arXiv subjects

Federico Panichi

Publications and source records attributed to Federico Panichi.

8 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.

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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.

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arXiv2018

The architecture and formation of the Kepler-30 planetary system

We study the orbital architecture, physical characteristics of planets, formation and long-term evolution of the Kepler-30 planetary system, detected and announced in 2012 by the KEPLER team. We show that the Kepler-30 system belongs to a particular class of very compact and quasi-resonant, yet long-term stable planetary systems. We re-analyse the light curves of the host star spanning Q1-Q17 quarters of the KEPLER mission. A huge variability of the Transit Timing Variations (TTV) exceeding 2 days is induced by a massive Jovian planet located between two Neptune-like companions. The innermost pair is near to the 2:1 mean motion resonance (MMR), and the outermost pair is close to higher order MMRs, such as 17:7 and 7:3. Our re-analysis of photometric data allows us to constrain, better than before, the orbital elements, planets' radii and masses, which are $9.2 \pm 0.1$, $536 \pm 5$, and $23.7 \pm 1.3$ Earth masses for Kepler-30b, Kepler-30c and Kepler-30d, respectively. The masses of the inner planets are determined within $\sim 1\%$ uncertainty. We infer the internal structures of the Kepler-30 planets and their bulk densities in a wide range from $ (0.19 \pm 0.01)$ g$\cdot$cm$^{-3}$ for Kepler-30d, $(0.96\pm0.15)$ g$\cdot$cm$^{-3}$ for Kepler-30b, to $(1.71 \pm 0.13)$ g$\cdot$cm$^{-3}$ for the Jovian planet Kepler-30c. We attempt to explain the origin of this unique planetary system and a deviation of the orbits from exact MMRs through the planetary migration scenario. We anticipate that the Jupiter-like planet plays an important role in determining the present dynamical state of this system.

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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 λt, while when the motion is regular (quasi-periodic) then REM increases as a power law in time, as t^α, where α and λ 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.

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arXiv2016

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 $σ(t)$ 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 $\exp\left(-σ^2(t)\right)$. Some numerical examples such as the anharmonic oscillator and the Hénon 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.

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arXiv2016

The origin and 9:7 MMR dynamics of the Kepler-29 system

We analyse the Transit Timing Variation (TTV) measurements of a~system of two super-Earths detected as Kepler-29, in order to constrain the planets' masses and orbital parameters. A dynamical analysis of the best-fitting configurations constrains the masses to be $\sim 6$ and $\sim 5$ Earth masses for the inner and the outer planets, respectively. The analysis also reveals that the system is likely locked in the 9:7~mean motion resonance. However, a variety of orbital architectures regarding eccentricities and the relative orientation of orbits is permitted by the observations as well as by stability constraints. We attempt to find configurations preferred by the planet formation scenarios as an additional, physical constraint. We show that configurations with low eccentricities and anti-aligned apsidal lines of the orbits are a natural and most likely outcome of the convergent migration. However, we show that librations of the critical angles are not necessary for the Kepler-29 system to be dynamically resonant, and such configurations may be formed on the way of migration as well. We argue, on the other hand, that aligned configurations with $e \gtrsim 0.03$ may be not consistent with the migration scenario.

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arXiv2015

The Laplace resonance in the Kepler-60 system

We investigate the dynamical stability of the Kepler-60 planetary system with three super-Earths. We first determine their orbital elements and masses by Transit Timing Variation (TTV) data spanning quarters Q1-Q16 of the KEPLER mission. The system is dynamically active but the TTV data constrain masses to ~4 Earth masses and orbits in safely wide stable zones. The observations prefer two types of solutions. The true three-body Laplace MMR exhibits the critical angle librating around 45 degrees and aligned apsides of the inner and outer pair of planets. In the Laplace MMR formed through a chain of two-planet 5:4 and 4:3 MMRs, all critical angles librate with small amplitudes of ~30 degrees and apsidal lines in planet's pairs are anti-aligned. The system is simultaneously locked in a three-body MMR with librations amplitude of ~10 degrees. The true Laplace MMR can evolve towards a chain of two-body MMRs in the presence of planetary migration. Therefore the three-body MMR formed in this way seems to be more likely state of the system. However, the true three-body MMR cannot be disregarded a priori and it remains a puzzling configuration that may challenge the planet formation theory.

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arXiv2015

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.

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