Claudia M. Giordano

Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits

In this paper we deal with an alternative technique to study global dynamics in Hamiltonian systems, the mean exponential growth factor of nearby orbits (MEGNO), that proves to be efficient to investigate both regular and stochastic components of phase space. It provides a clear picture of resonance structures, location of stable and unstable periodic orbits as well as a measure of hyperbolicity in chaotic domains which coincides with that given by the Lyapunov characteristic number. Here the MEGNO is applied to a rather simple model, the 3D perturbed quartic oscillator, in order to visualize the structure of its phase space and obtain a quite clear picture of its resonance structure. Examples of application to multi-dimensional canonical maps are also included.

A comparison of different indicators of chaos based on the deviation vectors: application to symplectic mappings

AbstractThe aim of this research work is to compare the reliability of several variational indicators of chaos in mappings. The Lyapunov Indicator; the Mean Exponential Growth factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI); the Fast Lyapunov Indicator (FLI); the Dynamical Spectra of stretching numbers and the corresponding Spectral Distance and the Relative Lyapunov Indicator (RLI), which is based on the evolution of the difference between two close orbits, have been included. The experiments presented herein allow us to reliably suggest a group of chaos indicators to analyze a general mapping. We show that a package composed of the FLI and the RLI (to analyze the phase portrait globally) and the MEGNO and the SALI (to analyze orbits individually) is good enough to make a description of the systems’ dynamics.

COMPARATIVE STUDY OF VARIATIONAL CHAOS INDICATORS AND ODEs' NUMERICAL INTEGRATORS

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., 2011b] 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., 2011b] for mappings to the 2D [Henon & 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 (GALI); the Fast Lyapunov Indicator (FLI) and its variant, the Orthogonal Fast Lyapunov Indicator (OFLI); the Spectral Distance (D) and the Dynamical Spectra of Stretching Numbers (SSNs). We also include in the record the Relative Lyapunov Indicator (RLI), which is not a variational indicator as the others. Then, we test a numerical technique to integrate Ordinary Differential Equations (ODEs) based on the Taylor method implemented by [Jorba & Zou, 2005] (called taylor), and we compare its performance with other two well-known efficient integrators: the [Prince & Dormand, 1981] implementation of a Runge–Kutta of order 7–8 (DOPRI8) and a Bulirsch–Stoer implementation. These tests are run under two very different systems from the complexity of their equations point of view: a triaxial galactic potential model and a perturbed 3D quartic oscillator. We first show that a combination of the FLI/OFLI, the MEGNO and the GALI2N succeeds in describing in detail most of the dynamical characteristics of a general Hamiltonian system. In the second part, we show that the precision of taylor is better than that of the other integrators tested, but it is not well suited to integrate systems of equations which include the variational ones, like in the computing of almost all the preceeding indicators of chaos. The result which induces us to draw this conclusion is that the computing times spent by taylor are far greater than the times consumed by the DOPRI8 and the Bulirsch–Stoer integrators in such cases. On the other hand, the package is very efficient when we only need to integrate the equations of motion (both in precision and speed), for instance to study the chaotic diffusion. We also notice that taylor attains a greater precision on the coordinates than either the DOPRI8 or the Bulirsch–Stoer.

Chaos detection tools: application to a self-consistent triaxial model

Fil: Maffione, Nicolas Pablo. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - La Plata. Instituto de Astrofisica La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Astronomicas y Geofisicas. Instituto de Astrofisica la Plata; Argentina

Testing a fast dynamical indicator: The MEGNO

Abstract To investigate non-linear dynamical systems, like for instance artificial satellites, Solar System, exoplanets or galactic models, it is necessary to have at hand several tools, such as a reliable dynamical indicator. The aim of the present work is to test a relatively new fast indicator, the Mean Exponential Growth factor of Nearby Orbits (MEGNO), since it is becoming a widespread technique for the study of Hamiltonian systems, particularly in the field of dynamical astronomy and astrodynamics, as well as molecular dynamics. In order to perform this test we make a detailed numerical and statistical study of a sample of orbits in a triaxial galactic system, whose dynamics was investigated by means of the computation of the Finite Time Lyapunov Characteristic Numbers (FT-LCNs) by other authors.

Chirikov and Nekhoroshev diffusion estimates: Bridging the two sides of the river

Abstract 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.

On the relevance of chaos for halo stars in the solar neighbourhood

In a previous paper based on dark matter only simulations we show that, in the approximation of an analytic and static potential describing the strongly triaxial and cuspy shape of Milky Way-sized haloes, diffusion due to chaotic mixing in the neighbourhood of the Sun does not efficiently erase phase space signatures of past accretion events. In this second paper we further explore the effect of chaotic mixing using multicomponent Galactic potential models and solar neighbourhood-like volumes extracted from fully cosmological hydrodynamic simulations, thus naturally accounting for the gravitational potential associated with baryonic components, such as the bulge and disc. Despite the strong change in the global Galactic potentials with respect to those obtained in dark matter only simulations, our results confirm that a large fraction of halo particles evolving on chaotic orbits exhibit their chaotic behaviour after periods of time significantly larger than a Hubble time. In addition, significant diffusion in phase space is not observed on those particles that do exhibit chaotic behaviour within a Hubble time.

Theory and Applications of the Mean Exponential Growth Factor of Nearby Orbits (MEGNO) Method

In this chapter we discuss in a pedagogical way and from the very beginning the 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.

CHAOTIC DIFFUSION IN MULTIDIMENSIONAL CONSERVATIVE MAPS

In the present paper, we provide results and discussions concerning the processes that lead to local and global chaotic diffusion in the phase space of multidimensional conservative systems. We investigate and provide a measure of the extent of the domain over which diffusion may occur. All these issues are thoroughly discussed by dealing with a multidimensional conservative map that would be representative of the dynamics of a resonance interaction, which is an important mechanism in many dynamical systems.