Abstract
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. © World Scientific Publishing Company.
Original language | English |
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Pages (from-to) | 3295-3302 |
Journal | International Journal of Bifurcation and Chaos in Applied Sciences and Engineering |
Volume | 17 |
DOIs | |
Publication status | Published - 1 Jan 2007 |
Keywords
- Heteroclinic loops
- Polynomial vector fields
- Symmetric periodic orbits