Invariant circles for homogeneous polynomial vector fields on the 2-dimensional sphere

2006 ◽  
Vol 55 (1) ◽  
pp. 63-81 ◽  
Author(s):  
Jaume Llibre ◽  
Claudio Pessoa
Keyword(s):  
Homogeneous Polynomial ◽  
Vector Fields ◽  
Dimensional Sphere ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Invariant Circles

scholarly journals GENERATION OF SYMMETRIC PERIODIC ORBITS BY A HETEROCLINIC LOOP FORMED BY TWO SINGULAR POINTS AND THEIR INVARIANT MANIFOLDS OF DIMENSIONS 1 AND 2 IN ℝ3

2007 ◽  
Vol 17 (09) ◽  
pp. 3295-3302 ◽  
Author(s):  
MONTSERRAT CORBERA ◽  
JAUME LLIBRE
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifold ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Main Tool ◽  
Singular Points ◽  
Dimensional Sphere ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Heteroclinic Loop

In this paper we will find continuous periodic orbits passing near infinity for a class of polynomial vector fields in ℝ3. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane Σ and that possess a "generalized heteroclinic loop" formed by two singular points e+ and e- at infinity and their invariant manifolds Γ and Λ. Γ is an invariant manifold of dimension 1 formed by an orbit going from e- to e+, Γ is contained in ℝ3 and is transversal to Σ. Λ is an invariant manifold of dimension 2 at infinity. In fact, Λ is the two-dimensional sphere at infinity in the Poincaré compactification minus the singular points e+ and e-. The main tool for proving the existence of such periodic orbits is the construction of a Poincaré map along the generalized heteroclinic loop together with the symmetry with respect to Σ.

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