Nekhoroshev stability

Of course, even in presence of Nekhoroshev stability, evolution of the actions is possible on the very long exponential times typical of Arnold d diffusion (Arnold 1964), but this kind of instability is outside the subject of this article (see Lega, Guzzo and Froeschle 2003 for a numerical study). 

Guzzo, M. (1998) Nekhoroshev stability of quasi-integrable Hamiltonian systems with singularities and degeneracy. PhD thesis. 

Guzzo, M., Knezevic, Z. and Milani, A. (2002) Probing the Nekhoroshev stability of Asteroids. Celest. Mech. Dyn. Astr., Volume 83, Issues 1-4, 2002. 

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]. 

Bounded motions on KAM tori, or Nekhoroshev-type long-time stability could, however, hardly explain such variety of time scales, because the trajectory on a KAM torus is confined on /V-dimensional subspaces in... 

Abstract These lectures are devoted to the main results of classical perturbation theory. We start by recalling the methods of Hamiltonian dynamics, the problem of small divisors, the series of Lindstedt and the method of normal form. Then we discuss the theorem of Kolmogorov with an application to the Sun-Jupiter-Saturn problem in Celestial Mechanics. Finally we discuss the problem of long-time stability, by discussing the concept of exponential stability as introduced by Moser and Littlewood and fully exploited by Nekhoroshev. The phenomenon of superexponential stability is also recalled. 

Benettin, G., Galgani, L., Giorgilli, A. (1985). A proof of Nekhoroshev s theorem for the stability times in nearly integrable Hamiltonian systems. Cel. Mech., 37 1-25. Benettin, G., Gallavotti, G. (1986). Stability of motions near resonances in quasi-integrable Hamiltonian systems. Joum. Stat. Phys., 44 293-338. 

Nekhoroshev, N. N. (1977). Exponential estimates of the stability time of near-integrable Hamiltonian systems. Russ. Math. Surveys, 32 1-65. 

According to Nekhoroshev (1977) and to Morbidelli and Giorgilli (1995), the old and crucial question of stability of a dynamical system turns out to be related to the structure and density of invariant tori which foliate the phase space. For instance the puzzle of the 2/1 gap of the asteroidal belt distribution was explained showing that the corresponding region of the phase space is a weak chaotic one (Nesvorny and Ferraz-Mello 1997). 

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. 

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 ... 

We are beginning to understand chaotic structure in a way not seen before. Numerical methods of measuring chaotic and regular behaviour such as Fast Liapunov Indicators, sup-maps, twist-angles, Frequency Map Analysis, fourier spectal analysis are providing lucid maps of the global dynamical behaviour of multidimensional systems. Fourier spectral analysis of orbits looks to be a powerful tool for the study of Nekhoroshev type stability. Identification of the main resonances and measures of the diffusion of trajectories can be found easily and quickly. Applied to the full N-body problem without simplification, use of these tools is beginning to explain the observed behaviour of real physical systems. 

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