Antonio Giorgilli

Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory

SommarioDa diversi anni gli esponenti caratteristici di Lyapunov sono divenuti di notevole interesse nello studio dei sistemi dinamici al fine di caratterizzare quantitativamente le proprietà di stocasticità, legate essenzialmente alla divergenza esponenziale di orbite vicine. Si presenta dunque il problema del calcolo esplicito di tali esponenti, già risolto solo per il massimo di essi. Nel presente lavoro si dà un metodo per il calcolo di tutti tali esponenti, basato sul calcolo degli esponenti di ordine maggiore di uno, legati alla crescita di volumi. A tal fine si dà un teorema che mette in relazione gli esponenti di ordine uno con quelli di ordine superiore. Il metodo numerico e alcune applicazioni saranno date in un sucessivo articolo.SummarySince several years Lyapunov Characteristic Exponents are of interest in the study of dynamical systems in order to characterize quantitatively their stochasticity properties, related essentially to the exponential divergence of nearby orbits. One has thus the problem of the explicit computation of such exponents, which has been solved only for the maximal of them. Here we give a method for computing all of them, based on the computation of the exponents of order greater than one, which are related to the increase of volumes. To this end a theorem is given relating the exponents of order one to those of greater order. The numerical method and some applications will be given in a forthcoming paper.

Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application

SommarioQuesto articolo, insieme con il precedente (Parte 1: Teoria, pubblicato in questa stessa rivista) è inteso a fornire un metodo esplicito per il calcolo di tutti gli esponenti caratteristici di Lvapunov per un sistema dinamico. Dopo la teoria generale su tali esponenti sviluppata nella prima parte, qui si illustra il metodo di calcolo (Capitolo A) e si danno esempi numerici per applicazioni di varietà in sè e per sistemi Hamiltoniani (Capitolo B).SummaryThe present paper, together with the previous one (Part 1: Theory, published in this journal) is intended to give an explicit method for computing all Lyapunov Characteristic Exponents of a dynamical system. After the general theory on such exponents developed in the first part, in the present paper the computational method is described (Chapter A) and some numerical examples for mappings on manifolds and for Hamiltonian systems are given (Chapter B).

On the Hamiltonian Interpolation of Near-to-the-Identity Symplectic Mappings with Application to Symplectic Integration Algorithms

We reconsider the problem of the Hamiltonian interpolation of symplectic mappings. Following Mosers scheme, we prove that for any mapping ψε, analytic and ε-close to the identity, there exists an analytic autonomous Hamiltonian system, Hε such that its time-one mapping ΦHε differs from ψε by a quantity exponentially small in 1/ε. This result is applied, in particular, to the problem of numerical integration of Hamiltonian systems by symplectic algorithms; it turns out that, when using an analytic symplectic algorithm of orders to integrate a Hamiltonian systemK, one actually follows “exactly,” namely within the computer roundoff error, the trajectories of the interpolating Hamiltonian Hε, or equivalently of the rescaled Hamiltonian Kε=ε-1Hε, which differs fromK, but turns out to be ε5 close to it. Special attention is devoted to numerical integration for scattering problems.

Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three body problem

Abstract We consider an n -degrees of freedom Hamiltonian system near an elliptic equilibrium point. The system is put in normal form (up to an arbitrary order and with respect to some resonance module) and estimates are obtained for both the size of the remainder and for the domain of convergence of the transformation leading to normal form. A bound to the rate of diffusion is thus found, and by optimizing the order of normalization exponential estimates of Nekhoroshevs type are obtained. This provides explicit estimates for the stability properties of the elliptic point, and leads in some cases to “effective stability,” i.e., stability up to finite but long times. An application to the stability of the triangular libration points in the spatial restricted three body is also given.

A PROOF OF NEKHOROSHEV'S THEOREM FOR THE STABILITY TIMES IN NEARLY INTEGRABLE HAMILTONIAN SYSTEMS

In the present paper we give a proof of Nekhoroshevs theorem, which is concerned with an exponential estimate for the stability times in nearly integrable Hamiltonian systems. At variance with the already published proof, which refers to the case of an unperturbed Hamiltonian having the generic property of steepness, we consider here the particular case of a convex unperturbed Hamiltonian. The corresponding simplification in the proof might be convenient for an introduction to the subject.

Superexponential stability of KAM tori

We study the dynamics in the neighborhood of an invariant torus of a nearly integrable system. We provide an upper bound to the diffusion speed, which turns out to be of superexponentially small size exp[-exp(1/σ)], σ being the distance from the invariant torus. We also discuss the connection of this result with the existence of many invariant tori close to the considered one.

Realization of holonomic constraints and freezing of high frequency degrees of freedom in the light of classical perturbation theory. Part II

As in Part I of this paper, we consider the problem of the energy exchanges between two subsystems, of which one is a system of ν harmonic oscillators, while the other one is any dynamical system ofn degrees of freedom. Such a problem is of interest both for the realization of holonomic constraints of classical mechanics, and for the freezing of the internal degrees of freedom in molecular collisions. The results of Part I, which referred to the particular case ν=1, are here extended to the more difficult case ν>1. For the rate of energy transfer we find exponential estimates of Nekhoroshevs type, namely of the form exp (λ*/λ)1/a, where λ is a positive real number giving the size of the involved frequencies, and λ* anda are constants. For the particularly relevant constanta we find in generala=1/ν however, in the particular case when the ν frequencies are equal (collision of identical molecules), we finda=1 independently of ν, as conjectured by Jeans in the year 1903.

Formal integrals for an autonomous Hamiltonian system near an equilibrium point

In this paper we give a new method to construct formal integrals for an autonomous Hamiltonian system near an equilibrium point. Our construction is reminiscent of the algorithms introduced by Hori and Deprit; we show however how it presents itself quite naturally if one follows the line of attack of Whittaker, Cherry, and Contopoulos.

A proof of Kolmogorov’s theorem on invariant tori using canonical transformations defined by the Lie method

SummaryIn this paper a proof is given of Kolmogorov’s theorem on the existence of invariant tori in nearly integrable Hamiltonian systems. The scheme of proof is that of Kolmogorov, the only difference being in the way canonical transformations near the identity are defined. Precisely, use is made of the Lie method, which avoids any inversion and thus any use of the implicit-function theorem. This technical fact eliminates a spurious ingredient and simplifies the establishment of a central estimate.RiassuntoNel presente lavoro si dimostra il teorema di Kolmogorov sull’esistenza di tori invarianti in sistemi Hamiltoniani quasi integrabili. Si usa lo schema di dimostrazione di Kolmogorov, con la sola variante del modo in cui si definiscono le trasformazioni canoniche prossime all’identità. Si usa infatti il metodo di Lie, che elimina la necessità d’inversioni e quindi dell’impiego del teorema delle funzioni implicite. Questo fatto tecnico evita un ingrediente spurio e semplifica il modo in cui si ottiene una delle stime principali.РезюмеВ этой работе предлагается доказательство теоремы Колмогорова о существовании инвариантных торов в квази-интегрируемых Гамильтоновых системах. Используется схема доказательства, предложенная Колмогоровым, единственное отличие состоит в способе, которым определяются канонические преобразования. В этой работе используется метод Ли, которыи исключает необходимость инверсии и, следовательно, использование теоремы для неявной функции. Этот технический прием исключает ложный ингрдеиент и упрощает получение главной оценки.

On the problem of energy equipartition for large systems of the Fermi-Pasta-Ulam type: analytical and numerical estimates

Abstract We report on some analytical and numerical results on the exchanges of energy in systems of the Fermi-Pasta-Ulam type, in the light of Nekhoroshevs theorem, with particular attention to the dependence of the estimates on the number n of degrees of freedom. For the ordinary FPU problem we look for a control of the single normal mode energies, and we find both the analytical and numerical estimates to agree in predicting that the energy exchanges of the single modes cannot be controlled in the thermodynamic limit. We consider then a modified FPU model, with alternating light and heavy particles, which appears as composed of two subsystems, of low (acoustic) frequency and of high (optical) frequency respectively. We try to control the exchange of the total energy of the high frequency modes up to times increasing exponentially with the frequency. In this case the numerical estimates are stronger than the available analytical ones, and give indications for nonequipartition with constants essentially independent of the number n of degrees of freedom.