Papers by Claudia M Giordano
Regular and Chaotic Dynamics
Physical Review E
The present work revisits and improves the Shannon entropy approach when applied to the estimatio... more The present work revisits and improves the Shannon entropy approach when applied to the estimation of an instability timescale for chaotic diffusion in multidimensional Hamiltonian systems. This formulation has already been proved efficient in deriving the diffusion timescale in 4D symplectic maps and planetary systems, when the diffusion proceeds along the chaotic layers of the resonance's web. Herein the technique is used to estimate the diffusion rate in the Arnold model, i.e., of the motion along the homoclinic tangle of the so-called guiding resonance for several values of the perturbation parameter such that the overlap of resonances is almost negligible. Thus differently from the previous studies, the focus is fixed on deriving a local timescale related to the speed of an Arnold diffusion-like process. The comparison of the current estimates with determinations of the diffusion time obtained by straightforward numerical integration of the equations of motion reveals a quite good agreement.
The European Physical Journal Special Topics
The European Physical Journal Special Topics, 2022
The reader can find in the literature a lot of different techniques to study the dynamics of a gi... more The reader can find in the literature a lot of different techniques to study the dynamics of a given system and also, many suitable numerical integrators to compute them. Notwithstanding the recent work of Maffione et al. (2011a) for mappings, a detailed comparison among the widespread indicators of chaos in a general system is still lacking. Such a comparison could lead to select the most efficient algorithms given a certain dynamical problem. Furthermore, in order to choose the appropriate numerical integrators to compute them, more comparative studies among numerical integrators are also needed. This work deals with both problems. We first extend the work of Maffione et al. (2011) for mappings to the 2D H\'enon & Heiles (1964) potential, and compare several variational indicators of chaos: the Lyapunov Indicator (LI); the Mean Exponential Growth Factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI) and its generalized version, the Generalized Alignment Index (GA...
The European Physical Journal Special Topics, 2022
Physica D: Nonlinear Phenomena, 2022
We model chaotic diffusion, in a symplectic 4D map by using the result of a theorem that was deve... more We model chaotic diffusion, in a symplectic 4D map by using the result of a theorem that was developed for stochastically perturbed integrable Hamiltonian systems. We explicitly consider a map defined by a free rotator (FR) coupled to a standard map (SM). We focus in the diffusion process in the action, I, of the FR, obtaining a semi--numerical method to compute the diffusion coefficient. We study two cases corresponding to a thick and a thin chaotic layer in the SM phase space and we discuss a related conjecture stated in the past. In the first case the numerically computed probability density function for the action I is well interpolated by the solution of a Fokker-Planck (F-P) equation, whereas it presents a non--constant time delay respect to the concomitant F-P solution in the second case suggesting the presence of an anomalous diffusion time scale. The explicit calculation of a diffusion coefficient for a 4D symplectic map can be useful to understand the slow diffusion observ...
Celestial Mechanics and Dynamical Astronomy
In the present effort, we revisit the Shannon entropy approach for the study of both the extent a... more In the present effort, we revisit the Shannon entropy approach for the study of both the extent and the rate of diffusion in a given dynamical system. In particular, we provide a theoretical and numerical study of the dependence of the formulation on the parameters of the method. We succeed in deriving not only a diffusion coefficient, $$D_{S}$$ D S , but also an estimate of the macroscopical instability time for the system under study. Dealing with a toy model, namely a 4D symplectic application that represents the dynamics around a junction of resonances of different order, and an a particular case of the planar three-body problem, the HD20003 planetary system, we obtain numerical evidence that $$D_{S}$$ D S is a robust measure of the diffusion rate, no significant dependence on the free parameter of the entropy formulation (the size of the elements of the partition) being observed. Moreover, successful results concerning the estimation of macroscopical instability times obtained from $$D_{S}$$ D S are presented in both cases.
Physica D: Nonlinear Phenomena
Monthly Notices of the Royal Astronomical Society
We show that diffusion due to chaotic mixing in the Neighbourhood of the Sun may not be as releva... more We show that diffusion due to chaotic mixing in the Neighbourhood of the Sun may not be as relevant as previously suggested in erasing phase space signatures of past Galactic accretion events. For this purpose, we analyse Solar Neighbourhood-like volumes extracted from cosmological simulations that naturally account for chaotic orbital behaviour induced by the strongly triaxial and cuspy shape of the resulting dark matter haloes, among other factors. In the approximation of an analytical static triaxial model, our results show that a large fraction of stellar halo particles in such local volumes have chaos onset times (i.e., the timescale at which stars commonly associated with chaotic orbits will exhibit their chaotic behaviour) significantly larger than a Hubble time. Furthermore, particles that do present a chaotic behaviour within a Hubble time do not exhibit significant diffusion in phase space.
The European Physical Journal Special Topics, 2016
The stability of the magnetic levitation showed by the Levitron was studied by M.V. Berry as a si... more The stability of the magnetic levitation showed by the Levitron was studied by M.V. Berry as a six degrees of freedom Hamiltonian system using an adiabatic approximation. Further, H.R. Dullin found critical spin rate bounds where the levitation persist and R.F. Gans et al. offered numerical results regarding the initial conditions' manifold where this occurs. In the line of this series of works, first, we extend the equations of motion to include dissipation for a more realistic model, and then introduce a mechanical forcing to inject energy into the system in order to prevent the Levitron from falling. A systematic study of the flying time as a function of the forcing parameters is carried out which yields detailed bifurcation diagrams showing an Arnold's tongues structure. The stability of these solutions were studied with the help of a novel method to compute the maximum Lyapunov exponent called MEGNO. The bifurcation diagrams for MEGNO reproduce the same Arnold's tongue structure.
In the present chapter we review as well as provide new results on the processes that lead to cha... more In the present chapter we review as well as provide new results on the processes that lead to chaotic diffusion in phase space of multidimensional Hamiltonian systems. It is well known that the simplest mechanisms leading to a transition from regularity to chaos, and therefore to diffusion in phase space, are the overlap of resonances, resonance crossings and Arnol'd diffusion-like processes. When dealing with nearly integrable Hamiltonian systems, chaos actually means the variation of the unperturbed integrals, which is usually called chaotic diffusion. Unfortunately, it does not yet exist any theory that could describe global diffusion in phase space. In other words, it is not possible to estimate either its routes or its extent. Though one could get accurate values of the Lyapunov exponents, the KS entropy or any other indicator of the stability of the motion, they only provide local values for the variation of the integrals. A given orbit in a chaotic component of phase space could have, for instance, a positive and large value for two of the Lyapunov exponents, however, this does not necessarily mean that the unperturbed integrals would change over a rather large domain. This is a natural consequence of the structure of phase space of almost all actual dynamical systems such as planetary systems or galaxies. Therefore, what is actually significant is the extent of the domain and the timescale over which diffusion may occur. In it is shown that in models similar to those suitable for the description of an elliptical galaxy, the time-scale over which diffusion becomes relevant is several orders of magnitude the Hubble time. On the other hand, in models corresponding to planetary or asteroidal dynamics, diffusion may occur in physical time-scales. All these issues as well as a relatively new fast indicator of the dynamics, the Mean Exponential Growth Factor of Nearby Orbits (MEGNO), are thoroughly discussed in this chapter by both numerical and theoretical means.
Astronomy and Astrophysics, Sep 29, 1996
The capture regions for planetary satellites are numerically explored in the frame of the restric... more The capture regions for planetary satellites are numerically explored in the frame of the restricted tree-body problem. Appropriate scaling relations for describing capture conditions in terms of mu are provided in the bidimensional problem. Also the tridimensional problem is considered and capture regions completely characterized for Jupiter, Uranus and the Earth. A discussion is given of conditions leading to capture
Lecture Notes in Physics, 2014
ABSTRACT In this chapter we discuss in a pedagogical way and from the very beginning the {\it Mea... more ABSTRACT In this chapter we discuss in a pedagogical way and from the very beginning the {\it Mean Exponential Growth factor of Nearby Orbits} (MEGNO) method, that has proven, in the last ten years, to be efficient to investigate both regular and chaotic components of phase space of a Hamiltonian system. It is a fast indicator that provides a clear picture of the resonance structure, the location of stable and unstable periodic orbits as well as a measure of hyperbolicity in chaotic domains which coincides with that given by the maximum Lyapunov characteristic exponent but in a shorter evolution time. Applications of the MEGNO to simple discrete and continuous dynamical systems are discussed and an overview of the stability studies present in the literature encompassing quite different dynamical systems is provided.
Physica A: Statistical Mechanics and its Applications, 2004
We revisit some aspects of the foundations of the q-non-extensive thermostatistical formalism, wh... more We revisit some aspects of the foundations of the q-non-extensive thermostatistical formalism, which are particularly relevant to its astronomical applications. We analyse some important features of escort mean values and escort distributions, in connection with the dynamical evolution of ensemble probability distributions associated with the statistical description of general classical dynamical systems. Furthermore, we compare the roles played by escort mean values and escort distributions when studying classical N -body systems at the levels of (i) the Liouville equation and (ii) the Vlasov equation.
An appropriate generalization of the Jacobi equation of motion for the polar moment of inertia I ... more An appropriate generalization of the Jacobi equation of motion for the polar moment of inertia I is considered in order to study the N-body problem with variable masses. Two coupled ordinary differential equations governing the evolution of I and the total energy E are obtained. A regularization scheme for this system of differential equations is provided. We compute some illustrative numerical examples, and discuss an average method for obtaining approximate analytical solutions to this pair of equations. For a particular law of mass loss we also obtain exact analytical solutions. The application of these ideas to other kind of perturbed gravitational N-body systems involving drag forces or a different type of mass variation is also considered.
Physical Review E, 2014
We model chaotic diffusion, in a symplectic 4D map by using the result of a theorem that was deve... more We model chaotic diffusion, in a symplectic 4D map by using the result of a theorem that was developed for stochastically perturbed integrable Hamiltonian systems. We explicitly consider a map defined by a free rotator (FR) coupled to a standard map (SM). We focus in the diffusion process in the action, I, of the FR, obtaining a semi-numerical method to compute the diffusion coefficient. We study two cases corresponding to a thick and a thin chaotic layer in the SM phase space and we discuss a related conjecture stated in the past. In the first case the numerically computed probability density function for the action I is well interpolated by the solution of a Fokker-Planck (F-P) equation, whereas it presents a non-constant time delay respect to the concomitant F-P solution in the second case suggesting the presence of an anomalous diffusion time scale. The explicit calculation of a diffusion coefficient for a 4D symplectic map can be useful to understand the slow diffusion observed in Celestial Mechanics and Accelerator Physics.
Physica D: Nonlinear Phenomena, 2014
We present theoretical and numerical results pointing towards a strong connection between the est... more We present theoretical and numerical results pointing towards a strong connection between the estimates for the diffusion rate along simple resonances in multidimensional nonlinear Hamiltonian systems that can be obtained using the heuristic theory of Chirikov and a more formal one due to Nekhoroshev. We show that, despite a wide-spread impression, the two theories are complementary rather than antagonist. Indeed, although Chirikov's 1979 review has thousands of citations, almost all of them refer to topics such as the resonance overlap criterion, fast diffusion, the Standard or Whisker Map, and not to the constructive theory providing a formula to measure diffusion along a single resonance. However, as will be demonstrated explicitly below, Chirikov's formula provides values of the diffusion coefficient which are quite well comparable to the numerically computed ones, provided that it is implemented on the so-called optimal normal form derived as in the analytic part of Nekhoroshev's theorem. On the other hand, Chirikov's formula yields unrealistic values of the diffusion coefficient, in particular for very small values of the perturbation, when used in the original Hamiltonian instead of the optimal normal form. In the present paper, we take advantage of this complementarity in order to obtain accurate theoretical predictions for the local value of the diffusion coefficient along a resonance in a specific 3DoF nearly integrable Hamiltonian system. Besides, we compute numerically the diffusion coefficient and a full comparison of all estimates is made for ten values of the perturbation parameter, showing a very satisfactory agreement.
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Papers by Claudia M Giordano