Phase portraits of quadratic systems without finite critical points

1996 ◽  
Vol 27 (2) ◽  
pp. 207-222 ◽  
Author(s):  
J.W. Reyn
Keyword(s):  
Critical Points ◽  
Phase Portraits ◽  
Quadratic Systems

scholarly journals Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families Information

2021 ◽  
Vol 20 ◽  
pp. 186-195
Author(s):  
Orge Rodríguez Contreras ◽  
Alberto Reyes Linero ◽  
Bladimir Blanco Montes ◽  
Primitivo B. Acosta Humánez
Keyword(s):  
Critical Points ◽  
Stable Manifold ◽  
Phase Portraits ◽  
Infinite Plane ◽  
Finite Plane ◽  
Quadratic Systems ◽  
Liénard Equations ◽  
The Stability ◽  
Transcritical Bifurcations

This article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of the critical points in the finite plane, its bifurcations, stable manifold and lastly, the stability of the critical points in the infinite plane, afterwards the phase portraits resulting from the analysis, moreover Algebraic aspects are also included such that hamiltonian cases and Galois differential groupes. It should be noted that these families have associated oscillating type problems given their similarity to the Liénard equations.

scholarly journals Quadratic systems with invariant straight lines of total multiplicity two having Darboux invariants

2015 ◽  
Vol 17 (03) ◽  
pp. 1450018 ◽  
Author(s):  
Jaume Llibre ◽  
Regilene D. S. Oliveira
Keyword(s):  
Polynomial Systems ◽  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Quadratic Systems ◽  
Total Multiplicity ◽  
Straight Lines

In this paper, we present the global phase portraits in the Poincaré disc of the planar quadratic polynomial systems which admit invariant straight lines with total multiplicity two and Darboux invariants.

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.

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