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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
On the detection of slow diffusion along resonances in
Hamiltonian systems
E. Lega 1 , C. Froeschl'e 1 , M. Guzzo 2
1 Observatoire de Nice, Bv. de l'Observatoire, B.P. 4229, 06304 Nice cedex 4,
France
2 Dipartimento di Matematica Pura ed Applicata, Universit`a degli Studi di
Padova, via Belzoni 7, 35131 Padova, Italy
It is now well known that deterministic systems can give rise to so--called
chaotic motion [13]. Sometimes, there has been the tendency to associate chaotic
motion with unstable motion. Yet many examples have been provided in the
literature of chaotic motions which seem to remain stable up to very long times
(see Benettin et al., 1985) [11].
Such behavior, detected in different fields of physics (beam--beam interaction,
asteroidal motion), is now known to be typical of a certain class of dynamical
systems.
Indeed, the representation of resonant motions given in (Benettin and
Gallavotti, 1986) in the framework of the stability result of Nekhoroshev
(Nekhoroshev, 1977) [20] shows that in quasi--integrable Hamiltonian systems
there typically exist resonant chaotic motions whose actions are bounded up to
an exponentially long time. Morbidelli and Froeschl'e (Morbidelli and Froeschl'e,
1996) [18] have shown, using a quite simple model, that the actions can remain
confined up to very long times despite the fact that the largest Lyapunov char­
acteristic exponent associated to the motion is quite large.
The existence of diffusive chaotic orbits has been heuristically shown in
(Chirikov, 1979) [5] as due to the overlapping of resonances. Let us remark that
such diffusion can be quite slow when the harmonics of the overlapping resonances
are small (see for example Morbidelli and Guzzo, 1997) [19]. Therefore, such a
slow diffusion is not easily recognized using even very long numerical integrations
looking at the variations of the actions. It is very difficult to distinguish between
the two different regimes with purely analytic tools although many improvements
in this direction have been recently obtained (Celletti and Chierchia, 1995, [2];
Celletti and Chierchia, 1997, [3]; Celletti et al. 2000, [4]; Locatelli and Giorgilli,
2000, [17]). Therefore, numerical tools have been developed in the last ten years
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(Laskar, 1990, Laskar et al., 1992, Lega and Froeschl'e, 1996, Contopoulos and
Voglis, 1997, Froeschl'e and Lega, 1998) [14], [15], [16], [6], [9] to investigate the
problem in an indirect way, i.e. by looking for the mathematical consequences of
the two regimes, which are different from the stability of the actions.
In a previous work we have used two tools recently introduced to investigate
the transition from the Nekhoroshev to the Chirikov regime, both in Hamiltonian
systems and symplectic maps. The first tool, called the Fast Lyapunov Indicator
(hereafter called FLI, Froeschl'e et al., 1997; Lega and Froeschl'e, 1997) [10], [16] is
related to the computation of the tangent map for a suitable choice of an initial
tangent vector, and allows us to distinguish rapidly not only slow chaos from
ordered motion (Froeschl'e et al., 1997) but also discriminates between regular
resonant motions and tori (Froeschl'e et al., 2000; Froeschl'e and Lega, 2000; Lega
and Froeschl'e, 2001) [1], [8], [7]. Indeed, such numerical studies show that the
computation of the FLI on a grid of regularly spaced initial conditions permits
detection of the geometry of resonances with quite short numerical integrations.
Let us remark that the definition of the FLI is very close to that of the Finite
Time Lyapunov Exponent (see for example Tang and Boozer, 1996). These two
indicators, defined independently, differ mainly in the dependence on the choice
of the initial tangent vector.
The second method, introduced in Guzzo and Benettin, 2001 [12], and called
``analytically filtered Fourier analysis'' (hereafter AFFA), is related to the rep­
resentation of the spectrum of a generic observable for systems which are in
the Nekhoroshev regime. It provides global information on the long--term sta­
bility properties of the system through computation of a few well chosen orbits.
Of course this requires less CPU time than grid--based calculations. Using both
methods we have measured an interval of transition, centerd on a given value of
the perturbation parameter ffl \Lambda , from the Nekhoroshev to the Chirikov regime for a
three degrees of freedom Hamiltonian system. The relationship between Nekhoro­
shev stability and diffusion as still to be explored numerically and this is the aim
of the present work. We know from the Nekhoroshev theorem that the effective
stability time is exponentially long with respect to the ratio ffl \Lambda
=ffl. This means
that diffusion can in principle be detected if the system is close to the transition
to the Chirikov regime, i.e. if ffl ' ffl \Lambda . Using the FLI charts we have been able to
select a set of resonant chaotic initial conditions for some values of ffl lower than ffl \Lambda
and to follow the motion of the corresponding orbits. We have observed diffusion
along resonant lines, for decreasing value of ffl up to ffl ' ffl \Lambda
=10. The measure of the
diffusion coefficient as a function of the perturbation parameter seems to follow
the expected exponential law.
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