Mercè Ollé

On the families of periodic orbits which bifurcate from the circular Sitnikov motions

In this paper we deal with the circular Sitnikov problem as a subsystem of the three-dimensional circular restricted three-body problem. It has a first analytical part where by using elliptic functions we give the analytical expressions for the solutions of the circular Sitnikov problem and for the period function of its family of periodic orbits. We also analyze the qualitative and quantitative behavior of the period function. In the second numerical part, we study the linear stability of the family of periodic orbits of the Sitnikov problem, and of the families of periodic orbits of the three-dimensional circular restricted three-body problem which bifurcate from them; and we follow these bifurcated families until they end in families of periodic orbits of the planar circular restricted three-body problem. We compare our results with the previous ones of other authors on this problem. Finally, the characteristic curves of some bifurcated families obtained for the mass parameter close to 1/2 are also described.

The motion of Saturn coorbital satellites in the restricted three-body problem

This paper provides a description of the motion of Saturn coorbital satellites Janus and Epimetheus by means of horseshoe periodic orbits in the framework of the planar restricted three-body problem for the mass parameter

Numerical continuation of families of homoclinic connections of periodic orbits in the RTBP

\mu =3.5\times 10^{-9}

Invariant curves near Hamiltonian-Hopf bifurcations of four-dimensional symplectic maps

. The mechanism of existence of such orbits for any value of

SECOND-SPECIES SOLUTIONS IN THE CIRCULAR AND ELLIPTIC RESTRICTED THREE-BODY PROBLEM

\mu > 0

A note on the elliptic restricted three-body problem

and the Jacobi constant C close to

Invariant manifolds of L3 and horseshoe motion in the restricted three-body problem

C(L_3)

Second-species solutions in the circular and elliptic restricted three-body problem. II : Numerical explorations

, L 3 being an adequate collinear equilibrium point, is analyzed from two different points of view and a systematic way to compute the horseshoe periodic orbits is also described.

Two Classes of Cycler Trajectories in the Earth-Moon System

The goal of this paper is the numerical computation and continuation of homoclinic connections of the Lyapunov families of periodic orbits (p.o.) associated with the collinear equilibrium points, L1, L2 and L3, of the planar circular restricted three-body problem (RTBP). We describe the method used that allows us to follow individual families of homoclinic connections by numerical continuation of a system of (nonlinear) equations that has as unknowns the initial condition of the p.o., the linear approximation of its stable and unstable manifolds and a point in a given Poincare section in which the unstable and stable manifolds match. For the L3 case, some comments are made on the geometry of the manifold tubes and the possibility of obtaining trajectories with prescribed itineraries.

Quantitative estimates on the normal form around a non-semi-simple 1:−1 resonant periodic orbit

In this paper, we give a numerical description of an extended neighbourhood of a fixed point of a symplectic map undergoing an irrational transition from linear stability to complex instability, i.e. the so-called Hamiltonian–Hopf bifurcation. We have considered both the direct and inverse cases.This study is based on numerical computation of the Lyapunov families of invariant curves near the fixed point. We show how these families, jointly with their invariant manifolds and the invariant manifolds of the fixed point, organize the phase space around the bifurcation.