Phase portraits for quadratic homogeneous polynomial vector fields on % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqr1ngB % PrgifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaqFr0x % c9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8fr % Fve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJX % wAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGqbaiab-jj8tnaaCaaa % leqabaGaaGOmaaaaaaa!4488! $$ \mathbb{S}^2 $$

2009 ◽  
Vol 58 (3) ◽  
pp. 361-406 ◽  
Author(s):  
Jaume Llibre ◽  
Claudio Pessoa
Keyword(s):  
Homogeneous Polynomial ◽  
Vector Fields ◽  
Phase Portraits ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

scholarly journals SYMMETRIC PERIODIC ORBITS NEAR HETEROCLINIC LOOPS AT INFINITY FOR A CLASS OF POLYNOMIAL VECTOR FIELDS

2006 ◽  
Vol 16 (11) ◽  
pp. 3401-3410 ◽  
Author(s):  
MONTSERRAT CORBERA ◽  
JAUME LLIBRE
Keyword(s):  
Phase Portrait ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Phase Portraits ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Open Set ◽  
Symmetric Periodic Orbits

For polynomial vector fields in ℝ3, in general, it is very difficult to detect the existence of an open set of periodic orbits in their phase portraits. Here, we characterize a class of polynomial vector fields of arbitrary even degree having an open set of periodic orbits. The main two tools for proving this result are, first, the existence in the phase portrait of a symmetry with respect to a plane and, second, the existence of two symmetric heteroclinic loops.

scholarly journals Centers of weight-homogeneous polynomial vector fields on the plane

2016 ◽  
Vol 145 (6) ◽  
pp. 2539-2555 ◽  
Author(s):  
Jaume Giné ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Homogeneous Polynomial ◽  
Vector Fields ◽  
Polynomial Vector ◽  
Polynomial Vector Fields
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