BUSCADOR RECOLECTA

Within the framework of the planar three-body problem we establish the existence of quasi-periodic motions and KAM 4-tori related to the co-orbital motion of two small moons about a large planet where the moons move in nearly circular orbits with almost equal radii. The approach is based on a combination of normal form and symplectic reduction theories and the application of a KAM theorem for high-order degenerate systems. To accomplish our results we need to expand the Hamiltonian of the three-body problem as a perturbation of two uncoupled Kepler problems. This approximation is valid in the region of phase space where co-orbital solutions occur., J. M. Cors was partially supported by grants MTM2016-77278-P (FEDER) and AGAUR
grant 2017 SGR 1617. J. F. Palacián and P. Yanguas have been partially supported by grants
MTM 2014-59433-C2-1-P and MTM 2017-88137-C2-1-P.

The well-known problem of the nonlinear stability of L4 and L5 in the circular spatial restricted three-body problem is revisited. Some new results in the light of the concept of Lie (formal) stability are presented. In particular, we provide stability and asymptotic estimates for three specific values of the mass ratio that remained uncovered. Moreover, in many cases we improve the estimates found in the literature., The authors are partially supported by Project MTM 2017-88137-C2-1-P of the Ministry
of Science, Innovation and Universities of Spain. D. Cárcamo-Díaz acknowledges support from
CONICYT PhD/2016-21161143. C. Vidal is partially supported by Fondecyt grant 1180288.

In this paper we focus on the classical dynamics of a neutral atom trapped in a doublewire waveguide in the presence of two uniform bias fields. Because the trapping region takes place in a plane perpendicular to the (parallel) wires, the dynamics is governed by
a two-degrees of freedom Hamiltonian where, besides the energy, the two bias fields are the relevant system’s parameters. An exhaustive study of the critical points of the potential energy surface, their stability and bifurcations is carried out, so that, two different
trapping regions are characterized. The dynamics in each of these regions is studied by applying classical perturbation theory, which provides an integrable approximation of the original Hamiltonian. The dynamics arising from this normalized Hamiltonian (stability of the equilibrium points, their bifurcations and the phase flow evolution) is then analyzed in a convenient set of phase variables. Poincaré surfaces of section to describe the structure and evolution of the phase space governed by the full Hamiltonian are also used. A complete agreement between the descriptions of the dynamics provided by the perturbation theory and the numerical studies is obtained., This work has been partly supported from the Spanish Ministry of Science and Innovation through the Project MTM2017-88137-CO (Subprojects MTM2017-88137-C2-1-P and MTM2017-88137-C2-2-P), and by University of La Rioja through Projects REGI 2018751 and REGI 2020/15.

In the framework of nonlinear stability of elliptic equilibria in Hamiltonian systems with n degrees of freedom we provide a criterion to obtain a type of formal stability, called Lie stability. Our result generalises previous approaches, as exponential stability in the sense of Nekhoroshev (excepting a few situations) and other classical results on formal stability of equilibria. In case of Lie stable systems we bound the solutions near the equilibrium over exponentially long times. Some examples are provided to illustrate our main contributions., The authors are partially supported by Projects MTM 2014-59433-C2-1-P of the Ministry of
Economy and Competitiveness of Spain and MTM 2017-88137-C2-1-P of the Ministry of Science,
Innovation and Universities of Spain. D. C.-D. acknowledges support from CONICYT PhD/2016-
21161143. C. Vidal is partially supported by Fondecyt, grant 1180288.

This is a post-peer-review, pre-copyedit version of an article published in Journal of Nonlinear Science (2018) 28:1293–1359. The final authenticated version is available online at https://doi.org/10.1007/s00332-018-9449-y, We study the dynamics of a family of perturbed three-degree-of-freedom Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry, and then, periodic solutions and KAM 3-tori of the full system are determined from the relative equilibria. Next, the oscillator symmetry is extended by
normalisation up to terms of degree 4 in rectangular coordinates; after truncation of
higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction in the two symmetries leads to a one-degree-of-freedom system that is completely analysed in the twice reduced space. All the relative equilibria together with the stability and parametric bifurcations are determined. Moreover, the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduced space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system., The authors are partially supported by Projects MTM 2011-28227-C02-01 of the Ministry of Science and
Innovation of Spain, MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of
Spain, and MTM 2017-88137-C2-1-P of the Ministry of Economy, Industry and Competitiveness of
Spain. D. Carrasco is also partially supported by Project DIUBB 165708 3/R, Universidad del Bío-Bío,
Chile and by FONDECYT Project 1181061, CONICYT (Chile).

We apply Reeb’s theorem to prove the existence of periodic orbits in the rotating Hénon–
Heiles system. To this end, a sort of detuned normal form is calculated that yields a reduced
system with at most four non degenerate equilibrium points. Linear stability and bifurcations
of equilibrium solutions mimic those for periodic solutions of the original system. We also
determine heteroclinic connections that can account for transport phenomena., This work has been partly supported from the Spanish Ministry of Science and Innovation
through the Projects MTM2014-59433-CO (Subprojects MTM2014-59433-C2-1-P and MTM2014-59433-
C2-2-P), MTM2017-88137-CO (Subprojects MTM2017-88137-C2-1-P and MTM2017-88137-C2-2-P), and
by University of La Rioja through Project REGI 2018751.

We consider the planar restricted N-body problem where the N−1 primaries are assumed to be in a central configuration whereas the infinitesimal particle escapes to infinity in a parabolic orbit. We prove the existence of transversal intersections between the stable and unstable manifolds of the parabolic orbits at infinity which guarantee the existence of a Smale’s horseshoe. This implies the occurrence of chaotic mo-tions, namely the oscillatory motions, that is, orbits for which the massless particle leaves every bounded region but it returns infinitely often to some fixed bounded region. Our achievement is based in an adequate scaling of the variables which allows us to write the Hamiltonian function as the Hamiltonian of the Kepler problem plus higher-order terms that depend on the chosen configuration. We compute the Melnikov function related to the first non-null perturbative term and characterize the cases where it has simple zeroes. Concretely, for some combinations of the configuration parameters, i.e. mass values and positions of the primaries, and for a specific value of a parameter related to the angular momentum vector, the Melnikov function vanishes, otherwise it has simple zeroes and the transversality condition is satisfied. When the Melnikov function corresponding to the principal part of the perturbation is zero we compute the next non-zero Melnikov function proving that it has simple zeroes. The theory is illustrated for various cases of restricted N-body problems, including the circular restricted three-body problem. No restrictions on the mass parameters are assumed., The authors have received partial support from Project Grant Red de Cuerpos Académicos de
Ecuaciones Diferenciales, Sistemas Dinámicos y Estabilización. PROMEP 2011-SEP, Mexico
and from Projects MTM 2014–59433–C2–1–P of the Ministry of Economy and Competitiveness
of Spain and MTM 2017-88137-C2-1-P of the Ministry of Economy, Industry and Competitiveness
of Spain.

This is a peer-reviewed, un-copyedited version of an article published in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1361-6544/ab1bc6., A family of perturbed Hamiltonians H = 1/2 (x^2 + X^2) − 1/2 (y^2 + Y^2)+1/2
(z^2 + Z^2) + 2[ (x^4 + y^4 + z^4) + (x^2 y^2 + x^2 z^2 + y^2 z^2)] in 1: −1:1 resonance
depending on two real parameters is considered. We show the existence and
stability of periodic solutions using reduction and averaging. In fact, there are
at most thirteen families for every energy level h < 0 and at most twenty six
families for every h > 0. The different types of periodic solutions for every
nonzero energy level, as well as their bifurcations, are characterised in terms
of the parameters. The linear stability of each family of periodic solutions,
together with the determination of KAM 3-tori encasing some of the linearly
stable periodic solutions is proved. Critical Hamiltonian bifurcations on the
reduced space are characterised. We find important differences with respect
to the dynamics of the 1:1:1 resonance with the same perturbation as the one
given here. We end up with an intuitive interpretation of the results from a
cosmological viewpoint., The authors are partially supported by Projects MTM 2011-28227-C02-01 of the Ministry of Science and Innovation of Spain, MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain and MTM 2017-88137-C2-1-P of the Ministry of Science, Innovation and Universities of Spain. C Vidal is partially supported by Project Fondecyt 1180288.

In the context of the spatial three-body problem and KAM theory, specifically in the regime where the
Hamiltonian function is split as the sum of two Keplerian terms plus a small perturbation, we deal with
quasi-periodic motions of the three bodies such that two of the three particles (the so-called inner bodies)
describe near-collision orbits. More precisely the inner bodies never collide, but they follow orbits that
are bounded straight lines or close to straight lines. The motion of the inner bodies occurs either near the
axis that is perpendicular to the invariable plane (i.e. the fixed plane orthogonal to the angular momentum
vector that passes through the centre of mass of the system) or near the invariable plane. The outer particle’s
trajectory has an eccentricity varying between zero and a value that is upper bounded by eM
2 < 1 and lies
near the invariable plane. The three bodies’ orbits fill in invariant 5-tori and when the inner particles move
in an axis perpendicular to the invariable plane, they correspond to new solutions of the three-body problem.
Our approach consists in a combination of a regularisation procedure with the construction of various
reduced spaces and the explicit determination of sets of symplectic coordinates. The various reduced spaces
we build depend on what symmetries are taken into consideration for the reduction. Moreover we apply an
iso-energetic theorem by Han, Li and Yi on the persistence of quasi-periodic solutions for Hamiltonian
systems with high-order proper degeneracy. All these elements allow us to calculate explicitly the torsions
for the possible combinations that the three particles’ motions can achieve, The authors have received partial support by Projects MTM 2011-28227-C02-01 of the Ministry of Science and Innovation of Spain; MTM 2014-59433-C2-1-P of the Ministry of Economy and Competitiveness of Spain; and MTM 2017-88137-C2-1-P of the Ministry of Economy, Industry and Competitiveness of Spain.

The goal of this paper is to investigate the effects that the replacement of the Coulomb potential by a soft-core Coulomb potential produces in the classical dynamics of a perturbed Rydberg hydrogen atom. As example, we consider a Rydberg hydrogen atom near a metal surface subjected to a constant electric field in the electron-extraction regime. Thence, the dynamics of the real perturbed Coulomb system, studied by applying the Levi–Civita regularization, is compared with that of the softened one. The results of this study show that the global behavior of the system is significantly altered when the original Coulomb potential part is replaced by a soft-core potential., This work has been partially supported by MTM2014-59433-C2-1-P and MTM2014-59433-C2-2-P of the Ministry of Economy
and Competitiveness of Spain and MTM2017-88137-C2-1-P and MTM2017-88137-C2-2-P of the Ministry of Economy, Industry and Competitiveness of Spain.

The paper considers the attitude nonlinear stability analysis of the spatial satellite problem and takes it one step further. A study of the Lie (formal) stability is presented and long-time estimates related to the Lie stable cases are provided. The connection with Nekhoroshev theory is also shown. Finally, KAM tori related to Lie stable, as well as unstable equilibria, are also calculated., This work was partially supported by Project MTM 2017-88137-C2-1-P of the Ministry of Science, Innovation and Universities of Spain. The work of the first author was supported by CONICYT PhD/2016-21161143. The work of the third author was partially supported by Fondecyt grant 1180288.

We present a procedure for the normalization of perturbed Keplerian problems in n dimensions based on Moser regularization of the Kepler problem and the invariants associated to the reduction process. The approach allows us not only to circumvent the problems introduced by certain classical variables used in the normalization of this kind of problems, but also to do both the normalization and reduction in one step. The technique is introduced for any dimensions and is illustrated for n = 2, 3 by relating Moser coordinates with Delaunay-like variables. The theory is applied to the spatial circular restricted three-body problem for the study of the existence of periodic and quasi-periodic solutions of rectilinear type., The authors have received partial support from Projects MTM 2014-59433-C2-1-P of the
Ministry of Economy and Competitiveness of Spain, from MTM 2017-88137-C2-1-P of the Ministry
of Economy, Industry and Competitiveness of Spain and from the Charles Phelps Taft Foundation.