arXiv subjects

Guido Gentile

Publications and source records attributed to Guido Gentile.

At least 19 recordsLinked to original sources

arXiv2025

On an Analytical Criterion for Detecting Intermittent Turbulent Behaviour of Solutions of Partial Differential Equations

A main question in the study of partial differential equations is the following: how do we understand the nature of the solutions and, in particular, how do we determine if a given solution shows turbulent or non-turbulent behaviour? Being able to answer such a question would be a major advance in the comprehension of the nature of turbulence. In this paper we focus on the case of intermittent turbulence and provide an analytical criterion, based on the crest factor, which captures the essential feature of the solutions. By computing the crest factor for the solutions of some classical equations, both linear and nonlinear, we illustrate the capability of the criterion for discerning between solutions exhibiting time-intermittent turbulence behaviour and solutions which either are not turbulent or show statistically stationary turbulence, like, for example, in the case described by Kolmogorov's theory.

math.AP↗

arXiv2025

Some subsets of set F in the diluted Hofstadter problem

We continue work started in [1] concerning integer sequences q(n), n in N, defined by q(n) = q(n-q(n-1)) + f(n), with q(1) = 1. Here, f(n), with f(1) = 0, is a given sequence. We define F as the set of semi-infinite sequence f such that the resulting sequence q exists. This requires that the term q(n-q(n-1)) be defined for all n, that is, 0 < q(n) < n+1 applies for all n in N.

math.NT↗

arXiv2025

Learning traffic flows: Graph Neural Networks for Metamodelling Traffic Assignment

The Traffic Assignment Problem is a fundamental, yet computationally expensive, task in transportation modeling, especially for large-scale networks. Traditional methods require iterative simulations to reach equilibrium, making real-time or large-scale scenario analysis challenging. In this paper, we propose a learning-based approach using Message-Passing Neural Networks as a metamodel to approximate the equilibrium flow of the Stochastic User Equilibrium assignment. Our model is designed to mimic the algorithmic structure used in conventional traffic simulators allowing it to better capture the underlying process rather than just the data. We benchmark it against other conventional deep learning techniques and evaluate the model's robustness by testing its ability to predict traffic flows on input data outside the domain on which it was trained. This approach offers a promising solution for accelerating out-of-distribution scenario assessments, reducing computational costs in large-scale transportation planning, and enabling real-time decision-making.

cs.LG↗

arXiv2025

Synchronization and averaging in partially hyperbolic systems with fast and slow variables

We study a family of dynamical systems obtained by coupling an Anosov map on the two-dimensional torus -- the chaotic system -- with the identity map on the one-dimensional torus -- the neutral system -- through a dissipative interaction. We show that the two systems synchronize: the trajectories evolve toward an attracting invariant manifold, and the full dynamics is conjugated to its linearization around the invariant manifold. As a byproduct, we obtain that there exists a unique exponentially mixing physical measure. When the interaction is small, the evolution of the variable which describes the neutral system is very close to the identity; hence, it appears as a slow variable with respect to the variable which describes the chaotic system, and which is wherefore named the fast variable. We demonstrate that, seen on a suitably long time scale, the slow variable effectively follows the solution of a deterministic differential equation obtained by averaging over the fast variable. More precisely, we prove that the invariant manifold is in probability close to the fixed point of the averaged dynamics and that the difference between the exact evolution of the slow variable, seen from the invariant manifold, and its averaged evolution is in probability exponentially decreasing for arbitrarily large times.

nlin.CD↗

arXiv2025

Asymptotically full measure sets of almost-periodic solutions for the NLS equation

We study the dynamics of solutions for a family of nonlinear Schroedinger equations on the circle, with a smooth convolution potential and Gevrey regular initial data. Our main result is the construction of an asymptotically full measure set of small-amplitude time almost-periodic solutions, which are dense on invariant tori. In regions corresponding to positive actions, we prove that such maximal invariant tori are Banach manifolds, which provide a Cantor foliation of the phase space. As a consequence, we establish that, for many small initial data, the Gevrey norm of the solution remains approximately constant for all time and hence the elliptic fixed point at the origin is Lyapunov statistically stable. This is first result in KAM Theory for PDEs that regards the persistence of a large measure set of invariant tori and hence may be viewed as a strict extension to the infinite dimensional setting of the classical KAM theorem.

math.AP↗

arXiv2024

Maximal tori in infinite-dimensional Hamiltonian systems: a Renormalization Group approach

We study the existence of infinite-dimensional invariant tori in a mechanical system of infinitely many rotators weakly interacting with each other. We consider explicitly interactions depending only on the angles, with the aim of discussing in a simple case the analyticity properties to be required on the perturbation of the integrable system in order to ensure the persistence of a large measure set of invariant tori with finite energy. The proof we provide of the persistence of the invariant tori implements the Renormalization Group scheme based on the tree formalism -- i.e. the graphical representation of the solutions of the equations of motion in terms of trees -- which has been widely used in finite-dimensional problems. The method is very effectual and flexible: it naturally extends, once the functional setting has been fixed, to the infinite-dimensional case with only minor technical-natured adaptations.

math.DS↗

arXiv2023

A diluted version of the problem of the existence of the Hofstadter sequence

We investigate the conditions on an integer sequence f(n), n 2 N, with f(1) = 0, such that the sequence q(n), computed recursively via q(n) = q(n - q(n - 1)) + f(n), with q(1) = 1, exists. We prove that f(n + 1) - f(n) in {0,1}, n > 0, is a sufficient but not necessary condition for the existence of sequence q. Sequences q defined in this way typically display non-trivial dynamics: in particular, they are generally aperiodic with no obvious patterns. We discuss and illustrate this behaviour with some examples.

math.NT↗

arXiv2023

Almost-periodic solutions to the NLS equation with smooth convolution potentials

We consider the one-dimensional NLS equation with a convolution potential and a quintic nonlinearity. We prove that, for most choices of potentials with polynomially decreasing Fourier coefficients, there exist almost-periodic solutions in the Gevrey class with frequency satisfying a Bryuno non-resonance condition. This allows convolution potentials of class $C^p$, for any integer $p$: as far as we know this is the first result where the regularity of the potential is arbitrarily large and not compensated by a corresponding smoothing of the nonlinearity.

math.AP↗

arXiv2021

Pseudo-synchronous solutions for dissipative non-autonomous systems

In the framework of KAM theory, the persistence of invariant tori in quasi-integrable systems is proved by assuming a non-resonance condition on the frequencies, such as the standard Diophantine condition or the milder Bryuno condition. In the presence of dissipation, most of the quasi-periodic solutions disappear and one expects, at most, only a few of them to survive together with the periodic attractors. However, to prove that a quasi-periodic solution really exists, usually one assumes that the frequencies still satisfy a Diophantine condition and, furthermore, that some external parameters of the system are suitably tuned with them. In this paper we consider a class of systems on the one-dimensional torus, subject to a periodic perturbation and in the presence of dissipation, and show that, however small the dissipation, if the perturbation is a trigonometric polynomial in the angles and the unperturbed frequencies satisfy a non-resonance condition of finite order, depending on the size of the dissipation, then a quasi-periodic solution exists with slightly perturbed frequencies provided the size of the perturbation is small enough. If on the one hand the maximal size of the perturbation is not uniform in the degree of the trigonometric polynomial, on the other hand all but finitely many frequencies are allowed and there is no restriction arising from the tuning of the external parameters. A physically relevant case, where the result applies, is the spin-orbit model, which describes the rotation of a satellite around its own axis, while revolving on a Keplerian orbit around a planet, in the case in which the dissipation is taken into account through the MacDonald torque.

math.DS↗

arXiv2020

Response solutions for strongly dissipative quasi-periodically forced systems with arbitrary nonlinearities and frequencies

We consider quasi-periodically systems in the presence of dissipation and study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term. When the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions are known to exist without assuming any non resonance condition on the frequency vector. We analyse the case of non-simple zeroes and, in order to deal with the small divisors problem, we confine ourselves to two-dimensional frequency vectors, so as to use the properties of continued fractions. We show that, if the order of the zero is odd (if it is even, in general no response solution exists), a response solution still exists provided the inverse of the parameter measuring the dissipation belongs to a set given by the union of infinite intervals depending on the convergents of the ratio of the two components of the frequency vector. The intervals may be disjoint and as a consequence we obtain the existence of response solutions in a set with "holes". If we want the set to be connected we have to require some non-resonance condition on the frequency: in fact, we need a condition weaker than the Bryuno condition usually considered in small divisors problems.

math.DS↗

arXiv2018

Forced quasi-periodic oscillations in strongly dissipative systems of any finite dimension

We consider a class of singular ordinary differential equations describing analytic systems of arbitrary finite dimension, subject to a quasi-periodic forcing term and in the presence of dissipation. We study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term, in the case of large dissipation. We assume the system to be conservative in the absence dissipation, so that the forcing term is --- up to the sign --- the gradient of a potential energy, and both the mass and damping matrices to be symmetric and positive definite. Further, we assume a non-degeneracy condition on the forcing term, essentially that the time-average of the potential energy has a strict local minimum. On the contrary, no condition is assumed on the forcing frequency; in particular we do not require any Diophantine condition. We prove that, under the assumptions above, a response solution always exist provided the dissipation is strong enough. This extends results previously available in literature in the one-dimensional case.

math.DS↗

arXiv2017

Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque

Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of the eccentricity of its orbit. However, the mathematical model originally introduced to study its spin-orbit evolution proved not to be entirely convincing, because of the expression commonly used for the tidal torque. Only recently, in a series of papers mainly by Efroimsky and Makarov, a different model for the tidal torque has been proposed, which has the advantages of being more realistic, and of providing a higher probability of capture in the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and consists of a superposition of kinks, so that both analytical and numerical computations turn out to be rather delicate: indeed, standard perturbation theory based on power series expansion cannot be applied and the implementation of a fast algorithm to integrate the equations of motion numerically requires a high degree of care. In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model: In particular, we present numerical and analytical results about the nature of the librations of Mercury's spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time.

math.DS↗

arXiv2016

Explicit estimates on the torus for the sup-norm and the crest factor of solutions of the Modified Kuramoto-Sivashinky Equation in one and two space dimensions

We consider the Modified Kuramoto-Sivashinky Equation (MKSE) in one and two space dimensions and we obtain explicit and accurate estimates of various Sobolev norms of the solutions. In particular, by using the sharp constants which appear in the functional interpolation inequalities used in the analysis of partial differential equations, we evaluate explicitly the sup-norm of the solutions of the MKSE. Furthermore we introduce and then compute the so-called crest factor associated with the above solutions. The crest factor provides information on the distortion of the solution away from its space average and therefore, if it is large, gives evidence of strong turbulence. Here we find that the time average of the crest factor scales like $λ^{(2d-1)/8}$ for $λ$ large, where $λ$ is the bifurcation parameter of the source term and $d=1,2$ is the space dimension. This shows that strong turbulence cannot be attained unless the bifurcation parameter is large enough.

math.AP↗

arXiv2016

Numerics for the spin orbit equation of Makarov with constant eccentricity

We present an algorithm for the rapid numerical integration of a time-periodic ODE with a small dissipation term that is $C^1$ in the velocity. Such an ODE arises as a model of spin-orbit coupling in a star/planet system, and the motivation for devising a fast algorithm for its solution comes from the desire to estimate probability of capture in various solutions, via Monte Carlo simulation: the integration times are very long, since we are interested in phenomena occurring on times similar to the formation time of the planets. The proposed algorithm is based on the High-order Euler Method (HEM) which was described in~\cite{hem}, and it requires computer algebra to set up the code for its implementation. The pay-off is an overall increase in speed by a factor of about $7.5$ compared to standard numerical methods. Means for accelerating the purely numerical computation are also discussed

math.CA↗

arXiv2016

Response solutions for forced systems with large dissipation and arbitrary frequency vectors

We study the behaviour of one-dimensional strongly dissipative systems subject to a quasi-periodic force. In particular we are interested in the existence of response solutions, that is quasi-periodic solutions having the same frequency vector as the forcing term. Earlier results available in the literature show that, when the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions can be proved to exist and to be attractive provided some Diophantine condition is assumed on the frequency vector. In this paper we show that the results extend to the case of arbitrary frequency vectors.

math.DS↗

arXiv2015

Construction of the Lyapunov spectrum in a chaotic system displaying phase synchronization

We consider a three-dimensional chaotic system consisting of the suspension of Arnold's cat map coupled with a clock via a weak dissipative interaction. We show that the coupled system displays a synchronization phenomenon, in the sense that the relative phase between the suspension flow and the clock locks to a special value, thus making the motion fall onto a lower dimensional attractor. More specifically, we construct the attractive invariant manifold, of dimension smaller than three, using a convergent perturbative expansion. Moreover, we compute via convergent series the Lyapunov exponents, including notably the central one. The result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model. The main novelty of the current construction relies in the computation of the Lyapunov spectrum, which consists of non-trivial analytic exponents. Some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed.

math-ph↗

arXiv2014

The high-order Euler method and the spin-orbit model A fast algorithm for solving differential equations with small, smooth nonlinearity

We present an algorithm for the rapid numerical integration of smooth, time-periodic differential equations with small nonlinearity, particularly suited to problems with small dissipation. The emphasis is on speed without compromising accuracy and we envisage applications in problems where integration over long time scales is required; for instance, orbit probability estimation via Monte Carlo simulation. We demonstrate the effectiveness of our algorithm by applying it to the spin-orbit problem, for which we have derived analytical results for comparison with those that we obtain numerically. Among other tests, we carry out a careful comparison of our numerical results with the analytically predicted set of periodic orbits that exists for given parameters. Further tests concern the long-term behaviour of solutions moving towards the quasi-periodic attractor, and capture probabilities for the periodic attractors computed from the formula of Goldreich and Peale. We implement the algorithm in standard double precision arithmetic and show that this is adequate to obtain an excellent measure of agreement between analytical predictions and the proposed fast algorithm.

math.NA↗

arXiv2014

Basins of attraction in forced systems with time-varying dissipation

We consider dissipative periodically forced systems and investigate cases in which having information as to how the system behaves for constant dissipation may be used when dissipation varies in time before settling at a constant final value. First, we consider situations where one is interested in the basins of attraction for damping coefficients varying linearly between two given values over many different time intervals: we outline a method to reduce the computation time required to estimate numerically the relative areas of the basins and discuss its range of applicability. Second, we observe that sometimes very slight changes in the time interval may produce abrupt large variations in the relative areas of the basins of attraction of the surviving attractors: we show how comparing the contracted phase space at a time after the final value of dissipation has been reached with the basins of attraction corresponding to that value of constant dissipation can explain the presence of such variations. Both procedures are illustrated by application to a pendulum with periodically oscillating support.

math.DS↗