16 research outputs found
Semi-analytical computation of heteroclinic connections between center manifolds with the parameterization method
This paper presents methodology for the computation of whole sets of
heteroclinic connections between iso-energetic slices of center manifolds of
center x center x saddle fixed points of autonomous Hamiltonian systems. It
involves: (a) computing Taylor expansions of the center-unstable and
center-stable manifolds of the departing and arriving fixed points through the
parameterization method, using a new style that uncouples the center part from
the hyperbolic one, thus making the fibered structure of the manifolds
explicit; (b) uniformly meshing iso-energetic slices of the center manifolds,
using a novel strategy that avoids numerical integration of the reduced
differential equations and makes an explicit 3D representation of these slices
as deformed solid ellipsoids; (c) matching the center-stable and
center-unstable manifolds of the departing and arriving points in a Poincar\'e
section. The methodology is applied to obtain the whole set of iso-energetic
heteroclinic connections from the center manifold of L2 to the center manifold
of L1 in the Earth-Moon circular, spatial Restricted Three-Body Problem, for
nine increasing energy levels that reach the appearance of Halo orbits in both
L1 and L2. Some comments are made on possible applications to space mission
design.Comment: 36 pages, 14 figure
Pseudo-heteroclinic connections between bicircular restricted four-body problems
In this paper, we show a mechanism to explain transport from the outer to the inner Solar system. Such a mechanism is based on dynamical systems theory. More concretely, we consider a sequence of uncoupled bicircular restricted four-body problems –BR4BP –(involving the Sun, Jupiter, a planet and an infinitesimal mass), being the planet Neptune, Uranus and Saturn. For each BR4BP, we compute the dynamical substitutes of the collinear equilibrium points of the corresponding restricted three-body problem (Sun, planet and infinitesimal mass), which become periodic orbits. These periodic orbits are unstable, and the role that their invariant manifolds play in relation with transport from exterior planets to the inner ones is discussed.Peer ReviewedPostprint (published version
Fast numerical computation of Lissajous and quasi-halo libration point trajectories
In this paper we present a methodology for the automatic generation of quasi–periodic libration point trajectories
(Lissajous and quasi–halo) of the Spatial, Circular Restricted Three–Body Problem. This methodology is based
on the computation of a mesh of orbits which, using interpolation strategies, gives an accurate quantitative representation
of the full set of libration point orbits. This representation, when combined with the one obtained using
Poincar´e maps, provides a useful tool for the design of missions to libration points fulfilling specific requirements.
The same methodology applies to stable and unstable manifolds as well. This paper extends and improves results
presented in [10].Postprint (published version
Semianalytical computation of heteroclinic connections between center manifolds with the parameterization method
This paper presents a methodology for the computation of whole sets of heteroclinic connections between isoenergetic slices of center manifolds of center x center saddle fixed points of autonomous Hamiltonian systems. It involves (a) computing Taylor expansions of the center-unstable and center-stable manifolds of the departing and arriving fixed points through the parameterization method, using a new style that uncouples the center part from the hyperbolic one, thus making the fibered structure of the manifolds explicit; (b) uniformly meshing isoenergetic slices of the center manifolds, using a novel strategy that avoids numerical integration of the reduced differential equations and makes an explicit three-dimensional representation of these slices as deformed solid ellipsoids; (c) matching the center-stable and center-unstable manifolds of the departing and arriving points in a PoincarĂ© section. The methodology is applied to obtain the whole set of isoenergetic heteroclinic connections from the center manifold of L2 to the center manifold of L1 in the Earth-Moon circular, spatial restricted three-body problem, for nine increasing energy levels that reach the appearance of halo orbits in both L1 and L2. Some comments are made on possible applications to space mission design.This work has been supported by the Spanish grants MCINN-AEI PID2020-118281GB-C31 and PID2021-125535NB-I00, the Catalan grant 2021 SGR 0013, the Spanish State Research Agency, through the Severo Ochoa and MarĂa de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M).With funding from the Spanish government through the "Severo Ochoa Centre of Excellence" accreditation (CEX2020-001084-M).Peer reviewe
Numerical study of the geometry of the phase space of the Augmented Hill Three-Body problem
The Augmented Hill Three-Body problem is an extension of the classical Hill problem that, among other applications, has been used to model the motion of a solar sail around an asteroid. This model is a 3 degrees of freedom (3DoF) Hamiltonian system that depends on four parameters. This paper describes the bounded motions (periodic orbits and invariant tori) in an extended neighbourhood of some of the equilibrium points of the model. An interesting feature is the existence of equilibrium points with a 1:1 resonance, whose neighbourhood we also describe. The main tools used are the computation of periodic orbits (including their stability and bifurcations), the reduction of the Hamiltonian to centre manifolds at equilibria, and the numerical approximation of invariant tori. It is remarkable how the combination of these techniques allows the description of the dynamics of a 3DoF Hamiltonian system
Pseudo-heteroclinic connections between bicircular restricted four-body problems
In this paper, we show a mechanism to explain transport from the outer to the inner Solar system. Such a mechanism is based on dynamical systems theory. More concretely, we consider a sequence of uncoupled bicircular restricted four-body problems -BR4BP -(involving the Sun, Jupiter, a planet and an infinitesimal mass), being the planet Neptune, Uranus and Saturn. For each BR4BP, we compute the dynamical substitutes of the collinear equilibrium points of the corresponding restricted three-body problem (Sun, planet and infinitesimal mass), which become periodic orbits. These periodic orbits are unstable, and the role that their invariant manifolds play in relation with transport from exterior planets to the inner ones is discussed
HAmsys 2014
In this talk we give an explanation of transport in the
solar system based in dynamical systems theory. More concretely
we consider (as a first approximation) different bicircular problems
(i.e. Sun, Jupiter, a planet and an infinitesimal mass), we take
sl natural periodic orbits which are unstable and we study
their invariant manifolds as well as the existence
of possible heteroclinic connections. The role that these
particular trajectories
play in relation with transport from exterior planets
to the inner ones is discussed. Finally, some comments
concerning a more realistic model of the Solar System are given
and dynamical substitutes of invariant objects (from simpler models)
are obtainedPostprint (published version
Fast numerical computation of Lissajous and quasi-halo libration point trajectories
In this paper we present a methodology for the automatic generation of quasi–periodic libration point trajectories
(Lissajous and quasi–halo) of the Spatial, Circular Restricted Three–Body Problem. This methodology is based
on the computation of a mesh of orbits which, using interpolation strategies, gives an accurate quantitative representation
of the full set of libration point orbits. This representation, when combined with the one obtained using
Poincar´e maps, provides a useful tool for the design of missions to libration points fulfilling specific requirements.
The same methodology applies to stable and unstable manifolds as well. This paper extends and improves results
presented in [10]
The parameterization method for invariant manifolds: from rigorous results to effective computations
This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems