GLADS 2022

On the persistence of periodic tori for symplectic twist maps

Alfonso Sorrentino | University of Rome Tor Vergata

In this talk I shall discuss the persistence of Lagrangian periodic tori for symplectic twist maps of the 2d-dimensional annulus and the rigidity of completely integrable maps. This is based on a joint work with Marie-Claude Arnaud and Jessica Elisa Massetti.

Near quasi-integrable Hamiltonian impact systems

Vered Rom Kedar | The Weizmann Institute

Near-integrability is usually associated with smooth small perturbations of smooth integrable Hamiltonian systems or smooth deformations of billiards in confocal quadrics.  Combinations of integrable Hamiltonian flows with impacts that respect the symmetries of the integrable structure, globally or respectively, locally, provide additional classes of non-smooth integrable or, respectively, quasi-integrable systems [1, 2, 3]. A piecewise smooth return map that describes the dynamics near a tangent torus of the non-smooth integrable systems under small perturbations is derived and studied [4]. For the quasi-integrable case, we show that there are connected energy surfaces that include genus 2 level-sets as well as level sets of genus higher than 2. The return maps for such systems are families of interval exchange maps that are piecewise symplectic. For some cases it is possible to prove that the interval exchange maps are ergodic for a full measure set of actions [5]. Perturbations of such systems lead to iso-energy return maps that are near-quasi-integrable piecewise smooth symplectic maps with fascinating dynamics (in progress).

[1]     M. Pnueli and V. Rom-Kedar, On Near Integrability of Some Impact Systems, SIAM J. Appl. Dyn. Syst., 17(4), 2707–2732,  https://doi.org/10.1137/18M1177937, 2018.

[2]    M. Pnueli and V. Rom-Kedar, On the structure of Hamiltonian impact systems, Nonlinearity 34 (4), 2611.  doi:10.1088/1361-6544/abb450, 2021.

[3]    L. Becker, S. Elliott, B. Firester, S. Gonen Cohen, M. Pnueli and V. Rom-Kedar, Impact Hamiltonian systems and polygonal billiards,  to appear in the Proceedings of the MSRI 2018 Fall semester on Hamiltonian System https://arxiv.org/abs/2001.03726, 2020.

[4]   M. Pnueli and V. Rom-Kedar, Near tangent dynamics in a class of Hamiltonian impact systems, to apear in SIAM  J. Appl. Dyn. Syst, arXiv: https://arxiv.org/abs/2110.11637,  2022.

[5]    K. Fraczek and V. Rom-Kedar, Non-uniform ergodic properties of Hamiltonian flows with impacts, Ergodic Theory Dynamical Systems, DOI:10.1017/etds.2021.106,
67, 2022.    

Towards the ultimate frontier of (finite dimensional) KAM Theory

Luigi Chierchia | Università Roma III

Dropping bodies to fill the Hill region

Richard Montgomery | University Santa Cruz

From dynamical chaos to logical chaos and vice-versa: Explored and unexplored paths

Eva Miranda | Universitat Politècnica de Catalunya and CRM

One of the more exciting contributions of Tere M. Seara is the study of instability and chaos. Chaos was coined by Edward Lorenz in 1961 with the simple statement “Chaos: When the present determines the future, but the approximate present does not approximately determine the future”.A different sort of chaos was unveiled by Cris Moore in 1990 with a 2D Turing-type dynamical systems using generalized shifts. The existence of a Turing machine associated with the dynamical system added a new intrigue to the plot: the undecidability of the halting problem (established by Alan Turing himself back in 1936) yielded the impossibility of logical predictions in the new models.  Those 2D systems based on mappings on the square Cantor set, however, are not physical. In this talk, I will give 3D physical (and/or “almost” physical) constructions of logical chaos using fluids. Against all odds,  the main ingredient of this construction is geometrical and relies on powerful techniques from contact topology. Many questions around such construction are pending including the connection among different levels of complexity (dynamical and logical) and the (in)existence of a hierarchy among them.  I will end up my talk with some new challenges and open questions. One of them is a joint project with Tere M. Seara.

Main references:

I. Baldomà, S. Ibañez, and T.M. Seara, Hopf-zero singularities truly unfold chaos.
Commun. Nonlinear Sci. Numer. Simul. 84 (2020), 105162, 19 pp.

R. Cardona, E. Miranda and D. Peralta-Salas,Computability and Beltrami fields in Euclidean space  https://arxiv.org/abs/2111.03559

R. Cardona, E. Miranda,  D. Peralta-Salas and F. Presas,  Constructing Turing complete Euler flows in dimension 3. Proc. Natl. Acad. Sci. USA 118 (2021), no. 19, Paper No. e2026818118, 9 pp.

Lorenz, Edward N.Deterministic nonperiodic flow.  J. Atmospheric Sci. 20 (1963), no. 2, 130–141.

C. Moore, Generalized shifts: Unpredictability and undecidability in dynamical systems. Nonlinearity 4, 199 (1991).

 C. Moore, Unpredictability and undecidability in dynamical systems. Phys. Rev. Lett. 64, 2354 (1990).

Turing, A.M. (1937) [Delivered to the Society November 1936]. “On Computable Numbers, with an Application to the Entscheidungsproblem” Proceedings of the London Mathematical Society. 2. Vol. 42. pp. 230–65.

Quasi-periodic vortex patches of 2d-Euler

Massimiliano Berti | SISSA

I will discuss the bifurcation of quasi-periodic solutions of the 2d-Euler equations close to rotating Kirkhoff ellipses, for a Borel set of eccentricities of asymptotically full Lebesgue measure. Joint work with Z. Hazzaina and N. Masmoudi

The zero level set of Lyapunov exponents

Lorenzo Díaz | PUC-Rio

We consider skew products whose base dynamics is given by a shift and the fiber dynamics are circle diffeomorphisms. The dynamics is transitive and there are intermingled regions where the fiber dynamics is contracting and expanding, respectvely. The dynamics also exhibits many nonhyperbolic (fiber Lyspunov exponent equal to zero) ergodic measures, some of them with positive entropy. We discuss the aproximation (in the weak * topology and in entropy) of nonhyprbolic measures by hyperbolic ones. We use this description to study the entropy spectrum of Lyapunov exponents. We will discuss applications of our methods to elliptic cocycles and partially hyperbolic diffeomorphisms.

Works in collaboration with K. Gelfert (UFRJ, Brazil) and M. Rams (IMPAN, Poland)

Statistical properties of dispersive billiards

Viviane Baladi | Sorbonne Université

We will survey results obtained in the past decade on statistical properties of two-dimensional dispersive billiards (Sinai billiards), both in the discrete and continuous time setting, for various equilibrium states, including the physical measure and the MME. Since we will not consider the setting of open billiards, the singularities of the map or flow need to be addressed, and this is a major technical difficulty.

Interactions between the three—body problem and the Euler problem

Gabriella Pinzari | Università di Padova

The planar Euler problem has a first integral which, in the limit when the mass of one of the attracting centers vanishes, has a pendulum-like dynamics. On the other hand, the three—body problem Hamiltonian differs from the one of the Euler problem for a kinetic term. Therefore, the natural question arises whether, in the three—body problem there is a memory of the dynamics of the Euler problem. We show that, under suitable assumptions, this is the case. 

This talk is based on joint works with Qinbo Chen, Jerome Daquin and Sara Di Ruzza.

Oscillatory motions and parabolic manifolds at infinity in the planar circular restricted three body problem

Maciej Capiński | AGH University of Science and Technology

In the Restricted Planar Circular 3 Body Problem, if the trajectory of the body of zero mass is defined for all time, it can have the following four types of asymptotic motion when time tends to infinity forward or backward in time: bounded, parabolic (goes to infinity with asymptotic zero velocity), hyperbolic (goes to infinity with asymptotic positive velocity) or oscillatory (the position of the body is unbounded but goes back to a compact region of phase space for a sequence of arbitrarily large times). We consider realistic mass ratio for the Sun-Jupiter pair and Jacobi constant which allows the massless body to cross Jupiter’s orbit. This is a non-perturbative regime. We prove the existence of all possible combinations of past and future final motions. In particular, we obtain the existence of oscillatory motions. All the constructed trajectories cross the orbit of Jupiter but avoid close encounters with it. The proof relies on analyzing the stable and unstable invariant manifolds of infinity and their intersections. We construct orbits shadowing these invariant manifolds by the method of correctly aligned windows.

Joint work with: Marcel Guardia, Pau Martín, Tere Seara and Piotr Zgliczynski

 Structural stability in complex analytic systems

 Núria Fagella | Universitat de Barcelona

We discuss the concept of structural stability in holomorphic families of meromorphic maps, in the spirit of the celebrated theorem of Mañé-Sad-Sullivan and Lyubich in the 1980’s for rational maps.We show that, for functions with an essential singularity at infinity, a new type of bifurcation occurs which consists of periodic points disappearing to infinity. Nevertheless, under some finiteness conditions, we shaw that J-structurally stable maps are open and dense in the appropiate parameter space (of arbitrary dimension).

Joint work with: Marcel Guardia, Pau Martín, Tere Seara and Piotr Zgliczynski