**Organizers**: Marco Sansottera and Giulio Baù.

## Upcoming Talks

- TBA

## Previous Talks

**Monday 20 July 2**020, 2.30 pm (GMT + 2)**Giacomo Lari**(University of Pisa, Department of Mathematics)*On the stability of the Laplace resonance under tidal effects*

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**Abstract**

The strong tidal dissipation between Jupiter and Io leads to significant migration of all the moons involved in the so-called Laplace resonance (Io, Europa, and Ganymede). Using an accurate averaged dynamical model of the Galilean moons developed to investigate their future long-term behavior, we show that the Laplace resonance is preserved despite the tidal effects, at least until Ganymede approaches the 2:1 resonant region with Callisto (about 1.5 billion years from now, assuming the current estimation of the tidal dissipation in the system). The resonant encounter can then result into two distinct outcomes: (A) a chain of three 2:1 two-body resonances (Io–Europa, Europa–Ganymede, and Ganymede–Callisto), or (B) a resonant chain involving the 2:1 two-body resonance Io–Europa and a pure 4:2:1 three-body resonance between Europa, Ganymede and Callisto. In case A, the Laplace resonance is always preserved and the eccentricities remain confined to small values below 0.01. In case B, the Laplace resonance is generally disrupted and the eccentricities of Ganymede and Callisto can increase up to about 0.1, making this configuration unstable and driving the system into new resonances. This is a joint work with M. Saillenfest and M. Fenucci.

**Monday 6 July 2020, 2:30 pm (GTM + 2)****Chiara Caracciolo***Librational KAM tori in the secular dynamics of the Upsilon-Andromedae planetary system*

(University of Rome “Tor Vergata”, Department of Mathematics)

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**Abstract**

Abstract: We investigate the stability of the secular motion of the two most massive planets in the Upsilon-Andromedae planetary system. Such a system has some remarkable characteristics: both the major planets move in rather eccentric orbits (e ~ 0.3) with a relevant mutual inclination (~30**°**); moreover, since their masses are about ten times larger than the Jupiter one and the mutual distance is relatively short (being the semi-major axes ~0.8AU and ~2.5AU, resp.), there is a strong interaction between them. We study the secular approximation at order 2 in the masses of the planetary three-body problem. In this framework, using a normal form approach, we show the existence of librational KAM tori, whose configuration is such that the pericenters of the two orbits are in libration around the anti-alignment, namely in a state of apsidal locking. In practice, we start from the construction of a suitable elliptic lower dimensional torus, i.e. a solution where the pericenters stay completely anti-aligned; then, we construct tori of maximal dimension, by focusing on the one which corresponds to initial conditions compatible with the observations.

This work is made in joint collaboration with U. Locatelli, M. Sansottera and M. Volpi.

**Monday 22 June 2020, 2:30 pm (GTM + 2)****Veronica Danesi***Variation on Kolmogorov’s theorem: KAM with knobs*

(University of Milan, Department of Mathematics)

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**Abstract**

In this talk I will reconsider the proof of the Kolmogorov’s theorem, introducing a variation on the handling of the frequencies and avoiding the so-called translation step. The motivation behind the development of this approach has its origin in the problem of persistence of lower dimensional elliptic invariant tori under sufficiently small perturbation.

This is a joint work with M. Sansottera.

**Monday 8 June 2020, 2:30 pm (GTM + 2)****Marco Fenucci***Numerical methods for the computation of symmetric periodic orbits of the N-body problem*

(University of Belgrade, Faculty of Mathematics)

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**Abstract**

In the last two decades, several periodic solutions of the N-body problem with equal masses have been found as minimizer of the Lagrangian action. The first and the most famous periodic orbit of this kind is the figure-eight solution of the three-body problem, whose existence has been proved by A. Chenciner and R. Montgomery in 2000. Here three equal masses follow an eight shaped curve, with the same time law and a constant shift in phase. Orbits with this property are called choreographies. Motivated by the above example, in this talk we first describe some numerical methods for the computation of symmetric periodic orbits, then we see how their stability can be studied, using also rigorous numerical techniques. Moreover, since some orbits are found as minimizers of the Lagrangian action, we give an idea of how local minimality properties can be studied with numerical computations. After, we show how to apply the numerical methods to the computation of periodic orbits of the*N*-body problem and the Coulomb (*N*+1)-body problem with the symmetry of Platonic polyhedra, and see how they provided clues for rigorous proofs of existence.

**Monday 25 May 2020, 2:30 pm (GTM + 2)****Joan Gimeno***Delay perturbations of an ODE*

(Università degli Studi di Roma “Tor Vergata”, Dipartimento di Matematica)

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**Abstract**

In this talk we will start introducing some basics results on Delay Differential Equations (DDEs) in order to focus our attention on the state-dependent DDEs (SDDEs). To bypass the phase space notion for SDDEs, we will consider a very singular perturbation of an ODE to provide an a-posteriori theorem of the existence of a parametrization of a subfamily of solution of the infinite stable invariant manifold of the new system. We will also discuss a numerical implementation stressing the hardest part in the coding of the algorithm. In particular, the results admit a straightforward extension for advanced or even mixed differential equations as well as smooth dependence on parameters. Similar techniques allow us to prove the persistence of periodic solutions under this kind of singular perturbation with only mild assumptions on the original ODE.

This is a joint work with J. Yang and R. de la Llave.

**Monday 11 May 2020, 2:30 pm (GTM + 2)****Rocío Isabel Páez***The Levi-Civita Hamiltonian normalization, the analytical construction of large Lyapunov orbits and their manifolds*

(Università degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”)

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**Abstract**

In this seminar, I will introduce and describe at full extent the construction of the normalized Levi-Civita Hamiltonian, a fundamental tool for the study of the invariant manifolds emanating from planar Lyapunov orbits, in particular for large values of the energy. We will start from the Hamiltonian of the CR3BP in Cartesian coordinates, whose radius of convergence after being expanded at $L_1$ or $L_2$ is limited in amplitude by $|1 – \mu – x L_1|$, and therefore it is inadequate for dealing with large amplitudes. We will introduce regularized variables, so as to construct the Levi-Civita Hamiltonian, and we will investigate if regularizations prior to the normalization scheme allow us to overcome this limit. Finally, we will discuss the results obtained from the study of the normalized Hamiltonian, on which we notice variations in the structure of the tubes manifolds, allowing circulations in either sense and even collisions.

This research has been done in collaboration with M. Guzzo.

**Monday 4 May 2020, 2:30 pm (GTM + 2)****Sara Di Ruzza***Symbolic dynamics in a binary asteroid system*

(Università degli Studi di Padova, Dipartimento di Matematica “Tullio Levi-Civita”)

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**Abstract**

We consider a system of three point masses undergoing Newtonian attraction. Two of them (“binary asteroids”) have equal mass, while the third one (“planet”) is much heavier. The three masses are constrained on a plane. The two lighter particles orbit one to the other, while the trajectory of the third particle is external to the two, and far. We look at the secular motions of this system, meaning that, from a reference frame centred with one of the asteroids, we average out the position of the other asteroid, and look at the movements of the eccentricity and the pericenter of its instantaneous ellipse, as determined by the attraction by the planet. In the limit when the planet describes a circular trajectory with infinite radius, the asteroidal ellipse periodically squeezes to a segment while the pericenter oscillates about an equilibrium. The question we ask concerns the onset of chaos, once the planet exercises its attraction at a large, but finite distance from the asteroids. Our analysis is purely numerical. Based on covering relations as in a recent paper by A. Gierzkiewicz and P. Zgliczynski (2019), a topological horseshoe is highlighted, indicating symbolic dynamics.

Joint work with Jerome Daquin and Gabriella Pinzari.