Planar quadratic differential systems with invariant straight lines of total multiplicity four

2008 ◽  
Vol 68 (4) ◽  
pp. 681-715 ◽  
Author(s):  
Dana Schlomiuk ◽  
Nicolae Vulpe
Keyword(s):  
Quadratic Differential ◽  
Differential Systems ◽  
Total Multiplicity ◽  
Straight Lines

THE GEOMETRY OF QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS WITH A WEAK FOCUS AND AN INVARIANT STRAIGHT LINE

2010 ◽  
Vol 20 (11) ◽  
pp. 3627-3662 ◽  
Author(s):  
JOAN C. ARTÉS ◽  
JAUME LLIBRE ◽  
DANA SCHLOMIUK
Keyword(s):  
Phase Portrait ◽  
Bifurcation Diagram ◽  
Quadratic Differential ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Differential Systems ◽  
Weak Focus ◽  
Bifurcation Set ◽  
Total Multiplicity ◽  
Straight Lines

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than a thousand papers were written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we conduct a global study of the class QWI of all real quadratic differential systems which have a weak focus and invariant straight lines of total multiplicity of at least two. This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real projective space of the parameters of this form. The bifurcation diagram yields 73 phase portraits for systems in QWI plus 26 additional phase portraits with the center at its border points. Algebraic invariants are used to construct the bifurcation set. We show that all systems in QWI necessarily have their weak focus of order one and invariant straight lines of total multiplicity exactly two. The phase portraits are represented on the Poincaré disk. The bifurcation set is algebraic and all points in this set are points of bifurcation of singularities. We prove that there is no phase portrait with limit cycles in this class but that there is a total of five phase portraits with graphics, four having the invariant line as a regular orbit and one phase portrait with an infinity of graphics which are all homoclinic loops inside a heteroclinic graphic with two singularities, both at infinity.

scholarly journals Large amplitude oscillations for a class of symmetric polynomial differential systems in R³

2007 ◽  
Vol 79 (4) ◽  
pp. 563-575 ◽  
Author(s):  
Jaume Llibre ◽  
Marcelo Messias
Keyword(s):  
Large Amplitude ◽  
Symmetric Polynomial ◽  
The Other ◽  
Poincaré Compactification ◽  
Heteroclinic Loop ◽  
Differential Systems ◽  
Heteroclinic Cycles ◽  
Polynomial Differential Systems ◽  
Global Study ◽  
Straight Lines

In this paper we study a class of symmetric polynomial differential systems in R³, which has a set of parallel invariant straight lines, forming degenerate heteroclinic cycles, which have their two singular endpoints at infinity. The global study near infinity is performed using the Poincaré compactification. We prove that for all n <FONT FACE=Symbol>Î</FONT> N there is epsilonn > 0 such that for 0 < epsilon < epsilonn the system has at least n large amplitude periodic orbits bifurcating from the heteroclinic loop formed by the two invariant straight lines closest to the x-axis, one contained in the half-space y > 0 and the other in y < 0.

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.

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