scholarly journals Symmetric periodic orbits near a heteroclinic loop formed by two singular points and their invariant manifolds of dimension 1 and 2

2006 ◽  
Vol 39 (50) ◽  
pp. 15313-15326 ◽  
Author(s):  
Montserrat Corbera ◽  
Jaume Llibre ◽  
Ernesto Pérez-Chavela
Keyword(s):  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Singular Points ◽  
Heteroclinic Loop ◽  
Symmetric Periodic Orbits

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 in the Anisotropic Kepler Problem

1983 ◽  
Vol 74 ◽  
pp. 263-270
Author(s):  
Josefina Casasaya ◽  
Jaume Llibre
Keyword(s):  
Energy Level ◽  
Periodic Orbits ◽  
Kepler Problem ◽  
Negative Energy ◽  
Velocity Curve ◽  
Zero Velocity ◽  
Anisotropic Kepler Problem ◽  
Symmetric Periodic Orbits ◽  
Group Of Symmetries

AbstractThe anisotropic Kepler problem has a group of symmetries with three generators; they are symmetries respect to zero velocity curve and the two axes of motion’s plane. For a fixed negative energy level it has four homothetic orbits. We describe the symmetric periodic orbits near these homothetic orbits. Full details and proofs will appear elsewhere (Casasayas-Llibre).

scholarly journals The Mechanism of Branching of Three-Dimensional Periodic Orbits from the Plane

1983 ◽  
Vol 74 ◽  
pp. 213-224
Author(s):  
I.A. Robin ◽  
V.V. Markellos
Keyword(s):  
Recent Work ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Three Dimensional ◽  
General Situation ◽  
Orbital Symmetry ◽  
Resonant Orbits ◽  
Symmetry Properties ◽  
Elliptic Restricted Problem ◽  
Symmetric Periodic Orbits

AbstractA linearised treatment is presented of vertical bifurcations of symmetric periodic orbits(bifurcations of plane with three-dimensional orbits) in the circular restricted problem. Recent work on bifurcations from vertical-critical orbits (av = ±1) is extended to deal with the v more general situation of bifurcations from vertical self-resonant orbits (av = cos(2Πn/m) for integer m,n) and it is shown that in this more general case bifurcating families of three-dimensional orbits always occur in pairs, the orbital symmetry properties being governed by the evenness or oddness of the integer m. The applicability of the theory to the elliptic restricted problem is discussed.

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