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Giorgio Turchetti

Publications and source records attributed to Giorgio Turchetti.

12 recordsLinked to original sources
arXiv2025

Lyapunov and Reversibility error invariant indicators

In this review, we present a survey of the Lyapunov Error and Reversibility Error (\cite{Faranda2012}), and we propose a generalization to make them invariant to the choice of initial conditions. We first define a process as the evolution in time of a a map or a flow, we then introduce the covariance matrix of a given process, and use their trace to compute LE and RE. The determinant of the covariance matrices is used to compute all invariant indicators of higher order. In this way, two set of invariant indicators are proposed within the framework introduced here, one for the Reversibility and one for the Lyapunov Errors, respectively. LE and RE which have been used in the literature, are the \textit{first-order} invariant indicators in their respective sets. The new sets of invariant indicators have the same fundamental meaning, the set for RE is used to characterize the dynamical evolution of a continuous small perturbation over an orbit, while LE is used to study the evolution of an initial displacement between two nearby orbits. We obtain again also the Reversibility Error Method (REM), which is a particular case of RE, where the additive noise is replaced by the round-off errors. REM has been proved to be a practical, accurate, and fast dynamical indicator and the results are comparable with the RE induced by a random noise of given amplitude. the behaviour of RE invariants depends on the sum of the positive Lyapunov exponents. We showcase the accuracy and reliability of those indicators on two well studied Hamiltonian. Within this new framework, a full dynamical characterization of a Hamiltonian system can be obtained. For instance, the Gibbs entropy, its asymptotic behaviour and a method to compute the fidelity over the perturbed orbit, are provided.

nlin.CD↗
arXiv2021

Propagation of rays in 2D and 3D waveguides: a stability analysis with Lyapunov and Reversibility fast indicators

Propagation of rays in 2D and 3D corrugated waveguides is performed in the general framework of stability indicators. The analysis of stability is based on the Lyapunov and Reversibility error. It is found that the error growth follows a power law for regular orbits and an exponential law for chaotic orbits. A relation with the Shannon channel capacity is devised and an approximate scaling law found for the capacity increase with the corrugation depth.

nlin.CD↗
arXiv2017

The Reversibility Error Method (REM): a new, dynamical fast indicator for planetary dynamics

We describe the Reversibility Error Method (REM) and its applications to planetary dynamics. REM is based on the time-reversibility analysis of the phase-space trajectories of conservative Hamiltonian systems. The round-off errors break the time reversibility and the displacement from the initial condition, occurring when we integrate it forward and backward for the same time interval, is related to the dynamical character of the trajectory. If the motion is chaotic, in the sense of non-zero maximal Characteristic Lyapunov Exponent (mLCE), then REM increases exponentially with time, as exp \lambda t, while when the motion is regular (quasi-periodic) then REM increases as a power law in time, as t^{\alpha}, where {\alpha} and {\lambda} are real coefficients. We compare the REM with a variant of mLCE, the Mean Exponential Growth factor of Nearby Orbits (MEGNO). The test set includes the restricted three body problem and five resonant planetary systems: HD 37124, Kepler-60, Kepler-36, Kepler-29 and Kepler-26. We found a very good agreement between the outcomes of these algorithms. Moreover, the numerical implementation of REM is astonishing simple, and is based on solid theoretical background. The REM requires only a symplectic and time-reversible (symmetric) integrator of the equations of motion. This method is also CPU efficient. It may be particularly useful for the dynamical analysis of multiple planetary systems in the KEPLER sample, characterized by low-eccentricity orbits and relatively weak mutual interactions. As an interesting side-result, we found a possible stable chaos occurrence in the Kepler-29 planetary system.

astro-ph.EP↗
arXiv2016

Rise time of proton cut-off energy in 2D and 3D PIC simulations

The Target Normal Sheath Acceleration (TNSA) regime for proton acceleration by laser pulses is experimentally consolidated and fairly well understood. However, uncertainties remain in the analysis of particle-in-cell (PIC) simulation results. The energy spectrum is exponential with a cut-off, but the maximum energy depends on the simulation time, following different laws in two and three dimensional (2D, 3D) PIC simulations, so that the determination of an asymptotic value has some arbitrariness. We propose two empirical laws for rise time of the cut-off energy in 2D and 3D PIC simulations, suggested by a model in which the proton acceleration is due to a surface charge distribution on the target rear side. The kinetic energy of the protons that we obtain follows two distinct laws, which appear to be nicely satisfied by PIC simulations. The laws depend on two parameters: the scaling time, at which the energy starts to rise, and the asymptotic cut-off energy. The values of the cut-off energy, obtained by fitting the 2D and 3D simulations for the same target and laser pulse, are comparable. This suggests that parametric scans can be performed with 2D simulations, since 3D ones are computationally very expensive. In this paper, the simulations are carried out for $a_0=3$ with the PIC code ALaDyn by changing the target thickness $L$ and the incidence angle $\alpha$. A monotonic dependence, on $L$ for normal incidence and on $\alpha$ for fixed $L$, is found, as in the experimental results for high temporal contrast pulses.

physics.plasm-ph↗
arXiv2015

Errors, Correlations and Fidelity for noisy Hamilton flows. Theory and numerical examples

We analyse the asymptotic growth of the error for Hamiltonian flows due to small random perturbations. We compare the forward error with the reversibility error, showing their equivalence for linear flows on a compact phase space. The forward error, given by the root mean square deviation $\sigma(t)$ of the noisy flow, grows according to a power law if the system is integrable and according to an exponential law if it is chaotic. The autocorrelation and the fidelity, defined as the correlation of the perturbed flow with respect to the unperturbed one, exhibit an exponential decay as $\exp\left(-\sigma^2(t)\right)$. Some numerical examples such as the anharmonic oscillator and the H\'enon Heiles model confirm these results. We finally consider the effect of the observational noise on an integrable system, and show that the decay of correlations can only be observed after a sequence of measurements and that the multiplicative noise is more effective if the delay between two measurements is large.

nlin.CD↗
arXiv2014

Fidelity and Reversibility in the Restricted Three Body Problem

We use the Reversibility Error Method and the Fidelity to analyze the global effects of a small perturbation in a non-integrable system. Both methods have already been proposed and used in the literature but the aim of this paper is to compare them in a physically significant example adding some considerations on the equivalence, observed in this case, between round-off and random perturbations. As a paradigmatic example we adopt the restricted planar circular three body problem. The cumulative effect of random perturbations or round-off leads to a divergence of the perturbed orbit from the reference one. Rather than computing the distance of the perturbed orbit from the reference one, after a given number n of iterations, a procedure we name the Forward Error Method (FEM), we measure the distance of the reversed orbit (n periods forward and backward) from the initial point. This approach, that we name Reversibility Error Method (REM), does not require the computation of the unperturbed map. The loss of memory of the perturbed map is quantified by the Fidelity decay rate whose computation requires a statistical average over an invariant region.

nlin.CD↗
arXiv2012

Extreme value statistics for dynamical systems with noise

We study the distribution of maxima (Extreme Value Statistics) for sequences of observables computed along orbits generated by random transformations. The underlying, deterministic, dynamical system can be regular or chaotic. In the former case, we will show that by perturbing rational or irrational rotations with additive noise, an extreme value law appears, regardless of the intensity of the noise, while unperturbed rotations do not admit such limiting distributions. In the case of deterministic chaotic dynamics, we will consider observables specially designed to study the recurrence properties in the neighbourhood of periodic points. Hence, the exponential limiting law for the distribution of maxima is modified by the presence of the extremal index, a positive parameter not larger than one, whose inverse gives the average size of the clusters of extreme events. The theory predicts that such a parameter is unitary when the system is perturbed randomly. We perform sophisticated numerical tests to assess how strong is the impact of noise level, when finite time series are considered. We find agreement with the asymptotic theoretical results but also non-trivial behaviour in the finite range. In particular our results suggest that in many applications where finite datasets can be produced or analysed one must be careful in assuming that the smoothing nature of noise prevails over the underlying deterministic dynamics.

math.DS↗
arXiv2012

Analysis of Round Off Errors with Reversibility Test as a Dynamical Indicator

We compare the divergence of orbits and the reversibility error for discrete time dynamical systems. These two quantities are used to explore the behavior of the global error induced by round off in the computation of orbits. The similarity of results found for any system we have analysed suggests the use of the reversibility error, whose computation is straightforward since it does not require the knowledge of the exact orbit, as a dynamical indicator. The statistics of fluctuations induced by round off for an ensemble of initial conditions has been compared with the results obtained in the case of random perturbations. Significant differences are observed in the case of regular orbits due to the correlations of round off error, whereas the results obtained for the chaotic case are nearly the same. Both the reversibility error and the orbit divergence computed for the same number of iterations on the whole phase space provide an insight on the local dynamical properties with a detail comparable with other dynamical indicators based on variational methods such as the finite time maximum Lyapunov characteristic exponent, the mean exponential growth factor of nearby orbits and the smaller alignment index. For 2D symplectic maps the differentiation between regular and chaotic regions is well full-filled. For 4D symplectic maps the structure of the resonance web as well as the nearby weakly chaotic regions are accurately described.

math.DS↗
arXiv2011

Generalized Extreme Value distribution parameters as dynamical indicators of Stability

We introduce a new dynamical indicator of stability based on the Extreme Value statistics showing that it provides an insight on the local stability properties of dynamical systems. The indicator perform faster than other based on the iteration of the tangent map since it requires only the evolution of the original systems and, in the chaotic regions, gives further information about the information dimension of the attractor. A numerical validation of the method is presented through the analysis of the motions in a Standard map.

math.DS↗
arXiv2011

Extreme Value distribution for singular measures

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems that have a singular measure. Using the block maxima approach described in Faranda et al. [2011] we show that, numerically, the Extreme Value distribution for these maps can be associated to the Generalised Extreme Value family where the parameters scale with the information dimension. The numerical analysis are performed on a few low dimensional maps. For the middle third Cantor set and the Sierpinskij triangle obtained using Iterated Function Systems, experimental parameters show a very good agreement with the theoretical values. For strange attractors like Lozi and H\`enon maps a slower convergence to the Generalised Extreme Value distribution is observed. Even in presence of large statistics the observed convergence is slower if compared with the maps which have an absolute continuous invariant measure. Nevertheless and within the uncertainty computed range, the results are in good agreement with the theoretical estimates.

math.DS↗
arXiv2011

Numerical convergence of the block-maxima approach to the Generalized Extreme Value distribution

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. [2006] have found analytical results.

math.DS↗