Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

We consider the planar restricted three-body problem and the collinear equilibrium point L3, as an example of a center×saddle equilibrium point in a Hamiltonian with two degrees of freedom.We explore numerically the existence of symmetric and non-symmetric homoclinic orbits to L3, when varying the mass parameter μ. Concerning the symmetric homoclinic orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely, given a transversal value of μ for which there is a 1-round SHO, say μ1, we show that for any m ≥ 2, there are countable sets of values of μ, tending to μ1, corresponding to m-round SHO. Some comments on related analytical results are also made.Peer ReviewedPostprint (published version

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

We consider the planar restricted three-body problem and the collinear equilibrium point L3, as an example of a center×saddle equilibrium point in a Hamiltonian with two degrees of freedom.We explore numerically the existence of symmetric and non-symmetric homoclinic orbits to L3, when varying the mass parameter μ. Concerning the symmetric homoclinic orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely, given a transversal value of μ for which there is a 1-round SHO, say μ1, we show that for any m ≥ 2, there are countable sets of values of μ, tending to μ1, corresponding to m-round SHO. Some comments on related analytical results are also made.Peer ReviewedPostprint (published version

Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L1 and the other around L2, with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L1 and L2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the "interior" and "exterior" Hill's regions and other resonant phenomena

The collinear four body problem

In this master thesis we will study the collinear four body problem from the numerical perspective. More precisely we will find different families of periodic, homoclinic and connection orbits and generalize the results. An homoclinic orbit is an orbit such that its limit forward in time and backward in time exist and are the same (in our case will be a periodic orbit). On the other hand a connection orbit is an orbit such that its limit forward in time is the total collision (collision of the four masses) and its limit backward in time is a periodic orbit. These orbits play an essential role in the dynamics of the system

Study and simulation of the planar and circular restricted three-body problem

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2015, Director: Antoni Benseny ArdiacaWe study the Planar and Circular Restricted Three-Body Problem, as an idealization of the Three-Body Problem. We follow a dynamical systems approach. Once the main characteristics of the problem have been described, we try to explore a little bit the chaos of the system. Without pretending to be systematic, we focus on final evolutions. In particular, the parabolic final evolutions are used to show evidence of chaos, as they correspond to the invariant manifolds of the periodic orbit at infinity, which intersect non-tangentially in a certain Poincaré section, giving rise to transversal homoclinic points of the associated Poincaré map. Furthermore, we try to outline some differences between the integrable Kepler Problem and the non-integrable Planar and Circular Restricted Three-Body Problem, which can be explained by the splitting of the two previously mentioned invariant manifolds. The use of numerical methods has been fundamental for the realization of this work