On the computation of reducible invariant tori on a parallel computer

We present an algorithm for the computation of reducible invariant tori of discrete dynamical systems that is suitable for tori of dimensions larger than 1. It is based on a quadratically convergent scheme that approximates, at the same time, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. The Floquet matrix describes the linearization of the dynamics around the torus and, hence, its linear stability. The algorithm presents a high degree of parallelism, and the computational effort grows linearly with the number of Fourier modes needed to represent the solution. For these reasons it is a very good option to compute quasi-periodic solutions with several basic frequencies. The paper includes some examples (flows) to show the efficiency of the method in a parallel computer. In these flows we compute invariant tori of dimensions up to 5, by taking suitable sections

Transport and invariant manifolds near L3 in the Earth-Moon Bicircular model

This paper focuses on the role of $\mathrm{L}_3$ to organise trajectories for a particle going from Earth to Moon and viceversa, and entering or leaving the Earth-Moon system. As a first model, we have considered the planar Bicircular problem to account for the gravitational effect of the Sun on the particle. The first step has been to compute a family of hyperbolic quasi-periodic orbits near $\mathrm{L}_3$. Then, the computation of their stable and unstable manifolds provides connections between Earth and Moon, and also generates trajectories that enter and leave the Earth-Moon system. Finally, by means of numerical simulations based on the JPL ephemeris we show that these connections can guide the journey of lunar ejecta towards the Earth

On the station keeping of a Solar sail in the Elliptic Sun-Earth system

In this work we focus on the dynamics of a solar sail in the Sun Earth Elliptic Restricted Three-Body Problem with solar radiation pressure. The considered situation is the motion of a sail close to the $L_{1}$ point, but displacing the equilibrium point with the sail so that it is possible to have continuous communication with the Earth. In previous works we derived a station keeping strategy for this situation but using the Circular RTBP as a model. In this paper we discuss the effect of the eccentricity in the region close to the sail-displaced $L_{1}$ point of the Circular RTBP. Then we show how to use the information on this dynamics to design a station keeping strategy. Finally, we apply these results to the GeoStorm mission, including errors in the sail orientation and on the estimation of the position of the sail in the simulations

Using invariant manifolds to capture an asteroid near the L3 point of the Earth-Moon Bicircular model

This paper focuses on the capture of Near-Earth Asteroids (NEAs) in a neighbourhood of the $\mathrm{L}_{3}$ point of the Earth-Moon system. The dynamical model for the motion of the asteroid is the planar Earth-Moon-Sun Bicircular problem (BCP). It is known that the $\mathrm{L}_{3}$ point of the Restricted Three-Body Problem is replaced, in the BCP, by a periodic orbit of centre $\times$ saddle type, with a family of mildly hyperbolic tori that is born from the elliptic direction of this periodic orbit. It is remarkable that some pieces of the stable manifolds of these tori escape (backward in time) the Earth-Moon system and become nearly circular orbits around the Sun. In this work we compute this family of invariant tori and also high order approximations to their stable/unstable manifolds. We show how to use these manifolds to compute an impulsive transfer of a NEA to an invariant tori near $\mathrm{L}_{3}$. As an example, we study the capture of the asteroid $2006 \mathrm{RH} 120$ in its approach of 2006. We show that there are several opportunities for this capture, with different costs. It is remarkable that one of them requires a $\Delta v$ as low as 20 $\mathrm{m} / \mathrm{s}$

On the stabilizing effect of Solar Radiation Pressure in the Earth-Moon system

Solar sails change the natural dynamics of systems: The typical trajectories are displaced and changed because of the effect of Solar Radiation Pressure (SRP). Moreover, if the effectivity of the sail is large enough, the instability of certain orbits can be diminished and even removed. In this paper we modify two models for the motion of a probe in the Earth-Moon system that include the effect of Sun’s gravity to take also into account the effect of SRP. These models, the Bicircular Problem (BCP) and the Quasi-Bicircular Problem (QBCP), are periodic perturbations of the Earth-Moon Restricted Three Body Problem (RTBP). The models are modified to consider the effect of the SRP upon a Solar Sail. We provide examples of periodic orbits that are stabilized (or made less unstable) due to the effect of SRP

Effective Reducibility of Quasi-Periodic Linear Equations close to Constant Coefficients

Let us consider the differential equation $$\dot{x}=(A+\varepsilon Q(t,\varepsilon))x, \;\;\;\; |\varepsilon|\le\varepsilon_0,$$ where A is an elliptic constant matrix and Q depends on time in a quasi-periodic (and analytic) way. It is also assumed that the eigenvalues of A and the basic frequencies of Q satisfy a diophantine condition. Then it is proved that this system can be reduced to $$\dot{y}=(A^{*}(\varepsilon)+\varepsilon R^{*}(t,\varepsilon))y, \;\;\;\; |\varepsilon|\le\varepsilon_0,$$ where $R^{*}$ is exponentially small in $\varepsilon$, and the linear change of variables that performs such a reduction is also quasi-periodic with the same basic frequencies as Q. The results are illustrated and discussed in a practical example

Families of Halo-like invariant tori around L2 in the Earth-Moon Bicircular Problem

The Bicircular Problem (BCP) is a periodic time dependent perturbation of the Earth-Moon Restricted Three-Body Problem that includes the direct gravitational effect of the Sun. In this paper we use the BCP to study the existence of Halo-like orbits around $L_{2}$ in the Earth-Moon system taking into account the perturbation of the Sun. By means of computing families of 2D invariant tori, we show that there are at least two different families of Halo-like quasi-periodic orbits around $L_{2}$

Classification of linear skew-products of the complex plane and an affine route to fractalization

Linear skew products of the complex plane, \left.\begin{array}{l} \theta \mapsto \theta+\omega \\ z \mapsto a(\theta) z \end{array}\right\} where $\theta \in \mathrm{T}, z \in \mathbb{C}, \frac{\omega}{2 \pi}$ is irrational, and $\theta \mapsto a(\theta) \in \mathbb{C} \backslash\{0\}$ is a smooth map, appear naturally when linearizing dynamics around an invariant curve of a quasi-periodically forced complex map. In this paper we study linear and topological equivalence classes of such maps through conjugacies which preserve the skewed structure, relating them to the Lyapunov exponent and the winding number of $\theta \mapsto a(\theta) .$ We analyze the transition between these classes by considering one parameter families of linear skew products. Finally, we show that, under suitable conditions, an affine variation of the maps above has a non-reducible invariant curve that undergoes a fractalization process when the parameter goes to a critical value. This phenomenon of fractalization of invariant curves is known to happen in nonlinear skew products, but it is remarkable that it also occurs in simple systems as the ones we present

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

On Quasiperiodic Perturbations of Ordinary Differential Equations

In this work we study several topics concerning quasi-periodic time-dependent perturbations of ordinary differential equations. This kind of equations appear as models in many applied problems of Celestial Mechanics, and we have used, as an illustration, the study of the behaviour near the equilateral libration points of the real Earth-Moon system. Let us introduce this problem as a motivation. As a first approximation, suppose that the Earth and Moon arc revolving in circular orbits around their centre of masses, neglect the effect of the rest of the solar system and neglect the spherical terms coming from the Earth and Moon (of course, all the effects minor than the above mentioned) as the relativistic corrections, must be neglected). With this, we can write the equations of motion of an infinitesimal particle (by infinitesimal we mean that the particle is influenced by the Earth and Moon, but it does not affect them) by means of Newton's Jaw. The study of the motion of that particle is the so-called Restricted Three Body Problem (RTBP). Usually, in order to simplify the equations, the units of length, time and mass are chosen so that the angular velocity of rotation, the sum of masses of the bodies and the gravitational constant are all equal to one. With these normalized units, the distance between the bodies is also equal to one. If these equations of motion are written in a rotating frame leaving fixed the Earth and Moon (these main bodies are usually called primaries), it is known that the system has five equilibrium points. Two of them can be found as the third vertex of equilateral triangles having the Earth and Moon as vertices, and they are usually called equilateral libration points.It is also known that, when the mass parameter "mi" (the mass of the small primary in the normalized units) is less than the Routh critical value "mi"(R) = 1/2(1 - square root (23/27) = 0.03852 ... (this is true in the Earth-Moon case) these points are linearly stable. Applying the KAM theorem to this case we can obtain that there exist invariant tori around these points. Now, if we restrict the motion of the particle to the plane of motion of the primaries we have that, inside each energy level, these tori split the phase space and this allows to prove that the equilateral points are stable (except for two values, "mi" = "mi"2 and "mi"= "mi"3 with low order resonances). In the spatial case, the invariant tori do not split the phase space and, due to the possible Arnold diffusion, these points can be unstable. But Arnold diffusion is a very slow phenomenon and we can have small neighbourhoods of "practical stability", that is, the particle will stay near the equilibrium point for very long time spans.Unfortunately, the real Earth-Moon system is rather complex. In this case, due to the fact that that the motions of the Earth and the Moon are non circular (even non elliptical) and the strong influence of the Sun, the libration points do not exist as equilibrium points, and we need to define "instantaneous" libration points as the ones forming an equilateral triangle with the Earth and the Moon at each instant. If we perform some numerical integrations starting at (or near) these points we can see that the solutions go away after a short period of time, showing that these regions are unstable.Two conclusions can be obtained from this fact. First: if we are interested in keeping a spacecraft there, we will need to use some kind of control. Second: the RTBP is not a good model for this problem} because the behaviour displayed by it is different from the one of the real system.For these reasons, an improved model has been developed in order to study this problem. This model includes the main perturbations (due to the solar effect and to the noncircular motion of the Moon), assuming that they are quasi-periodic. This is a very good approximation for time spans of some thousands of years. It is not clear if this is true for longer time spans, but this matter will not be considered in this work. This model is in good agreement with the vector field of the solar system directly computed by means of the JPL ephemeris, for the time interval for which the JPL model is available.The study of this kind of models is the main purpose of this work.First of all, we have focused our attention on linear differential equations with constant coefficients, affected by a small quasi-periodic perturbation. These equations appear as variational equations along a quasi-periodic solution of a general equation and they also serve as an introduction to nonlinear problems.The purpose is to reduce those systems to constant coefficients ones by means of a quasi-periodic change of variables, as the classical Floquet theorem does for periodic systems. It is also interesting to nave a way to compute this constant matrix, as well as the change of variables. The most interesting case occurs when the unperturbed system is of elliptic type. Other cases, as the hyperbolic one, have already been studied. We have added a parameter ("epsilon") in the system, multiplying the perturbation, such that if "epsilon" is equal to zero we recover the unperturbed system. In this case we have found that, under suitable hypothesis of non-resonance, analyticity and non-degeneracy with respect to "epsilon", it is possible to reduce the system to constant coefficients, for a cantorian set of values of "epsilon". Moreover, the proof is constructive in an iterative way. This means that it is possible to find approximations to the reduced matrix as well as to the change of variables that performs such reduction. These results are given in Chapter 1.The nonlinear case is now going to be studied. We have then considered an elliptic equilibrium point of an autonomous ordinary differential equation, and we have added a small quasi-periodic perturbation, in such a way that the equilibrium point does not longer exist. As in the linear case, we have put a parameter ("epsilon") multiplying the perturbation. There is some "practical" evidence that there exists a quasi-periodic orbit, having the same basic frequencies that the perturbation, such that, when the perturbation goes to zero, this orbit goes to the equilibrium point. Our results show that, under suitable hypothesis, this orbit exists for a cantorian set of values of "epsilon". We have also found some results related to the stability of this orbit. These results are given in Chapter 2.A remarkable case occurs when the system is Hamiltonian. Here it is interesting to know what happens to the invariant tori near these points when the perturbation is added. Note that the KAM theorem can not be applied directly due to the fact that the Hamiltonian is degenerated, in the sense that it has some frequencies (the ones of the perturbation) that have fixed values and they do not depend on actions in a diffeomorphic way. In this case, we have found that some tori still exist in the perturbed system. These tori come from the ones of the unperturbed system whose frequencies are non-resonant with those of the perturbation. The perturbed tori add these perturbing frequencies to the ones they already had. This can be described saying that the unperturbed tori are "quasi-periodically dancing" under the "rhythm" of the perturbation. These results can also be found in Chapter 2 and Appendix C.The final point of this work has been to perform a study of the behaviour near the instantaneous equilateral libration points of the real Earth-Moon system. The purpose of those computations has been to find a way of keeping a spacecraft near these points in an unexpensive way. As it has been mentioned above in the real system these points are not equilibrium points, and their neighbourhood displays unstability. This leads us to use some control to keep the spacecraft there. It would be useful to have an orbit that was always near these points, because the spacecraft could be placed on it. Thus, only a station keeping would be necessary. The simplest orbit of this kind that we can compute is the one that replaces the equilibrium point. In Chapter 3, this computation has been carried out first for a planar simplified model and then for a spatial model. Then, the solution found for this last model has been improved, by means of numerical methods, in order to have a real orbit of the real system (here, by real system we mean the model of solar system provided by the JPL tapes). This improvement has been performed for a given (fixed) time-span. That is sufficient for practical purposes. Finally, an approximation to the linear stability of this refined orbit has been computed, and a very mild unstability has been found, allowing for an unexpensive station keeping. These results are given in Chapter 3 and Appendix A.Finally, in Appendix B the reader can find the technical details concerning the way of obtaining the models used to study the neighbourhood of the equilateral points. This has been jointly developed with Gerard Gomez, Jaume Llibre, Regina Martinez, Josep Masdemont and Carles Simó.We study several topics concerning quasi-periodic time-dependent perturbations of ordinary differential equations. This kind of equations appear in many applied problems of Celestial Mechanics, and we have used, as an illustration, the study of the behaviour near the Lagrangian points of the real Earth-Moon system. For this purpose, a model has been developed. It includes the main perturbations (due to the Sun and Moon), assuming that they are quasi-periodic.Firstly, we deal with linear differential equations with constant coefficients, affected by a small quasi-periodic perturbation, trying to reduce then: to constant coefficients by means of a quasi-periodic change of variables. The most interesting case occurs when the unperturbed system is of elliptic type. We have added a parameter "epsilon" in the system, multiplying the perturbation, such that if "epsilon" is equal to zero we recover the unperturbed system. In this case, under suitable hypothesis of non-resonance, analyicity and non degeneracy with respect to "epsilon", it is possible to reduce the system to constant coefficients, for a cantorian set of values of "epsilon".In the nonlinear case, we have considered an elliptic equilibrium point of an autonomous differential equation, and we have added a small quasi-periodic perturbation, in such a way that the equilibrium point does not exist. As in the linear case, we have put a parameter ("epsilon") multiplying the perturbation. Then, for a cantorian set of "epsilon", there exists a quasi-periodic orbit having the same basic frequencies as the perturbation, going to the equilibrium point when t: goes to zero. Some results concerning the stability of this orbit are stated. When the system is Hamiltonian, we have found that some tori still exist in the perturbed system. These tori come from the ones of the unperturbed system whose frequencies are non-resonant with those of the perturbation, adding these perturbing frequencies to the ones they already had.Finally, a study of the behaviour near the Lagrangian points of the real Earth-Moon system is presented. The purpose has been to find the orbit replacing the equilibrium point. This computation has been carried out first for the model mentioned above and then it has been improved numerically, in order to have a real orbit of the real system. Finally, a study of the linear stability of this refined orbit has been done