arXiv subjects

Alessandra Celletti

Publications and source records attributed to Alessandra Celletti.

At least 19 recordsLinked to original sources

arXiv2026

Effective stability estimates close to resonances with applications to rotational dynamics

We consider nearly-integrable Hamiltonian systems defined over a non-resonant domain. In the neighborhood of resonances, we use Nekhoroshev-like estimates to provide effective stability bounds for the action variables over long time. The applicability conditions of these estimates allow some freedom in the choice of parameters. Hence, we develop an optimization algorithm for choosing parameters that maximize the stability time. To further improve the stability estimates, we use perturbation theory to reduce the norm of the perturbing function. We implement this procedure (effective stability estimates and perturbation theory) to analyze the stability of sequences of irrational (Diophantine) frequencies converging to frequencies corresponding to resonances. We consider two applications to models describing problems of rotational dynamics in Celestial Mechanics: the spin-orbit problem, described by a 1D time-dependent Hamiltonian, and the spin-spin-orbit model, described by a 2D time-dependent Hamiltonian. We show stability results for orbits close to the main resonances associated with such models.

math.DS↗

arXiv2025

The dynamics of the spin-spin problem in Celestial Mechanics

This work investigates different models of rotational dynamics of two rigid bodies with the shape of an ellipsoid, moving under their gravitational influence. The focus of this study is on their behavior, their linear stability, and numerical investigation of the main resonances. We assume that the spin axes of the two bodies are perpendicular to the orbital plane and coinciding with the direction of their shortest physical axis. In the basic approximation, we assume that the orbits of the centers of mass are Keplerian and we retain the lowest order of the potential, according to which the rotational motions of the two bodies are decoupled, the so-called spin-orbit problem. When considering highest order approximation of the potential, the rotational motions become coupled giving rise to the so-called spin-spin problem. Finally, we release the assumption that the orbit is Keplerian, which implies that the rotational dynamics is coupled to the variation of the orbital elements. The resulting system is called the full spin-spin problem. We also consider the above models under the assumption that one or both bodies are non rigid; the dissipative effect is modeled by a linear function of the rotational velocity, depending on some dissipative and drift coefficients. We consider three main resonances, namely the (1:1,1:1), (3:2,3:2), (1:1,3:2) resonances and we start by analyzing the linear stability of the equilibria in the conservative and dissipative settings (after averaging and keeping only the resonant angle), showing that the stability depends on the value of the orbital eccentricity. By a numerical integration of the equations of motion, we compare the spin-orbit and spin-spin problems to highlight the influence of the coupling term of the potential in the conservative and dissipative case...

math.DS↗

arXiv2025

The dynamics around the collinear points of the elliptic three-body problem: A normal form approach

We study the dynamics of the collinear points in the planar, restricted three-body problem, assuming that the primaries move on an elliptic orbit around a common barycenter. The equations of motion can be conveniently written in a rotating pulsating barycentric frame, taking the true anomaly as independent variable. We consider the Hamiltonian modeling this problem in the extended phase space and we imple ment a normal form to make a center manifold reduction. The normal form provides an approximate solution for the Cartesian coordinates, which allows us to construct several kinds of orbits, most notably planar and vertical Lyapunov orbits, and halo orbits. We compare the analytical results with a numerical simulation, which requires special care in the selection of the initial conditions.

math.DS↗

arXiv2025

Stability and bifurcations of resonances in ring's dynamics

We use perturbation theory and bifurcation theory to analyze the dynamical behavior of resonances, associated to a model describing a particle moving within a ring around a celestial object. The central body is modeled as a homogeneous triaxial ellipsoid, rotating about its shortest physical axis at a constant angular velocity. It is assumed that the massless ring particle moves within the equatorial plane of the ellipsoid. The dynamics of the particle is studied using epicyclic variables, that lead to a straightforward definition of corotation and Lindblad resonances. These resonances are associated to a Hamiltonian function with two degrees of freedom, for which we compute appropriate expansions for the normal form and the resonant Hamiltonian. Initially, the normal form is verified to be non--degenerate, thereby guaranteeing the existence of invariant KAM tori, providing the stability of the resonances, through their confinement in phase space. Subsequently, two exemplary test cases are examined: a nearly spherical ellipsoid and a highly aspherical ellipsoid. Furthermore, this study concentrates on three principal resonances: corotation, 1:2, and 1:3, for which we present results concerning their dynamical behavior obtained analyzing the Hamiltonian formulation of the model and the resonant normal form. Specifically, we examine the phase space structure, the amplitude of libration around the resonances, and the occurrence of bifurcations. Remarkably, in none of the two studied test cases the 1:3 resonance presents evidence of bifurcations for relevant values of the eccentricity. Our dynamical study thus supports a greater probability of selecting the $1:3$ resonance in comparison to the other resonances.

math-ph↗

arXiv2023

Dalle leggi di Keplero alla materia oscura attraverso la storia di donne scienziate

The universe is composed of ordinary matter (for example, stars, planets, ourselves), dark matter and dark energy. The estimated percentage of dark matter is about $27\%$ of the entire universe, while $5\%$ is ordinary matter and the remaining part is dark energy. This article introduces the fundamental ingredients that led to conjecture the existence of dark matter and to study its properties. To this result have contributed, directly or indirectly, numerous female mathematicians and astronomers, who we quote to give them the right credit.

physics.pop-ph↗

arXiv2022

Breakdown of rotational tori in 2D and 4D conservative and dissipative standard maps

We study the breakdown of rotational invariant tori in 2D and 4D standard maps by implementing three different methods. First, we analyze the domains of analyticity of a torus with given frequency through the computation of the Lindstedt series expansions of the embedding of the torus and the drift term. The Pad\'e approximants provide the shape of the analyticity domains by plotting the poles of the polynomial at the denominator of the approximants. Secondly, we implement a Newton method to construct the embedding of the torus; the breakdown threshold is then estimated by looking at the blow-up of the Sobolev norms of the embedding. Finally, we implement an extension of Greene method to get information on the breakdown threshold of an invariant torus with irrational frequency by looking at the stability of the periodic orbits with periods approximating the frequency of the torus. We apply these methods to 2D and 4D standard maps. The 2D maps can either be conservative (symplectic) or dissipative ( more precisely, conformally symplectic, namely a dissipative map with the geometric property to transform the symplectic form into a multiple of itself). The 4D maps are obtained coupling $(i)$ two symplectic standard maps, or $(ii)$ two conformally symplectic standard maps, or $(iii)$ a symplectic and a conformally symplectic standard map. Concerning the results, Pad\'e and Newton methods perform well and provide reliable and consistent results (although we implemented Newton method only for symplectic and conformally symplectic maps). Our implementation of the extension of Greene method is inconclusive, since it is computationally expensive and delicate, especially in 4D non-symplectic maps, also due to the existence of Arnold tongues.

math.DS↗

arXiv2022

Accurate computations up to break-down of quasi-periodic attractors in the dissipative spin-orbit problem

We consider a Celestial Mechanics model: the spin-orbit problem with a dissipative tidal torque, which is a singular perturbation of a conservative system. The goal of this paper is to show that it is possible to compute quasi-periodic attractors accurately and reliably for parameter values extremely close to the breakdown. Therefore, it is possible to obtain information on mathematical phenomena at breakdown. The method we use incorporates the same time numerical and rigorous improvements. Among them (i) the formalism is based on studying the time-one map of the spin-orbit problem (which reduces the dimensionality of the problem) and has mathematical advantages; (ii) very accurate integration of the ODE (high order Taylor methods implemented with extended precision) for the map at its jets; (iii) a very efficient KAM method for maps which computes the attractor and its tangent spaces ( quadratically convergent step with low storage requirements, and low operation count); (iv) the algorithms are backed by a rigorous a-posteriori KAM Theorem, which establishes that if the algorithm, produces a very approximate solution of functional equation with reasonable condition numbers. then there is a true solution nearby; and (v) the continuation algorithm is guaranteed to reach arbitrarily close to the border of existence if it is given enough computer resources. As a byproduct of the accuracy that we maintain till breakdown, we study several scale invariant observables of the tori used in the renormalization group of infinite dimensional spaces. In contrast with previously studied simple models, the behavior at breakdown of the spin-orbit problem does not satisfy standard scaling relations which implies that the spin-orbit problem is not described by a hyperbolic fixed point of a renormalization operator.

math.DS↗

arXiv2021

Nekhoroshev estimates for the orbital stability of Earth's satellites

We provide stability estimates, obtained by implementing the Nekhoroshev theorem, in reference to the orbital motion of a small body (satellite or space debris) around the Earth. We consider a Hamiltonian model, averaged over fast angles, including the $J_2$ geopotential term as well as third-body perturbations due to Sun and Moon. We discuss how to bring the Hamiltonian into a form suitable for the implementation of the Nekhoroshev theorem in the version given by P\"oschel(1993) for the `non-resonant' regime. The manipulation of the Hamiltonian includes i) averaging over fast angles, ii) a suitable expansion around reference values for the orbit's eccentricity and inclination, and iii) a preliminary normalization allowing to eliminate particular terms whose existence is due to the non-zero inclination of the invariant plane of secular motions known as the `Laplace plane'. After bringing the Hamiltonian to a suitable form, we examine the domain of applicability of the theorem in the action space, translating the result in the space of physical elements. We find that the necessary conditions for the theorem to hold are fulfilled in some non-zero measure domains in the eccentricity and inclination plane (e, i) for a body's orbital altitude (semi-major axis) up to about 20000 km. For altitudes around 11000 km we obtain stability times of the order of several thousands of years in domains covering nearly all eccentricities and inclinations of interest in applications of the satellite problem, except for narrow zones around some so-called `inclination-dependent' resonances. On the other hand, the domains of Nekhoroshev stability recovered by the present method shrink in size as the semi-major axis a increases (and the corresponding Nekhoroshev times reduce to hundreds of years), while the stability domains practically all vanish for a > 20000 km.

astro-ph.EP↗

arXiv2021

Proper Elements for Space Debris

Proper elements are quasi-invariants of a Hamiltonian system, obtained through a normalization procedure. Proper elements have been successfully used to identify families of asteroids, sharing the same dynamical properties. We show that proper elements can also be used within space debris dynamics to identify groups of fragments associated to the same break-up event. The proposed method allows to reconstruct the evolutionary history and possibly to associate the fragments to a parent body. The procedure relies on different steps: (i) the development of a model for an approximate, though accurate, description of the dynamics of the space debris; (ii) the construction of a normalization procedure to determine the proper elements; (iii) the production of fragments through a simulated break-up event. We consider a model that includes the Keplerian part, an approximation of the geopotential, and the gravitational influence of Sun and Moon. We also evaluate the contribution of Solar radiation pressure and the effect of noise on the orbital elements. We implement a Lie series normalization procedure to compute the proper elements associated to semi-major axis, eccentricity and inclination. Based upon a wide range of samples, we conclude that the distribution of the proper elements in simulated break-up events (either collisions and explosions) shows an impressive connection with the dynamics observed immediately after the catastrophic event. The results are corroborated by a statistical data analysis based on the check of the Kolmogorov-Smirnov test and the computation of the Pearson correlation coefficient.

astro-ph.EP↗

arXiv2021

The spin-spin problem in Celestial Mechanics

We study the dynamics of two homogeneous rigid ellipsoids subject to their mutual gravitational influence. We assume that the spin axis of each ellipsoid coincides with its shortest physical axis and is perpendicular to the orbital plane. Due to such assumptions, the problem is planar and depends on particular parameters of the ellipsoids, most notably, the equatorial oblateness and the flattening with respect to the shortest physical axes. We consider two models for such configuration: while in the full model, there is a coupling between the orbital and rotational motions, in the Keplerian model, the centers of mass of the bodies are constrained to move on coplanar Keplerian ellipses. The Keplerian case, in the approximation that includes the coupling between the spins of the two ellipsoids, is what we call spin-spin problem, that is a generalization of the classical spin-orbit problem. In this paper we continue the investigations of [Mis21] on the spin-spin problem by comparing it with the spin-orbit problem and also with the full model. Beside detailing the models associated to the spin-orbit and spin-spin problems, we introduce the notions of standard and balanced resonances, which lead us to investigate the existence of periodic and quasi-periodic solutions. We also give a qualitative description of the phase space and provide results on the linear stability of solutions for the spin-orbit and spin-spin problems. We conclude by providing a comparison between the full and the Keplerian models with particular reference to the interaction between the rotational and orbital motions.

math.DS↗

arXiv2021

KAM quasi-periodic tori for the dissipative spin-orbit problem

We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold. We consider the spin-orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative tidal torque that can be modeled as a time-dependent friction, which depends linearly upon the velocity. Our goal is to fix a frequency and compute the embedding of a smooth attractor with this frequency. This task requires to adjust a drift parameter. The goal of this paper is to provide numerical calculations of the condition numbers and verify that, when they are applied to the numerical solutions, they will lead to the existence of the torus for values of the parameters extremely close to the parameters of breakdown. Computing reliably close to the breakdown allows to discover several interesting phenomena, which we will report in [CCGdlL20a]. The numerical calculations of the condition numbers presented here are not completely rigorous, since we do not use interval arithmetic to estimate the round off error and we do not estimate rigorously the truncation error, but we implement the usual standards in numerical analysis (using extended precision, checking that the results are not affected by the level of precision, truncation, etc.). Hence, we do not claim a computer-assisted proof, but the verification is more convincing that standard numerics. We hope that our work could stimulate a computer-assisted proof.

math.DS↗

arXiv2021

Efficient and accurate KAM tori construction for the dissipative spin-orbit problem using a map reduction

We consider the dissipative spin-orbit problem in Celestial Mechanics, which describes the rotational motion of a triaxial satellite moving on a Keplerian orbit subject to tidal forcing and "drift". Our goal is to construct quasi-periodic solutions with fixed frequency, satisfying appropriate conditions. With the goal of applying rigorous KAM theory, we compute such quasi-periodic solution with very high precision. To this end, we have developed a very efficient algorithm. The first step is to compute very accurately the return map to a surface of section (using a high order Taylor's method with extended precision). Then, we find an invariant curve for the return map using recent algorithms that take advantage of the geometric features of the problem. This method is based on a rapidly convergent Newton's method which is guaranteed to converge if the initial error is small enough. So, it is very suitable for a continuation algorithm. The resulting algorithm is quite efficient. We only need to deal with a one dimensional function. If this function is discretized in $N$ points, the algorithm requires $O(N \log N) $ operations and $O(N) $ storage. The most costly step (the numerical integration of the equation along a turn) is trivial to parallelize. The main goal of the paper is to present the algorithms, implementation details and several sample results of runs. We also present both a rigorous and a numerical comparison of the results of averaged and not averaged models.

math.NA↗

arXiv2021

Semi-analytical estimates for the orbital stability of Earth's satellites

Normal form stability estimates are a basic tool of Celestial Mechanics for characterizing the long-term stability of the orbits of natural and artificial bodies. Using high-order normal form constructions, we provide three different estimates for the orbital stability of point-mass satellites orbiting around the Earth. i) We demonstrate the long term stability of the semimajor axis within the framework of the $J_2$ problem, by a normal form construction eliminating the fast angle in the corresponding Hamiltonian and obtaining $H_{J_2}$ . ii) We demonstrate the stability of the eccentricity and inclination in a secular Hamiltonian model including lunisolar perturbations (the 'geolunisolar' Hamiltonian $H_{gls}$), after a suitable reduction of the Hamiltonian to the Laplace plane. iii) We numerically examine the convexity and steepness properties of the integrable part of the secular Hamiltonian in both the $H_{J_2}$ and $H_{gls}$ models, which reflect necessary conditions for the holding of Nekhoroshev's theorem on the exponential stability of the orbits. We find that the $H_{J_2}$ model is non-convex, but satisfies a 'three-jet' condition, while the $H_{gls}$ model restores quasi-convexity by adding lunisolar terms in the Hamiltonian's integrable part.

math.DS↗

arXiv2020

KAM theory for some dissipative systems

Dissipative systems play a very important role in several physical models, most notably in Celestial Mechanics, where the dissipation drives the motion of natural and artificial satellites, leading them to migration of orbits, resonant states, etc. Hence the need to develop theories that ensure the existence of structures such as invariant tori or periodic orbits and device efficient computational methods. In this work we concentrate on the existence of invariant tori for the specific case of dissipative systems known as "conformally symplectic" systems, which have the property that they transform the symplectic form into a multiple of itself. To give explicit examples of conformally symplectic systems, we will present two different models: a discrete system, the standard map, and a continuous system, the spin-orbit problem. In both cases we will consider the conservative and dissipative versions, that will help to highlight the differences between the symplectic and conformally symplectic dynamics. For such dissipative systems we will present a KAM theorem in an a-posteriori format. The method of proof is based on extending geometric identities originally developed in [39] for the symplectic case. Besides leading to streamlined proofs of KAM theorem, this method provides a very efficient algorithm which has been implemented. Coupling an efficient numerical algorithm with an a-posteriori theorem, we have a very efficient way to provide rigorous estimates close to optimal. Indeed, the method gives a criterion (the Sobolev blow up criterion) that allows to compute numerically the breakdown. We will review this method as well as an extension of J. Greene's method and present the results in the conservative and dissipative standard maps. Computing close to the breakdown, allows to discover new mathematical phenomena such as the "bundle collapse mechanism".

math.DS↗

arXiv2020

KAM estimates for the dissipative standard map

From the beginning of KAM theory, it was realized that its applicability to realistic problems depended on developing quantitative estimates on the sizes of the perturbations allowed. In this paper we present results on the existence of quasi-periodic solutions for conformally symplectic systems in non-perturbative regimes. We recall that, for conformally symplectic systems, finding the solution requires also to find a "drift parameter". We present a proof on the existence of solutions for values of the parameters which agree with more than three figures with the numerically conjectured optimal values. The verification of the estimates presented here is not completely rigorous since we do not control the round-off error. Nevertheless, running with different precision shows very little difference in the results. Given the high precision of the calculation and the simplicity of the estimates, this does not seem to affect the results. A full verification should be done implementing interval arithmetic. We make available the approximate solutions, the highly efficient algorithms to generate them (incorporating high precision based on the MPFR library) and the routines used to verify the applicability of the theorem.

math.DS↗

arXiv2019

Resonances in the Earth's Space Environment

We study the presence of resonances in the region of space around the Earth. We consider a massless body (e.g, a dust particle or a small space debris) subject to different forces: the gravitational attraction of the geopotential, the effects of Sun and Moon. We distinguish different types of resonances: tesseral resonances are due to a commensurability involving the revolution of the particle and the rotation of the Earth, semi-secular resonances include the rates of variation of the mean anomalies of Moon and Sun, while secular resonances just depend on the rates of variation of the arguments of perigee and the longitudes of the ascending nodes of the perturbing bodies. We characterize such resonances, giving precise statements on the regions where the resonances can be found and provide examples of some specific commensurability relations.

astro-ph.EP↗

arXiv2019

Accurate modelling of the low-order secondary resonances in the spin-orbit problem

We provide an analytical approximation to the dynamics in each of the three most important low order secondary resonances (1:1, 2:1, and 3:1) bifurcating from the synchronous primary resonance in the gravitational spin-orbit problem. To this end we extend the perturbative approach introduced in Gkolias et. al. (2016), based on normal form series computations. This allows to recover analytically all non-trivial features of the phase space topology and bifurcations associated with these resonances. Applications include the characterization of spin states of irregular planetary satellites or double systems of minor bodies with irregular shapes. The key ingredients of our method are: i) the use of a detuning parameter measuring the distance from the exact resonance, and ii) an efficient scheme to `book-keep' the series terms, which allows to simultaneously treat all small parameters entering the problem. Explicit formulas are provided for each secondary resonance, yielding i) the time evolution of the spin state, ii) the form of phase portraits, iii) initial conditions and stability for periodic solutions, and iv) bifurcation diagrams associated with the periodic orbits. We give also error estimates of the method, based on analyzing the asymptotic behavior of the remainder of the normal form series.

astro-ph.EP↗

arXiv2019

Whiskered KAM Tori of Conformally Symplectic Systems

We investigate the existence of whiskered tori in some dissipative systems, called \sl conformally symplectic \rm systems, having the property that they transform the symplectic form into a multiple of itself. We consider a family $f_μ$ of conformally symplectic maps which depend on a drift parameter $μ$. We fix a Diophantine frequency of the torus and we assume to have a drift $μ_0$ and an embedding of the torus $K_0$, which satisfy approximately the invariance equation $f_{μ_0} \circ K_0 - K_0 \circ T_ω$ (where $T_ω$ denotes the shift by $ω$). We also assume to have a splitting of the tangent space at the range of $K_0$ into three bundles. We assume that the bundles are approximately invariant under $D f_{μ_0}$ and that the derivative satisfies some "rate conditions". Under suitable non-degeneracy conditions, we prove that there exists $μ_\infty$, $K_\infty$ and splittings, close to the original ones, invariant under $f_{μ_\infty}$. The proof provides an efficient algorithm to construct whiskered tori. Full details of the statements and proofs are given in [CCdlL18].

math.DS↗