Algebraic integrals of quadratic systems with a weak focus

1990 ◽  
pp. 373-384 ◽  
Author(s):  
Dana Schlomiuk
Keyword(s):  
Quadratic Systems ◽  
Weak Focus

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order

2004 ◽  
Vol 56 (2) ◽  
pp. 310-343 ◽  
Author(s):  
Jaume Llibre ◽  
Dana Schlomiuk
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Topological Classification ◽  
Phase Portraits ◽  
Third Order ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Weak Focus ◽  
Affine Invariants ◽  
Geometric Concepts

AbstractIn this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.

THE GEOMETRY OF QUADRATIC DIFFERENTIAL SYSTEMS WITH A WEAK FOCUS OF SECOND ORDER

2006 ◽  
Vol 16 (11) ◽  
pp. 3127-3194 ◽  
Author(s):  
JOAN C. ARTÉS ◽  
JAUME LLIBRE ◽  
DANA SCHLOMIUK
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Second Order ◽  
Phase Portraits ◽  
Nonlinear Phenomena ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Weak Focus ◽  
Bifurcation Set ◽  
The Family

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this class. In this article we make an interdisciplinary global study of the subclass [Formula: see text] which is the closure within real quadratic differential systems, of the family QW2 of all such systems which have a weak focus of second order. This class [Formula: see text] also includes the family of all quadratic differential systems possessing a weak focus of third order and topological equivalents of all quadratic systems with a center. The bifurcation diagram for this class, done in the adequate parameter space which is the three-dimensional real projective space, is quite rich in its complexity and yields 373 subsets with 126 phase portraits for [Formula: see text], 95 for QW2, 20 having limit cycles but only three with the maximum number of limit cycles (two) within this class. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by a set of points which we suspect to be analytic corresponding to global separatrices which have connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections. The global geometry of this class [Formula: see text] reveals interesting bifurcations phenomena; for example, all phase portraits with limit cycles in this class can be produced by perturbations of symmetric (reversible) quadratic systems with a center. Many other nonlinear phenomena displayed here form material for further studies.

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 Weak Focus, Limit Cycles, and Bifurcations for Bounded Quadratic Systems

1995 ◽  
Vol 115 (1) ◽  
pp. 193-223 ◽  
Author(s):  
C.Z. Li ◽  
J. Llibre ◽  
Z.F. Zhang
Keyword(s):  
Limit Cycles ◽  
Quadratic Systems ◽  
Weak Focus

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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