Nekhoroshev theorem

In addition to computer simulations, what drives the research in this direction is elaborated perturbation theories developed almost simultaneously. In particular, the Kolmogorov-Arnold-Moser (KAM) theorem, which has shown the existence of invariant tori under a small perturbation to completely inte-grable systems, and the Nekhoroshev theorem, which has proved exponentially long-time stability of trajectories close to completely integrable ones, are landmarks in this field. Although a lot of works have been done, there still remain unsolved important questions, and the Hamiltonian system is being studied as one of important branches in the theory of dynamical systems [3-5]. 

One explanation for anomalous diffusion in Hamiltonian dynamics is the presence of self-similar invariant sets or hierarchical structures formed in phase space that play the role of partial barriers. They slow down the normal diffusion. A different explanation for intermittent behavior is given by the existence of deformed and approximate adiabatic invariants in phase space. They are shown in terms of elaborated perturbation theories such as the KAM and Nekhoroshev theorems. 

M. Guzzo and G. Benettin, A Spectral Formulation of the Nekhoroshev Theorem and Its Relevance for Numerical and Experimental Analysis, preprint. 

We can expect that the stagnant motions of which statistical properties are discussed in the previous section reflect a universal structure of the interface between torus and chaos. Here we discuss a conjecture concerning the universality in the stagnant layer based on the Nekhoroshev theorem, which proves the onset of a new time scale accompanied by the collapse of tori [9]. 

A theorem demonstrated by Morbidelli and Giorgilli (1995 a,b), using both the results of KAM and Nekhoroshev theorem, gives to KAM tori... 

As we said in the introduction, the crucial question of stability of a dynamical system is related to the structure and density of invariant tori which foliate the phase space. This is in fact the geometrical representation of the Nekhoroshev theorem (1977). We recall that in a quasi-integrable system with Hamiltonian ... 

We have observed the phenomenon down to e = 0.001. According to Nekhoroshev theorem, the actions are stable for a time which is exponentially long with the inverse of the perturbation parameter. For such a low value of e, we have detected a very slow diffusion after / = 2 109, while for = 0.002 we had to wait only for a time of t = 2 108 we are evidently in the presence of an exponential law. 

Abstract We review results about the Fourier Analysis of chaotic solutions of quasi-integrable systems based on the Nekhoroshev theorem. We describe also an application to Asteroids stability. 

Keywords Fourier Analysis, Nekhoroshev theorem, Asteroids. 

In this article we discuss the problem of understanding the long-term stability properties of a solution of a quasi-integrable Hamiltonian system by means of a Fourier analysis on a short observation time. Precisely, even for resonant chaotic motions, we will show how the combined use of Fourier analysis and Nekhoroshev theorem allows to understand the stability properties on a time T exp(T), where T is a suitable observation time, of the order of the resonant period. To be definite, we will refer to quasi-integrable Hamiltonian systems with Hamiltonian of the form ... 

In quasi-integrable systems we do not find only KAM tori, but also resonant motions, and among resonant motions we find the chaotic ones. If e is small and h satisfies a suitable geometric condition (convexity of h is sufficient) the Nekhoroshev theorem proves the exponential stability of the actions for all initial conditions, including the resonant ones. More precisely, there exist positive constants eo, a,b,Io,to such that if < so, for any (I(0),ip(0)) e B x T it is I(t) — /(0) < Io a for any time t satisfying the exponential estimate ... 

The fact that the spectrum of an observable of a system computed on a chaotic solution has not the peak structure can be considered a numerical evidence that the system does not satisfy the hypotheses of Nekhoroshev theorem in the neighborhood of that solution, i.e. the Nekhoroshev theorem does not prevent fast evolution of the actions . 

All these examples concerned the non-degenerate case. In the next section we describe the spectral formulation of Nekhoroshev theorem also for degenerate cases, providing a method which can be directly applied to the real systems of Celestial Mechanics. 

A straightforward application of the usual non-degenerate version of the Nekhoroshev theorem to Hamiltonian (19) allows to prove that if h is quasi-convex (with respect to the I) and e is suitably small then the non-degenerate actions I are exponentially stable, if in the meantime... 

Guzzo, M. and Benettin, G. (2001) A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis. Discrete and Continuous Dynamical Systems-Series B, Volume 1, n. 1. 

Morbidelli, A. and Guzzo, M. (1997) The Nekhoroshev theorem and the asteroid belt dynamical system. Celest. Mech. Dyn. Astron., vol. 65, 107-136. 

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