integral Curves

SynopsisThe class of quadratic systems having a parabola composed of integral curves is examined. Canonical forms are found for the members of this class, and conditions are obtained, using the Bendixson's Criterion and the Poincaré–Bendixson Theorem, for the existence or non-existence of limit cycles, in the case where there is a limit cycle “inside” the parabola (that is, in the convex component of its compliment).

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.

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.

FOUR NORMAL SIZE LIMIT CYCLES IN TWO-DIMENSIONAL QUADRATIC SYSTEMS

2011 ◽  
Vol 21 (02) ◽  
pp. 425-429 ◽  
Author(s):  
G. A. LEONOV
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Two Dimensional ◽  
Quadratic Systems ◽  
Weak Focus ◽  
First Order ◽  
Size Limit ◽  
Existence Criterion ◽  
Finite Disturbance

The existence criterion of three normal size limit cycles in quadratic systems with a weak focus of first order is obtained. Further, giving a finite disturbance for weak focus, the fourth normal size limit cycle is obtained. Bifurcation of appearance of two limit cycles via semistable cycle is given.

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