Lagrangian descriptors and their applications to deterministic chaos

We present our recent contributions to the theory of Lagrangian descriptors for discriminating ordered and deterministic chaotic trajectories. The class of Lagrangian Descriptors we are dealing with is based on the Euclidean length of the orbit over a finite time window. The framework is free of tangent vector dynamics and is valid for both discrete and continuous dynamical systems. We review its last advancements and touch on how it illuminated recently Dvorak's quantities based on maximal extent of trajectories' observables, as traditionally computed in planetary dynamics.Comment: Submitted as part of the proceedings of the Complex Planetary Systems II - Kalvi-IAU Symposium 382. 3 figures. Limited to 6 page

Drift and its mediation in terrestrial orbits

The slow deformation of terrestrial orbits in the medium range, subject to lunisolar resonances, is well approximated by a family of Hamiltonian flow with $2.5$ degree-of-freedom. The action variables of the system may experience chaotic variations and large drift that we may quantify. Using variational chaos indicators, we compute high-resolution portraits of the action space. Such refined meshes allow to reveal the existence of tori and structures filling chaotic regions. Our elaborate computations allow us to isolate precise initial conditions near specific zones of interest and study their asymptotic behaviour in time. Borrowing classical techniques of phase- space visualisation, we highlight how the drift is mediated by the complement of the numerically detected KAM tori.Comment: 22 pages, 11 figures, 1 table, 52 references. Comments and feedbacks greatly appreciated. This article is part of the Research Topic `The Earth-Moon System as a Dynamical Laboratory', confer https://www.frontiersin.org/research-topics/5819/the-earth-moon-system-as-a-dynamical-laborator

Detection of separatrices and chaotic seas based on orbit amplitudes

The Maximum Eccentricity Method (MEM) is a standard tool for the analysis of planetary systems and their stability. The method amounts to estimating the maximal stretch of orbits over sampled domains of initial conditions. The present paper leverages on the MEM to introduce a sharp detector of separatrices and chaotic seas. After introducing the MEM analogue for nearly-integrable action-angle Hamiltonians, i.e., diameters, we use low-dimensional dynamical systems with multi-resonant modes and junctions, supporting chaotic motions, to recognise the drivers of the diameter metric. Once this is appreciated, we present a second-derivative based index measuring the regularity of this application. This quantity turns to be a sensitive and robust indicator to detect separatrices, resonant webs and chaotic seas. We discuss practical applications of this framework in the context of $N$-body simulations for the planetary case affected by mean-motion resonances, and demonstrate the ability of the index to distinguish minute structures of the phase space, otherwise undetected with the original MEM.Comment: Under review at Celestial Mechanics and Dynamical Astronomy. 8 Figures, 59 references, 17 pages. Comments and feedback welcom

From order to chaos in Earth satellite orbits

We consider Earth satellite orbits in the range of semi-major axes where the perturbing effects of Earth's oblateness and lunisolar gravity are of comparable order. This range covers the medium-Earth orbits (MEO) of the Global Navigation Satellite Systems and the geosynchronous orbits (GEO) of the communication satellites. We recall a secular and quadrupolar model, based on the Milankovitch vector formulation of perturbation theory, which governs the long-term orbital evolution subject to the predominant gravitational interactions. We study the global dynamics of this two-and-a-half degrees-of-freedom Hamiltonian system by means of the fast Lyapunov indicator (FLI), used in a statistical sense. Specifically, we characterize the degree of chaoticity of the action space using angle-averaged normalized FLI maps, thereby overcoming the angle dependencies of the conventional stability maps. Emphasis is placed upon the phase-space structures near secular resonances, which are of first importance to the space debris community. We confirm and quantify the transition from order to chaos in MEO, stemming from the critical inclinations, and find that highly inclined GEO orbits are particularly unstable. Despite their reputed normality, Earth satellite orbits can possess an extraordinarily rich spectrum of dynamical behaviors, and, from a mathematical perspective, have all the complications that make them very interesting candidates for testing the modern tools of chaos theory.Comment: 30 pages, 9 figures. Accepted for publication in the Astronomical Journa

A new analysis of the three--body problem

In the recent papers~[18],~[5], respectively, the existence of motions where the perihelions afford periodic oscillations about certain equilibria and the onset of a topological horseshoe have been proved. Such results have been obtained using, as neighbouring integrable system, the so--called two--centre (or {\it Euler}) problem and a suitable canonical setting proposed in~[16],~[17]. Here we review such results.Comment: 27 pages, 3 figure

The dynamical structure of the MEO region: long-term stability, chaos, and transport

It has long been suspected that the Global Navigation Satellite Systems exist in a background of complex resonances and chaotic motion; yet, the precise dynamical character of these phenomena remains elusive. Recent studies have shown that the occurrence and nature of the resonances driving these dynamics depend chiefly on the frequencies of nodal and apsidal precession and the rate of regression of the Moon's nodes. Woven throughout the inclination and eccentricity phase space is an exceedingly complicated web-like structure of lunisolar secular resonances, which become particularly dense near the inclinations of the navigation satellite orbits. A clear picture of the physical significance of these resonances is of considerable practical interest for the design of disposal strategies for the four constellations. Here we present analytical and semi-analytical models that accurately reflect the true nature of the resonant interactions, and trace the topological organization of the manifolds on which the chaotic motions take place. We present an atlas of FLI stability maps, showing the extent of the chaotic regions of the phase space, computed through a hierarchy of more realistic, and more complicated, models, and compare the chaotic zones in these charts with the analytical estimation of the width of the chaotic layers from the heuristic Chirikov resonance-overlap criterion. As the semi-major axis of the satellite is receding, we observe a transition from stable Nekhoroshev-like structures at three Earth radii, where regular orbits dominate, to a Chirikov regime where resonances overlap at five Earth radii. From a numerical estimation of the Lyapunov times, we find that many of the inclined, nearly circular orbits of the navigation satellites are strongly chaotic and that their dynamics are unpredictable on decadal timescales.Comment: Submitted to Celestial Mechanics and Dynamical Astronomy. Comments are greatly appreciated. 28 pages, 15 figure

Dynamical properties of the Molniya satellite constellation: long-term evolution of orbital eccentricity

The aim of this work is to analyze the orbital evolution of the mean eccentricity given by the Two-Line Elements (TLE) set of the Molniya satellites constellation. The approach is bottom-up, aiming at a synergy between the observed dynamics and the mathematical modeling. Being the focus the long-term evolution of the eccentricity, the dynamical model adopted is a doubly-averaged formulation of the third-body perturbation due to Sun and Moon, coupled with the oblateness effect on the orientation of the satellite. The numerical evolution of the eccentricity, obtained by a two-degree-of-freedom model assuming different orders in the series expansion of the third-body effect, is compared against the mean evolution given by the TLE. The results show that, despite being highly elliptical orbits, the second order expansion catches extremely well the behavior. Also, the lunisolar effect turns out to be non-negligible for the behavior of the longitude of the ascending node and the argument of pericenter. The role of chaos in the timespan considered is also addressed. Finally, a frequency series analysis is proposed to show the main contributions that can be detected from the observational data

A new analysis of the three-body problem

Intherecentpapers[5,18],respectively,theexistenceofmotionswhere the perihelions afford periodic oscillations about certain equilibria and the onset of a topological horseshoe have been proved. Such results have been obtained using, as neighbouring integrable system, the so-called two-centre (or Euler) problem and a suitable canonical setting proposed in [16, 17]. Here we review such results