Department of Mathematics Tullio Levi-Civita
University of Padova
The motion of the inner planets within the Solar System exhibits chaotic behavior, which precludes deterministic predictions of their future positions beyond a few tens of millions of years. Nonetheless, numerical simulations reveal a remarkable stability in their orbits over timescales comparable to the age of the Solar System. The likelihood of Mercury's eccentricity exceeding 0.7, potentially resulting in catastrophic events such as close encounters or collisions with either Venus or the Sun, is estimated to be only about 1% over the next 5 billion years.
In this talk, I will discuss recent advancements that highlight the hierarchical structure of chaos in the inner Solar System. I will show how certain symmetries characterize the strongest resonant interactions responsible for orbital chaos. These symmetries are weakly broken by secondary resonances, leading to the existence of quasi-conserved quantities that relate to the longest timescales of the dynamics, as notably evidenced by a principal component analysis of orbital solutions. By providing long-term memory of the current orbital configuration, these quasi-integrals of motion effectively constrain the chaotic diffusion of the inner orbits, playing a crucial role in their statistical stability over the lifetime of the Solar System.
Biological and bio-inspired locomotion is usually described by recognizing periodic patterns, or gaits, in the movement of limbs or other body parts. But is the evolution of the system actually periodic? Or more properly, relative-periodic, since, presumably, each cycle will propel the animal (or robot) a little bit forward? The answer is often no, due, for instance, to inertia or elasticity. However, we might expect the behaviour to converge asymptotically to a relative-periodic one. In this talk we will introduce this issue considering, as a case study, a dynamic model of rectilinear crawling locomotion.
We study the existence of a global periodic attractor for the reduced dynamics of the model, corresponding to an asymptotically relative-periodic motion of the crawler. The main result is of Massera-type, namely we show that the existence of a bounded solution implies the existence of the global periodic attractor for the reduced dynamics. Additional conditions and a counterexample for the existence of a bounded solution (and therefore of the attractor) will be briefly discussed. We conclude surveying the issue for some related models.
One of the main concepts of the fourth industrial revolution is the digital twin which helps to build a bridge between the physical and the digital world and becomes a fundamental component to check the status of machines, perform predictions, or make decisions to modify their behavior. In this talk, I will present a mathematical model of bread leavening in a warm chamber by coupling heat transfer, yeast growth, and carbon dioxide production and diffusion with the deformation of baking paste. The corresponding system of partial differential equations will be analyzed and discretized by using a semi-implicit Euler method and FEM in the time and space domain, respectively. Numerical simulations will be employed to identify the energy consumption necessary to achieve a target volume under different settings of the leavening chamber and different concentrations of yeast in the baking paste, thereby providing a tool for identifying cost effective protocols.Finally, I will show how the previous model will be useful in the perspective of building a digital twin of bread leavening to the end of avoiding energy waste.
In this talk we deal, for the classical N-body problem, with the existence of action minimizing half entire expansive solutions with prescribed limit shape and initial configuration, tackling the cases of hyperbolic, parabolic and hyperbolic-parabolic arcs in a unified manner. Our approach is based on the minimization of a renormalized Lagrangian action defined on a suitable functional space. With this new strategy, we are able to confirm the already-known results of existence of both hyperbolic and parabolic solutions, and we prove for the first time the existence of hyperbolic-parabolic solutions for any prescribed asymptotic expansion in a suitable class. Associated with each element of this class, we find a viscosity solution of the Hamilton-Jacobi equation as a linear correction of the value function. Besides, we also manage to give a precise description of the growth of parabolic and hyperbolic-parabolic solutions. This work is in collaboration with Susanna Terracini.
The celebrated Poincaré-Birkhoff theorem on area-preserving maps of the annulus is of fundamental importance in the fields of Hamiltonian dynamics and symplectic topology. In this talk I will formulate a twist condition, inspired by the Poincaré-Birkhoff theorem, which applies to the asymptotically linear Hamiltonian systems of Amann, Conley and Zehnder. When this twist condition is satisfied, together with some technical assumptions, the existence of infinitely many periodic points is obtained.
Mathematical billiards in strictly convex domains with smooth boundaries serve as concrete examples of twist maps on the cylinder, where the dynamics exhibit "almost integrable" behavior near the boundary of the domain. Building on this framework, Lazutkin established the existence of a Cantor set of positive measure that includes zero, within which the billiard maps feature invariant curves corresponding to certain rotation numbers. Furthermore, these invariant curves evolve smoothly as the rotation number changes, in a Whitney sense. In this presentation, I will discuss a generalization of this result for billiards with analytic boundaries, a joint work with Frank Trujillo and Vadim Kaloshin, inspired by recent contributions from Carminati, Marmi, Sauzin, and Sorrentino. This extension shows that the Cantor set of rotation numbers can be extended into the complex plane, with its complex counterpart containing structures known as "diamonds." This finding opens up new perspectives on length spectral rigidity.
In this talk, based on joint work with Marcelo Alves, I will present three new theorems on the dynamics of geodesic flows of closed Riemannian surfaces, proved using the curve shortening flow. The first result is the stability, under C^0-small perturbations of the Riemannian metric, of certain flat links of closed geodesics. The second one is a forced existence theorem for closed geodesics on orientable closed Riemannian surfaces. The third theorem asserts the existence of Birkhoff sections for the geodesic flow of any closed orientable Riemannian surface.
According to Gutzwiller, "understanding the motions of the Moon is possibly the first problem that excited the scientific curiosity of humankind". In fact, the attempts to make reliable and accurate
ephemerides for the Moon were a key factor in the initial developments of exact sciences. I will briefly recall the main players in the early development of lunar theory, and will describe some recent results concerning periodic orbits in the lunar problem.
In this talk we present evidence of the stability of a simplified model of the Solar System, a flat (Newtonian) Sun-Jupiter-Saturn system with realistic data: masses of the Sun and the planets, their semi-axes, eccentricities and (apsidal) precessions of the planets close to the real ones. The evidence is based on convincing numerics that a KAM theorem can be applied to the Hamiltonian equations of the model to produce quasiperiodic motion (on an invariant torus) with the appropriate frequencies. To do so, we first use KAM numerical schemes to compute translated tori to continue from the Kepler approximation (two uncoupled two-body problems) up to the actual Hamiltonian of the system, for which the translated torus is an invariant torus. Second, we use KAM numerical schemes for invariant tori to refine the solution giving the desired torus. Lastly, the convergence of the KAM scheme for the invariant torus is (numerically) checked by applying several times a KAM iterative lemma, from which we obtain that the final torus (numerically) satisfies the existence conditions given by a KAM theorem. Joint work with Alex Haro (Universitat de Barcelona).
In the framework of many body quantum dynamics we show some quantitative estimates to study the mean field asymptotics. The goal is to recover the flow of Hartree equation together with a discussion on different notions of convergence and sharp time dependence in the estimates.
This simple complex analytic mapping in the title induces a well-known correspondence between the planar Hooke and Kepler problems, and regularizes the planar Kepler problems with additional regular perturbations. This mapping has several extensions, and has more and more applications in dynamical systems. In this talk I shall illustrate its significance by presenting some recent results concerning enumerations of generalized periodic orbits in a periodically-forced Kepler problem and the construction and geometry of several new types of integrable mechanical billiards. The talk is based on several joint works with A.Boscaggin (Turin)-R.Ortega (Granada), with A. Takeuchi (Augsburg), and with D. Jaud (Holzkirchen).
Slender structures are continuous bodies characterised by the fact that their length is significantly larger than their transverse dimensions; this entails, in particular, that they can undergo large displacements even in the linear response regime. In this talk, we will discuss a method for the simulation of the dynamics of these mechanical systems, which is intrinsically nonlinear and possibly dissipative.
First, I will present a description of slender structures modelled as viscoelastic Cosserat rods, in which a rod is viewed as a strand of the special Euclidean group SE(3), and the evolution equations are expressed in terms of Lie algebraic quantities. This perspective allows to account for the geometric nonlinearities of the rod, while benefitting from the linear structure of the phase space. Then, a suitable discretisation scheme will be introduced for the numerical simulation of the nonlinear dynamics of these structures, and some examples will test the effectiveness of this method. References:
1. G. G. Giusteri, E. Miglio, N. Parolini, M. Penati, R. Zambetti. Simulation of viscoelastic Cosserat rods based on the geometrically exact dynamics of special Euclidean strands, Int. J. Numer. Methods. Eng. 123(2), 396-410 (2022)
2. S. Galasso, L. Santelli, R. Zambetti, G. G. Giusteri. Simulation of drill-string systems with fluid--structure and contact interactions in realistic geometries, submitted to Computational Mechanics.
A revisitation of the old standing problem of constructing first integrals for a nonlinear system of oscillators. Equivalently: the dynamics in a neighborhood of an elliptic equilibrium point. After Poincare', who proved that in general an integrable Hamiltonian system subject to a small perturbation does not possess a first integral independent of the Hamiltonian, attention has been focused on elliptic equilibria. I will describe two methods: a direct one (not often exploited) which consists in solving by series the equation for a first integral; and an indirect one (more common) which exploits the transformation of the Hamiltonian into a normal form, for example formally integrable. The formal solution, however, raises the question of investigating the (non) convergence of the formal series so constructed. This entails a discussion of the impact of small divisors. I will try to shed some light on this problem, thus identifying the mechanism that leads to divergence of the formal series.