Welcome Room: Conference Room of INdAM

The chaotic motion of the Solar System. How close is our model to the real Solar System

• Jacques Laskar (LTE-Observatoire de Paris)

Room: Conference Room of INdAM

Lagrangian tori exponentially close to whiskered tori in generic analytic mechanical systems

• Luigi Chierchia (Università Roma Tre)

Room: Conference Room of INdAM I will discuss the existence of maximal KAM tori which are exponentially close to whiskered secular tori appearing generically at simple resonances in nearly integrable mechanical systems. Joint work with Luca Biasco.

Coffee Break Room: Conference Room of INdAM

On the Arnold diffusion mechanism in Medium Earth Orbit

• Mar Giralt (UPC Barcelona & Obs de Paris)

Room: Conference Room of INdAM Motivated by the need of preserving the operational orbital regions around the Earth, natural perturbations can be exploited to lead the satellites towards an atmospheric reentry at the end of life. In this way, it is possible to dilute the collision probability in the long term and reduce the disposal cost. In the case of the Medium Earth Orbit (MEO) region, home of the navigation satellites (like GPS and Galileo), the main driver is the third-body perturbation. In this work, we show how an Arnold diffusion mechanism can trigger the eccentricity growth in MEO, so that the pericenter altitude drops into the atmospheric drag domain. Focusing on the case of Galileo, we consider a hierarchy of Hamiltonian models, assuming that the main perturbations on the motion of the spacecraft are the oblateness of the Earth and the gravitational attraction of the Moon. First, the Moon is assumed to lay on the ecliptic plane and periodic orbits and associated stable and unstable invariant manifolds are computed for various energy levels, in the neighborhood of a given resonance. Along each invariant manifold, the eccentricity increases naturally, achieving its
maximum at the first intersection between them. This growth is, however, not sufficient to achieve reentry. By moving to a more realistic model, where the inclination of the Moon is taken into account, the problem becomes non-autonomous and the satellite is able to move along different energy levels. Under the ansatz of transversality of the stable and unstable manifolds in the autonomous case, checked numerically, Poincaré-Melnikov techniques are applied to show how the Arnold diffusion can be attained, by constructing a sequence of homoclinic orbits that connect invariant tori at different energy levels on the normally hyperbolic invariant manifold. This is a joint work with E.M Alessi, I. Baldomá and M. Guardia.

Preliminary orbits with over-determined systems of Keplerian conservation laws

• Giovanni Gronchi (Università di Pisa)

Room: Conference Room of INdAM The modern asteroid surveys are producing very large databases of optical observations. Linking very short arcs (VSAs) of such observations we can compute the asteroid orbits. Searching for efficient linkage algorithms is an interesting mathematical problem. We show how this problem can be faced using the first integrals of Kepler's problem. Using all these integrals we find an overdetermined polynomial system of 9 equations in 6 variables, that is generically inconsistent, i.e. it does not have solutions, not even in the complex field, also when the two VSAs belong to the same observed body. We show how we can select two polynomial subsystems $\mathcal{S}_1, \mathcal{S}_2$ that are still overdetermined, but consistent, and define two algebraic varieties with the lowest number of points (9 and 18 respectively). Moreover, we search for compromise solutions of both $\mathcal{S}_1$ and $\mathcal{S}_2$ using the concept of approximated gcd. This is a joint work with Clara Grassi.

LUNCH

Critical points at infinity for the energy of the n-body problem at a given value of the angular momentum.

• Alain Albouy (LTE-Observatoire de Paris)

Room: Conference Room of INdAM (with Holger Dullin). In any motion of the n-body problem, seen in the Galilean frame of the center of mass, the energy H and the angular momentum vector L are constant. The integral manifolds are the submanifolds of the phase space defined by fixing H and L. If we fix L and let H vary, the submanifold will change topology at values which we call singular. The critical values are singular values, but for example H=0 is a singular value which is not critical, already in the 2-body problem. In [1], developing works by Smale, Cabral, Easton and Simó, I gave a list of values which includes all the singular values in the 3-dimensional n-body problem. I conjectured that this list does not contain artefacts but only singular values. This was later proved in the 3-body case by McCord, Meyer and Wang, by homology computations. What is still missing is a general argument in the n-body case proving that the values in the list are all significant. In an ongoing work with Holger Dullin we convinced ourselves that some specific results about the gradient flow of H restricted to L=L_0 and some general results about the critical points at infinity should give such a general argument. I will explain examples of such results. [1] Alain Albouy, Integral manifolds of the N-body problem Inventiones Mathematicae, 114 (1993), 463-488 [2] Andreas Knauf and Nikolay Martynchuk, Topology change of level sets in Morse theory, Arkiv for Matematik, 58 (2020), 333–356 [3] Christopher K. McCord, The Integral Manifolds of the 4 Body Problem with Equal Masses: Bifurcations at Relative Equilibria, Journal of Dynamics and Differential Equations, (2024) [4] Alain Albouy, Holger R. Dullin, Bounded orbits for three bodies in ℝ^4, Geometric Mechanics, 1 (2024)

Coffee Break Room: Conference Room of INdAM

Paradifferential approach to classical KAM theory

• Chengyang Shao (IHES, Univ. Paris Saclay)

Room: Conference Room of INdAM In this talk, we aim to present an alternative formulation of classical KAM theory via paradifferential calculus. A powerful tool introduced by J.-M. Bony during the 80s, paradifferential calculus is widely used in delicate analysis of nonlinearities. Based on a tailored definition of paraproduct operators, it turns out that dynamical conjugacy problems involving "small denominators" can be transformed into fixed point forms. Compared to the well-known KAM iteration, this alternative scheme is numerically easier to manipulate with. Hopefully, this numerical scheme could contribute to “KAM theory with realistic parameters”. The talk is based on joint works with Thomas Alazard, Jacques Fejoz and Kuang Huang.

Energy cascades for quantum harmonic oscillators in dimension 2

• Beatrice Langella (SISSA-Trieste)

Room: Conference Room of INdAM In this talk I will consider a class of linear Schrödinger equations, whose Hamiltonian is given by bounded, time periodic perturbations of the 2-dimensional Harmonic oscillator. Under complete resonance assumptions, I will show that a generic class of perturbations admits solutions whose Sobolev norms undergo infinite growth as time tends to infinity. The proof combines pseudo-differential normal form, Mourre theory, and techniques from dynamical systems, in order to identify the good class of perturbations admitting energy cascades. This is a joint work with A. Maspero and M. T. Rotolo.

Proper elements and effective stability in Celestial Mechanics

• Alessandra Celletti (Università Tor Vergata, Roma)

Room: Conference Room of INdAM We consider two problems relying on perturbative methods in Celestial Mechanics: the computation of proper elements for the space debris problem and effective stability estimates close to resonances in rotational dynamics. We show that perturbative methods can be integrated with Machine Learning techniques, specifically to investigate the dynamics of groups of objects for the classification and clustering of space debris generated by break-up events of artificial satellites. We use Nekhoroshev-like estimates to provide effective stability bounds close to resonances in the the spin-orbit problem, described by a 1D time-dependent Hamiltonian, and the spin-spin-orbit model, described by a 2D time-dependent Hamiltonian. Presentation below.

Nekhoroshev Theorem for time quasiperiodic perturbations of P-Steep integrable systems

• Dario Bambusi (Università degli Studi di Milano)

Room: Conference Room of INdAM Nekhoroshev theorem in its original form ensures stability over exponentially long times of perturbations of integrable Hamiltonian systems under a generic nondegeneracy condition introduced by Nekhoroshev and called Steepness. Here we consider the case of a perturbation which depends quasiperiodically on time and prove that if the frequencies of the forcing are Diophantine and the unperturbed integrable system fulfills a generic nondegeneracy condition generalizing steepness, then the actions are stable over exponentially long times. Previous results only dealt with perturbations of convex integrable systems. The proof is based on a generalization of the technique by Nekhoroshev as improved by Guzzo Chierchia Benettin and on some new ideas. Joint work with Santiago Barbieri, Mar Giralt and Beatrice Langella.

Coffee Break Room: Conference Room of INdAM

Uniqueness of hyperbolic Busemann functions in the N-body problem.

• Andrea Venturelli (Université d'Avignon)

Room: Conference Room of INdAM In a previous paper written with Ezequiel Maderna, we proved that in the newtonian N-body problem, given a starting configuration x_0 and a limit shape a without collisions denoted as a, there always exists a hyperbolic solution x(t) of the N-body problem such that x(0)=x_0 and x(t)/t converges to a as t tends to +\infty. We will say that x(t) is a hyperbolic motion with limit shape a. We obtain this solution as a calibrating curve of a weak KAM faible solution, that we call a Busemann function. In this talk i will show that if a is fixed, the Busemann function is unique, up to an additive constant. As a consequence, the set of configurations x_0 where the hyperbolic motion starting from x_0 (and with limit shape a) is unique has full measure in the configuration space. It is a joint work with Ezequiel Maderna.

Marchal's family: inclined co-orbitals in the three-body problem

• Alexandre Prieur (LTE-Observatoire de Paris)

Room: Conference Room of INdAM In the three-body problem, for all values of the masses, the vertical eigenvectors of the Lagrange relative equilibrium give rise to a family of spatial quasi-periodic orbits; these orbits are periodic in a rotating frame. For the equal masses' cases, this is the P12 family with its order-12 symmetry group, which ends at the figure Eight solution (studied in Chenciner-Fejoz-Montgomery (2005), Chenciner-Fejoz (2008) and Calleja, Garcia-Azpeitia, Hénot, Lessard and Mireles (2024)). In 2009, Christian Marchal studied this family in the special case of the average restricted secular problem. We propose a numerical study to follow this family in the general case. We show that the family discovered by Marchal exists in the full three-body problem (neither restricted nor secular) for a wide range of masses, and that the stability of its orbits evolves along the family, sometimes leading to stable systems for masses exceeding the Gascheau's value. We also link this family to P12: in the equal masses case, they are identical up to a bifurcation, where P12 continues towards the Eight and the Marchal branch goes to collisions.

LUNCH

Spectral Rigidity in Billiard Dynamics and Geometry.

• Alfonso Sorrentino (Università Tor Vergata, Roma)

Room: Conference Room of INdAM This talk explores spectral rigidity phenomena appearing in billiard dynamics, with a focus on convex planar billiard-like systems, which include Birkhoff, Symplectic, Outer and Outer-length billiards. I will present some new rigidity results for the associated action spectrum, which are linked to pointwise isoperimetric-type inequalities of Mather’s beta function. Notably, we will observe surprising distinctions in the case of Outer billiards. This work is joint with Misha Bialy and Stefano Baranzini. Time permitting, I will also discuss analogous results for the stable norm of Riemannian metrics, based on joint work with Anna Florio and Martin Leguil.

Coffee Break Room: Conference Room of INdAM

Typical almost-periodic solutions for the 1d NLS with external parameters

• Michela Procesi (Università Roma Tre)

Room: Conference Room of INdAM In the study of close to integrable Hamiltonian PDEs, a fundamental question is to understand the behaviour of “typical” solutions. With this in mind it is natural to study the persistence of almost-periodic solutions and infinite dimensional invariant tori, which are in fact typical in the integrable case. In this talk, I shall consider a family of NLS equations parametrized by a smooth convolution potential and prove that for “most” choices of the parameter there is a full measure set of Gevrey initial data that give rise to almost-periodic solutions whose hulls are invariant Tori.

KAM tori in a particular class of extrasolar planetary models

• Ugo Locatelli (Università Tor Vergata, Roma)

Room: Conference Room of INdAM Since the birth of KAM theory, planetary three-body problems were indicated as a natural benchmark to study its applicability to realistic systems of physical interest. It is well known that KAM statements proved in a purely analytical way completely fail such a challenging purpose. This the reason why, since the last two decades of the previous century Computer-Assisted Proofs (hereafter CAPs) are commonly used in this context. We revisit this general problem in the case of a few planar models of extrasolar systems hosting one star and two exoplanets; we consider them with values of the parameters that are in agreement with the observations. The existence of invariant KAM tori in correspondence with orbital motions is investigated by using a publicly available software code, that is specially designed to perform CAPs. Such an approach can be successful if and only if the problem is described by a Hamiltonian (in action-angle coordinates) that is close enough to a Kolmogorov normal form. We describe the bare minimum of preliminary canonical transformations, which are needed to bring the Hamiltonian in a suitable form to start a CAP. This strategy is implemented in such a way to rigorously prove the existence of KAM tori for three exoplanetary systems (namely HD11964, HD142 and HD4732). A rather simple argument allows us to give a characterization of the planetary systems for which our proof scheme has good chance of success. This talk is based on a joint work with C. Caracciolo.

Conference dinner

Exotic rotation domains and Herman rings for quadratic Hénon maps

• Raphael Krikorian (Ecole Polytechnique)

Room: Conference Room of INdAM Quadratic Hénon maps are polynomial automorphism of $\mathbb{C}^2$ of the form $h:(x,y)\mapsto (\lambda^{1/2}(x^2+c)-\lambda y,x)$. They have constant Jacobian equal to $\lambda$ and they admit two fixed points. If $\lambda$ is on the unit circle (one says the map $h$ is conservative) these fixed points can be elliptic or hyperbolic. In the elliptic case, a simple application of Siegel Theorem shows (under a Diophantine assumption) that $h$ admits many quasi-periodic orbits with two frequencies in the neighborhood of its fixed points. Surprisingly, in some hyperbolic cases, S. Ushiki observed some years ago what seems to be quasi-periodic orbits though no Siegel disks exist. I will explain why this is the case. This theoretical framework also predicts and mathematically proves, in the dissipative case ($\lambda$ of module less than 1), the existence of (attractive) Herman rings. These Herman rings, which were not observed before, can be produced in numerical experiments.

Lyapunov unstable approximation of integrable Hamiltonians

• Jaime Paradela (University of Maryland)

Room: Conference Room of INdAM We prove that any real-analytic Hamiltonian with a locally integrable, non-degenerate and not locally convex elliptic equilibrium, can be perturbed within the real analytic category, while keeping the Birkhoff normal form at the equilibrium unchanged up to any arbitrary order, so that the equilibrium becomes Lyapunov unstable. Joint work with B. Fayad, M. Saprykina and T.M. Seara.

Coffee Break Room: Conference Room of INdAM

A common first integral in problems of celestial and statistical mechanics

• Gabriella Pinzari (Università degli Studi di Padova)

Room: Conference Room of INdAM A few years ago, I realized that a single average of the inverse distance of two Keplerian bodies admits as a first integral an intriguing combination of the angular momentum and perihelion anomaly of the averaged planet. Almost at the same time, J. Jauslin and G. Gallavotti discovered a first integral in a version of the Boltzmann problem, a model of statistical mechanics proposed by Boltzmann to represent quite an opposite property: ergodicity. G. Felder later gave a geometrical description based on Poncelet property. In my talk, I shall discuss this and related topics. Part of the talk is based on joint work with L. Zhao.

Normally hyperbolic invariant manifolds and scattering maps for conformally symplectic systems

• Tere Seara (UPC, Barcelona)

Room: Conference Room of INdAM This work studies conformally symplectic diffeomorphisms defined in a symplectic manifold, that is, diffeomorphisms which transform a symplectic form into a multiple of itself, through a conformal factor. We focus on Normally hyperbolic invariant manifolds (NHIM) and their (un)stable manifolds, which are important landmarks that organize long-term dynamical behaviorb We prove that a NHIM is symplectic if and only if the rates of hyperbolicity satisfy certain pairing rules and if and only if the rates and the conformal factor satisfy certain (natural) inequalities. Homoclinic excursions to NHIMs are quantitatively described by scattering maps, which give the trajectory asymptotic in the future as a function of the trajectory asymptotic in the past. We prove that the scattering map is symplectic even if the dynamics is dissipative. We also show that if the symplectic form is exact, then the scattering maps are exact, even if the dynamics is not exact.

LUNCH

Discussion Room: Conference Room of INdAM