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
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.
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
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
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Papers by Claudia M Giordano