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.

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 Topological enumeration of complex polynomial vector fields

2014 ◽  
Vol 35 (4) ◽  
pp. 1315-1344 ◽  
Author(s):  
J. TOMASINI
Keyword(s):  
Closed Form ◽  
Orthogonal Group ◽  
Vector Fields ◽  
Main Tool ◽  
Roots Of Unity ◽  
Complex Polynomial ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Appl Math ◽  
General Method

AbstractThe enumeration of combinatorial classes of the complex polynomial vector fields in$ \mathbb{C} $presented by K. Dias [Enumerating combinatorial classes of the complex polynomial vector fields in$ \mathbb{C} $.Ergod. Th. & Dynam. Sys. 33(2013), 416–440] is extended here to a closed form enumeration of combinatorial classes for degree$d$polynomial vector fields up to rotations of the$2(d- 1)\mathrm{th} $roots of unity. The main tool in the proof of this result is based on a general method of enumeration developed by V. A. Liskovets [Reductive enumeration under mutually orthogonal group actions.Acta Appl. Math. 52(1998), 91–120].

SIMPLE VECTOR FIELDS WITH COMPLEX BEHAVIOR

2006 ◽  
Vol 16 (02) ◽  
pp. 369-381 ◽  
Author(s):  
MANUELA A. D. AGUIAR ◽  
SOFIA B. S. D. CASTRO ◽  
ISABEL S. LABOURIAU
Keyword(s):  
Vector Fields ◽  
Invariant Manifolds ◽  
Symmetric Polynomial ◽  
Complex Behavior ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Construction Technique ◽  
Analytic Proof ◽  
Three Sphere ◽  
Transverse Intersection

We construct examples of vector fields on a three-sphere, amenable to analytic proof of properties that guarantee the existence of complex behavior. The examples are restrictions of symmetric polynomial vector fields in R4 and possess heteroclinic networks producing switching and nearby suspended horseshoes. The heteroclinic networks in our examples are persistent under symmetry preserving perturbations. We prove that some of the connections in the networks are the transverse intersection of invariant manifolds. The remaining connections are symmetry-induced. The networks lie in an invariant three-sphere and may involve connections exclusively between equilibria or between equilibria and periodic trajectories. The same construction technique may be applied to obtain other examples with similar features.

KNOTTED PERIODIC ORBITS AND CHAOTIC BEHAVIOR OF A CLASS OF THREE-DIMENSIONAL FLOWS

2011 ◽  
Vol 21 (09) ◽  
pp. 2505-2523 ◽  
Author(s):  
JIBIN LI ◽  
FENGJUAN CHEN
Keyword(s):  
Periodic Orbits ◽  
Invariant Torus ◽  
Vector Fields ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Symmetric Polynomial ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Perturbed Systems

This paper considers a class of three-dimensional systems constructed by rotating some planar symmetric polynomial vector fields. It shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on a family of invariant torus. For two three-dimensional systems, exact explicit parametric representations of the knotted periodic orbits are given. For their perturbed systems, the chaotic behavior is discussed by using two different methods.

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