Papers by Esther Barrabés
In this paper we present a methodology for the automatic generation of trajectories within the Li... more In this paper we present a methodology for the automatic generation of trajectories within the Lis- sajous family of 2D invariant tori around L1 of the Spatial, Circular Restricted Three-Body Problem for the Earth-Moon mass ratio. This methodology is based on the computation of a mesh of orbits which, using interpolation strategies, gives an accurate q uantitative representation of the
The spatial, circular RTBP General framework. The RTBP r 2 r 1 3 m = 0 m = 2 µ m = 1− 1 µ P =( −1... more The spatial, circular RTBP General framework. The RTBP r 2 r 1 3 m = 0 m = 2 µ m = 1− 1 µ P =( −1, 0, 0) µ 2 L 2 L 1 L 3 P =(x,y,z) O µ 1 P =( , 0, 0) y z x
In the present work, we deal with horseshoe motion in the frame of the Restricted Three-Body Prob... more In the present work, we deal with horseshoe motion in the frame of the Restricted Three-Body Problem (RTBP) for different values of the mass parameter mm. On one hand, we study numerically families of periodic horseshoe orbits for m small and how they are organised. We figure out the mechanism of the organisation of such families from the two-body problem (m = 0). On the other hand, we study the existence of horseshoe periodic orbits for other values of m. We claim that the behaviour of the invariant manifolds associated to the equilibrium point L3 as well as the existence of homoclinic orbits play an important role.
manifolds of L3 and horseshoe motion in the
Communications in Nonlinear Science and Numerical Simulation, 2015
ABSTRACT The main purpose of the paper is the study of the motion of a massless body attracted, u... more ABSTRACT The main purpose of the paper is the study of the motion of a massless body attracted, under the Newton’s law of gravitation, by two equal masses moving in parabolic orbits all over in the same plane, the planar parabolic restricted three-body problem. We consider the system relative to a rotating and pulsating frame where the equal masses (primaries) remain at rest. The system is gradient-like and has exactly ten hyperbolic equilibrium points lying on the boundary invariant manifolds corresponding to escape of the primaries in past and future time. The global flow of the system is described in terms of the final evolution (forwards and backwards in time) of the solutions. The invariant manifolds of the equilibrium points play a key role in the dynamics. We study the connections, restricted to the invariant boundaries, between the invariant manifolds associated to the equilibrium points. Finally we study numerically the connections in the whole phase space, paying special attention to capture and escape orbits.
Hip-Hop solutions of the 2N -body problem with equal masses are shown to exist using a topologica... more Hip-Hop solutions of the 2N -body problem with equal masses are shown to exist using a topological argument. These solutions are close to a planar regular 2N -gon homographic configuration with values of the eccentricity close to 1, plus a small vertical oscillations in which each mass.
SIAM Journal on Applied Dynamical Systems, 2010
ABSTRACT We study the dynamics of an extremely idealized model of a planetary ring. In particular... more ABSTRACT We study the dynamics of an extremely idealized model of a planetary ring. In particular, we study the motion of an infinitesimal particle moving under the gravitational influence of a large central body and a regular n-gon of smaller bodies as n tends to infinity. Our goal is to gain insight into the structure of thin, isolated rings.
SIAM Journal on Applied Dynamical Systems, 2008
Abstract. The existence of a new class of inclined periodic orbits of the collision restricted th... more Abstract. The existence of a new class of inclined periodic orbits of the collision restricted threebody problem is shown. The symmetric periodic solutions found are perturbations of elliptic kepler orbits and they exist only for special values of the inclination and are related ...
Qualitative Theory of Dynamical Systems, 2013
ABSTRACT The aim of this work is to provide an insight of an idealized model of a planetary ring.... more ABSTRACT The aim of this work is to provide an insight of an idealized model of a planetary ring. The model is a limit case of the planar circular restricted 1 + n body problem, where an infinitesimal particle moves under the gravitational influence of a large central body and n smaller bodies located on the vertices of a regular n-gon. When considering n tending to infinity, a model depending on one parameter is obtained. We study the main important structures of the problem depending on this parameter (equilibria, Hill’s regions, linear stability, …). We use Poincaré maps, for different values of the parameter, in order to predict the width of the ring and the richness of the dynamics that occur is discussed. This work is a continuation of the work presented in Barrabés by (SIAM J Appl Dyn Syst 9:634–658, 2010).
Physica D: Nonlinear Phenomena, 2012
ABSTRACT We consider the problem of the hydrogen atom interacting with a circularly polarized mic... more ABSTRACT We consider the problem of the hydrogen atom interacting with a circularly polarized microwave field, modeled as a perturbed Kepler problem. A remarkable feature of this system is that the electron can follow what we term erratic orbits before ionizing. In an erratic orbit the electron makes multiple large distance excursions from the nucleus with each excursion being followed by a close approach to the nucleus, where the interaction is large. Here we are interested in the mechanisms that explain this observation. We find that the manifolds associated with certain hyperbolic periodic orbits may play an important role, despite the fact that, in some respects, the dynamics is almost Keplerian. A study of some relevant invariant objects is carried out for different system parameters. The consequences of our findings for ionization of an electron by the external field are also discussed.
Physica D: Nonlinear Phenomena, 2010
We show the existence of families of hip–hop solutions in the equal–mass 2N–body problem which ar... more We show the existence of families of hip–hop solutions in the equal–mass 2N–body problem which are close to highly eccentric planar elliptic homographic motions of 2N bodies plus small perpendicular non–harmonic oscillations. By introducing a parameter ϵ, the homographic motion and the small amplitude oscillations can be uncoupled into a purely Keplerian homographic motion of fixed period and a vertical
Nonlinearity, 2006
In this paper, we consider horseshoe motion in the planar restricted three-body problem. On one h... more In this paper, we consider horseshoe motion in the planar restricted three-body problem. On one hand, we deal with the families of horseshoe periodic orbits (which surround three equilibrium points called L 3 , L 4 and L 5 ), when the mass parameter µ is positive and small; we describe the structure of such families from the two-body problem (µ = 0). On the other hand, the region of existence of horseshoe periodic orbits for any value of µ ∈ (0, 1/2] implies the understanding of the behaviour of the invariant manifolds of L 3 . So, a systematic analysis of such manifolds is carried out. As well the implications on the number of homoclinic connections to L 3 , and on the simple infinite and double infinite period homoclinic phenomena are also analysed. Finally, the relationship between the horseshoe homoclinic orbits and the horseshoe periodic orbits are considered in detail.
Nonlinearity, 2009
The goal of this paper is the numerical computation and continuation of families of homoclinic co... more The goal of this paper is the numerical computation and continuation of families of homoclinic connections of the Lyapunov families of periodic orbits (p.o.) associated with the collinear equilibrium points, L 1 , L 2 and L 3 , of the planar circular Restricted Three-Body Problem (RTBP). We describe the method used that allows 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 Poincaré section in which the unstable and stable manifolds match. For the L 3 case, some comments are made on the geometry of the manifold tubes and the possibility of obtaining trajectories with prescribed itineraries.
Nonlinearity, 2007
The fact that a continuous self-map of a tree has positive topolog- ical entropy is related to th... more The fact that a continuous self-map of a tree has positive topolog- ical entropy is related to the amount of difierent gods (greatest odd divisors) exhibited by its set of periods. Llibre & Misiurewicz (11) and Blokh (9) give generic upper bounds for the maximum number of gods that a zero entropy tree map f : T ¡! T can
Nonlinearity, 2013
ABSTRACT This paper is devoted to the numerical computation and continuation of families of heter... more ABSTRACT This paper is devoted to the numerical computation and continuation of families of heteroclinic connections between hyperbolic periodic orbits (POs) of a Hamiltonian system. We describe a method that requires the numerical continuation of a nonlinear system that involves the initial conditions of the two POs, the linear approximations of the corresponding manifolds and a point in a given Poincaré section where the unstable and stable manifolds match. The method is applied to compute families of heteroclinic orbits between planar Lyapunov POs around the collinear equilibrium points of the restricted three-body problem in different scenarios. In one of them, for the Sun–Jupiter mass parameter, we provide energy ranges for which the transition between different resonances is possible.
Journal of Guidance, Control, and Dynamics, 2010
ABSTRACT Anticipating the need for infrastructure in cislunar space to support telecommunications... more ABSTRACT Anticipating the need for infrastructure in cislunar space to support telecommunications, navigation, and crew and cargo transportation, the available periodic orbits that encounter both the Earth and the moon, termed cyclers, should be characterized and their utility should be evaluated using appropriate metrics. Using the planar circular restricted three-body problem to model spacecraft motion in the Earth moon system, a classification scheme based on resonant orbits and homoclinic connections is proposed to summarize two broad cycler classes. One class consists of high-energy near-elliptical cyclers; the second class consists of low-energy cyclers that use the dynamical structure associated with the collinear libration point between the Earth and moon. The first class is organized around resonant cyclers that can be easily characterized and are representative of the whole class. The second class of cyclers is organized by the homoclinic connections associated with the unstable Lyapunov family of periodic orbits around the libration point between the Earth and moon. The homoclinic connections bound sets of cyclers, provide close approximations of the cyclers, and can thus be used to identify their characteristics. Computational methods for both cycler classes, based on continuation and differential correction, are discussed and demonstrated.
International Journal of Mathematics and Mathematical Sciences, 2005
We answer the following question: given any n ∈ N, which is the minimum number of endpoints e n o... more We answer the following question: given any n ∈ N, which is the minimum number of endpoints e n of a tree admitting a zero-entropy map f with a periodic orbit of period n? We prove that e n = s 1 s 2 ··· s k − k i=2 s i s i+1 ··· s k , where n = s 1 s 2 ··· s k is the decomposition of n into a product of primes such that s i ≤ s i+1 for 1 ≤ i < k. As a corollary, we get a criterion to decide whether a map f defined on a tree with e endpoints has positive entropy: if f has a periodic orbit of period m with e m > e, then the topological entropy of f is positive.
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Papers by Esther Barrabés