Energy localization on q-tori, long term stability and the interpretation of FPU recurrences

We focus on two approaches that have been proposed in recent years for the explanation of the so-called FPU paradox, i.e. the persistence of energy localization in the `low-q' Fourier modes of Fermi-Pasta-Ulam nonlinear lattices, preventing equipartition among all modes at low energies. In the first approach, a low-frequency fraction of the spectrum is initially excited leading to the formation of `natural packets' exhibiting exponential stability, while in the second, emphasis is placed on the existence of `q-breathers', i.e periodic continuations of the linear modes of the lattice, which are exponentially localized in Fourier space. Following ideas of the latter, we introduce in this paper the concept of `q-tori' representing exponentially localized solutions on low-dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the FPU paradox.Comment: 38 pages, 7 figure

Effective stability of the Trojan asteroids

We study the spatial circular restricted problem of three bodies in the light of Nekhoroshev theory of stability over large time intervals. We consider in particular the Sun-Jupiter model and the Trojan asteroids in the neighborhood of the Lagrangian point $L_4$. We find a region of effective stability around the point $L_4$ such that if the initial point of an orbit is inside this region the orbit is confined in a slightly larger neighborhood of the equilibrium (in phase space) for a very long time interval. By combining analytical methods and numerical approximations we are able to prove that stability over the age of the universe is guaranteed on a realistic region, big enough to include one real asteroid. By comparing this result with the one obtained for the planar problem we see that the regions of stability in the two cases are of the same magnitude.Comment: 9 pages, 2 figures, Astronomy & Astrophysics in pres

FPU phenomenon for generic initial data

The well known FPU phenomenon (lack of attainment of equipartition of the mode--energies at low energies, for some exceptional initial data) suggests that the FPU model does not have the mixing property at low energies. We give numerical indications that this is actually the case. This we show by computing orbits for sets of initial data of full measure, sampled out from the microcanonical ensemble by standard Montecarlo techniques. Mixing is tested by looking at the decay of the autocorrelations of the mode--energies, and it is found that the high--frequency modes have autocorrelations that tend instead to positive values. Indications are given that such a nonmixing property survives in the thermodynamic limit. It is left as an open problem whether mixing obtains within time--scales much longer than the presently available ones

Persistence of Diophantine flows for quadratic nearly-integrable Hamiltonians under slowly decaying aperiodic time dependence

The aim of this paper is to prove a Kolmogorov-type result for a nearly-integrable Hamiltonian, quadratic in the actions, with an aperiodic time dependence. The existence of a torus with a prefixed Diophantine frequency is shown in the forced system, provided that the perturbation is real-analytic and (exponentially) decaying with time. The advantage consists of the possibility to choose an arbitrarily small decaying coefficient, consistently with the perturbation size.Comment: Several corrections in the proof with respect to the previous version. Main statement unchange

A Kolmogorov theorem for nearly-integrable Poisson systems with asymptotically decaying time-dependent perturbation

The aim of this paper is to prove the Kolmogorov theorem of persistence of Diophantine flows for nearly-integrable Poisson systems associated to a real analytic Hamiltonian with aperiodic time dependence, provided that the perturbation is asymptotically vanishing. The paper is an extension of an analogous result by the same authors for canonical Hamiltonian systems; the flexibility of the Lie series method developed by A. Giorgilli et al., is profitably used in the present generalisation.Comment: 10 page

Integrability and strong normal forms for non-autonomous systems in a neighbourhood of an equilibrium

The paper deals with the problem of existence of a convergent "strong" normal form in the neighbourhood of an equilibrium, for a finite dimensional system of differential equations with analytic and time-dependent non-linear term. The problem can be solved either under some non-resonance hypotheses on the spectrum of the linear part or if the non-linear term is assumed to be (slowly) decaying in time. This paper "completes" a pioneering work of Pustil'nikov in which, despite under weaker non-resonance hypotheses, the nonlinearity is required to be asymptotically autonomous. The result is obtained as a consequence of the existence of a strong normal form for a suitable class of real-analytic Hamiltonians with non-autonomous perturbations.Comment: 10 page

Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories

We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, that can be regarded as a major refinement of the approach first introduced by Lagrange. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a solution that is invariant up to order two in the masses; therefore, we investigate the stability of the secular system for rather small values of the eccentricities. First, we explicitly construct a Kolmogorov normal form, so as to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that is at basis of the analytic part of the Nekhoroshev's theorem, so as to show that there is a neighborhood of that torus for which the estimated stability time is larger than the lifetime of the Solar System. The size of such a neighborhood, compared with the uncertainties of the astronomical observations, is about ten times smaller.Comment: 31 pages, 2 figures. arXiv admin note: text overlap with arXiv:1010.260

A Semi-Analytic Algorithm for Constructing Lower Dimensional Elliptic Tori in Planetary Systems

We adapt the Kolmogorov's normalization algorithm (which is the key element of the original proof scheme of the KAM theorem) to the construction of a suitable normal form related to an invariant elliptic torus. As a byproduct, our procedure can also provide some analytic expansions of the motions on elliptic tori. By extensively using algebraic manipulations on a computer, we explicitly apply our method to a planar four-body model not too different with respect to the real Sun--Jupiter--Saturn--Uranus system. The frequency analysis method allows us to check that our location of the initial conditions on an invariant elliptic torus is really accurate.Comment: 31 pages, 4 figure

Improved convergence estimates for the Schröder-Siegel problem

We reconsider the Schröder–Siegel problem of conjugating an analytic map in ℂ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension n>1 . Assuming a condition which is equivalent to Bruno’s one on the eigenvalues λ1,…,λn of the linear part, we show that the convergence radius ρ of the conjugating transformation satisfies lnρ(λ)≥−CΓ(λ)+C′ with Γ(λ) characterizing the eigenvalues λ , a constant C′ not depending on λ and C=1 . This improves the previous results for n>1 , where the known proofs give C=2 . We also recall that C=1 is known to be the optimal value for n=1

Il flebile sussurro del caos nell'armonia dei pianeti

Si ripercorre lo sviluppo del problema della stabilita del Sistema Solare a par- ` tire dall\u2019opera di Keplero. Vengono trattati gli argomenti seguenti: (i) la scoperta da parte dello stesso Keplero della cosiddetta \u2018\u2018grande ineguaglianza\u2019\u2019 di Giove e Saturno; (ii) lo sviluppo della teoria delle perturbazioni a opera di Lagrange e Laplace e il problema delle risonanze; (iii) la scoperta dei moti caotici da parte di Poincare; (iv) il teorema di Kol- \ub4 mogorov sulla persistenza di moti quasi periodici e la teoria di Nekhoroshev sulla stabilita` per tempi esponenzialmente lunghi. Nella parte finale si da un breve resoconto di alcuni ` lavori recenti sull\u2019applicabilita dei teoremi di Kolmogorov e Nekhoroshev a modelli reali- ` stici del Sistema Solare, mettendo in evidenza il loro ruolo nella discussione del problema della stabilita.The historical development of the problem of stability of the Solar System is revisited, starting from the work of Kepler. The following topics are included: (i) the discovery of the so called \u2018\u2018great inequality\u2019\u2019 of Jupiter and Saturn by Kepler himself; (ii) the dawn of perturbation theory in the work of Lagrange and Laplace and the problem of resonances; (iii) the discovery of chaotic motions in the work of Poincar\ub4e; (iv) the theorem of Kolmogorov on persistence of quasi periodic motions and the theory of Nekhoroshev on stability over exponentially long times. Finally, an account is given concerning some recent work on the actual applicability of the theorems of Kolmogorov and Nekhoroshev to realistic models of the Solar System, thus pointing out their relevance in discussing the problem of stability