Phase portraits of the quadratic vector fields with a polynomial first integral

2006 ◽  
Vol 55 (3) ◽  
pp. 420-440 ◽  
Author(s):  
Belén García ◽  
Jesús S. Pérez Del Río ◽  
Jaume Llibre
Keyword(s):  
Vector Fields ◽  
First Integral ◽  
Phase Portraits ◽  
Quadratic Vector Fields ◽  
Polynomial First Integral

scholarly journals Quadratic systems with an invariant conic having Darboux invariants

2018 ◽  
Vol 20 (04) ◽  
pp. 1750033
Author(s):  
Jaume Llibre ◽  
Regilene Oliveira
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Complete Characterization ◽  
Phase Portraits ◽  
Affine Transformations ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Quadratic Vector Fields ◽  
Polynomial Differential Systems

The complete characterization of the phase portraits of real planar quadratic vector fields is very far from being accomplished. As it is almost impossible to work directly with the whole class of quadratic vector fields because it depends on twelve parameters, we reduce the number of parameters to five by using the action of the group of real affine transformations and time rescaling on the class of real quadratic differential systems. Using this group action, we obtain normal forms for the class of quadratic systems that we want to study with at most five parameters. Then working with these normal forms, we complete the characterization of the phase portraits in the Poincaré disc of all planar quadratic polynomial differential systems having an invariant conic [Formula: see text]: [Formula: see text], and a Darboux invariant of the form [Formula: see text] with [Formula: see text].

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