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

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

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 obtaine