Classification of regular and chaotic motions in Hamiltonian systems with deep learning.

This paper demonstrates the capabilities of convolutional neural networks (CNNs) at classifying types of motion starting from time series, without any prior knowledge of the underlying dynamics. The paper applies different forms of deep learning to problems of increasing complexity with the goal of testing the ability of different deep learning architectures at predicting the character of the dynamics by simply observing a time-ordered set of data. We will demonstrate that a properly trained CNN can correctly classify the types of motion on a given data set. We also demonstrate effective generalisation capabilities by using a CNN trained on one dynamic model to predict the character of the motion governed by another dynamic model. The ability to predict types of motion from observations is then verified on a model problem known as the forced pendulum and on a relevant problem in Celestial Mechanics where observational data can be used to predict the long-term evolution of the system.

Reconnecting groups of space debris to their parent body through proper elements.

Satellite collisions or fragmentations generate a huge number of space debris; over time, the fragments might get dispersed, making it difficult to associate them to the configuration at break-up. In this work, we present a procedure to back-trace the debris, reconnecting them to their original configuration. To this end, we compute the proper elements, namely dynamical quantities which stay nearly constant over time. While the osculating elements might spread and lose connection with the values at break-up, the proper elements, which have been already successfully used to identify asteroid families, retain the dynamical features of the original configuration. We show the efficacy of the procedure, based on a hierarchical implementation of perturbation theory, by analyzing the following four different case studies associated to satellites that underwent a catastrophic event: Ariane 44lp, Atlas V Centaur, CZ-3, Titan IIIc Transtage. The link between (initial and final) osculating and proper elements is evaluated through tools of statistical data analysis. The results show that proper elements allow one to reconnect the fragments to their parent body.

The dynamics of Laplace-like resonances.

The three inner Galilean satellites of Jupiter-Io, Europa, and Ganymede-are observed to move in a particular dynamical configuration, which is commonly known as the Laplace resonance. These satellites are characterized by a 2:1 ratio between the mean longitudes of Io-Europa and Europa-Ganymede. Another dynamical configuration, known as the de Sitter resonance, occurs when the longitude of Ganymede is fixed, instead of rotating like in the Laplace resonance. Besides studying the Laplace and de Sitter resonances, we also consider their generalizations to the case in which the mean longitudes of the first two satellites are in a ratio k:j, while those of the second and third satellites are in a ratio m:n with k,j,m,n∈Z+ and |j-k|, |n-m|≤2. We derive a model apt to describe such resonant configurations. We make an extensive study of the structural stability of the resonances; we show that the libration of the Laplace resonant angle is deeply affected by small variations of some quantities, most notably the semimajor axes and the oblateness. A remarkable result is that in several cases, the standard Laplace resonance of the Galilean satellites displays a regular behavior in comparison to other resonances characterized by different mean longitude ratios, which instead show a rather chaotic behavior even on short time scales. This result provides a motivation to support why the Galilean satellites are found in the actual Laplace resonance.

Breakdown of invariant attractors for the dissipative standard map.

We implement different methods for the computation of the breakdown threshold of invariant attractors in the dissipative standard mapping. A first approach is based on the computation of the Sobolev norms of the function parametrizing the solution. Then we look for the approximating periodic orbits and we analyze their stability in order to compute the critical threshold at which an invariant attractor breaks down. We also determine the domain of convergence of the dissipative standard mapping by extending the computations to the complex parameter space as well as by investigating a two-frequency model.

Cantori of the dissipative sawtooth map.

We investigate the existence of cantori for a dissipative version of the sawtooth map. Making use of an explicit parametric representation of the solution, we prove that cantori exist for any irrational value of the frequency. We also perform a numerical study aimed to determine some dynamical properties of the dissipative sawtooth map.

Regions of nonexistence of invariant tori for spin-orbit models.

The spin-orbit problem in celestial mechanics describes the motion of an oblate satellite moving on a Keplerian orbit around a primary body. We apply the conjugate points criterion for the nonexistence of rotational invariant tori. We treat both the conservative case and a case including a dissipative effect modeling a tidal torque generated by internal nonrigidity. As a by-product of the conjugate points criterion we obtain a global view of the dynamics, thanks to the introduction of a tangent orbit indicator, which allows us to discern the dynamical character of the motion.