The celebrated KAM and Nekhoroshev theorems provide essential informations about the long term dynamics of quasi–integrable Hamiltonian systems. In particular, long–term instability of the action variables can be observed only in the so–called Arnold web, which is the complement in the phase–space of all KAM invariant tori, and only on the very long times which depend exponentially on an inverse power of the perturbation parameter. Though the structure of the Arnold's web was clearly explained already on Arnold's 1963 article, its numerical detection with a precision sufficient to reveal exponentially slow diffusion of the actions through the web itself has become possible only in the last decade with the extensive computation of dynamical indicators. In this paper, we first review the detection method that allowed us to compute the Arnold web, and then we discuss its use to study the long–term diffusion through the web itself. We also show that the Arnold web of a quasi–integrable Hamiltonian system is useful to track the diffusion of orbits of weakly dissipative perturbations of the same Hamiltonian system.
The dynamics of quasi–integrable Hamiltonian systems is widely studied in the literature. On the one hand, many interesting problems of Physics are in the form of small Hamiltonian perturbations of integrable systems, on the other hand most questions concerning the long-term diffusion are open. Moreover, the application of perturbation theories to the systems of interest for Physics, for example, the systems representing the motions of the asteroids and planets of our Solar System, is complicated by the fact that the small parameters are not extremely small, and their Hamilton functions may be highly degenerate, so that a direct application of the KAM and Nekhoroshev theorems in their original formulation is not possible. One of the most studied problems concerns the orbits which are unstable on suitably long times. The precise definition of unstable orbits depends on the specific system, and is related to a significant change of the action variables. The existence of unstable orbits is a non-trivial problem only depending on the non-degeneracy properties of the integrable approximation (it is quite easy to provide analytic examples of unstable motions even for systems which satisfy the hypotheses of KAM theorem, but not those of Nekhoroshev theorem, see Ref. 40, Sec. 1.7). A paper of fundamental importance was published by Arnold in 1964.2 The paper first provided a quasi–integrable model system with non trivial long-term instability, afterwards called Arnold diffusion, occurring at any small value of the perturbation parameter in the so-called Arnold web, which is the complement in the phase-space of all KAM invariant tori. The use of the term “diffusion” may be here confusing. We remark that all the definitions of Arnold diffusion which have been provided up to now in the literature refer to a phenomenon of topological instability of the action variables. Therefore, what justifies the universal use of the term “diffusion”? The reason can be found in Ref. 2, where the mechanism proposed for proving such instability was based on heteroclinic intersections of a chain of hyperbolic tori. Heteroclinic intersections are well known to be related to some random description of deterministic dynamics. Nevertheless, the idea of random motions behind such heteroclinic chaos is still of topological nature, and up to now, there do not exist in the literature any mathematically rigorous characterizations of the statistical properties of Arnold diffusion. Because of the extremely long times characterizing the Arnold diffusion, Arnold's paper soon motivated a debate about the possibility of numerical detection of Arnold diffusion: few years after the first numerical detections of chaotic motions,30 the problem of the numerical detection of Arnold diffusion was discussed in Ref. 12, and in the following decades, there appeared many studies of numerical diffusion through resonances, such as Refs. 17, 31, 35, 37, and 50, up to our papers.19,25,29,36 Most of these papers are concerned with the detection of the resonances of the system using extensive computations of dynamical indicators on grids of initial conditions of the phase–space. We remark that, specifically in Celestial Mechanics, the numerical detection of the resonances of a system using dynamical indicators is one of the major tools for studying its long–term instability (for recent examples, see Refs. 22, 23, 38, 43–45, and 49). The quasi–integrable structure of many systems gives the resonances the structure of a web, the so called Arnold web. In our papers,19,25,29,36 we provided numerical evidence about the existence of sets of orbits diffusing through the Arnold web with remarkable topological and statistical properties. In particular, by choosing the initial conditions in the single resonances (far from the main multiple resonances), the average over all the orbits of the square of the distance of the action variables from their initial condition (see Sec. III B for precise discussion) is well fitted by a linear time dependence, as in the genuine diffusion processes, in the following two cases: (i) the time interval is suitably long to allow us to measure a slow drift of the orbits along the resonance, but in the same time the orbits do not cross other main multiple resonances (see Ref. 36); (ii) in much longer time intervals the orbits visit several different single and multiple resonances (see Ref. 25). We referred to case (i) as to local diffusion, and to case (ii) as to global diffusion. Therefore, in a regime of moderately small perturbation parameters, it was possible to relate the actions' instability to diffusion coefficients, and we found that the diffusion coefficient scales with the perturbation parameter faster than power laws. In our opinion, the motivation of numerical studies of Arnold diffusion goes farther than providing support or evidence to analytical studies, which nowadays are still far to provide the complete description of Arnold diffusion. An important added value of numerical studies, which cannot handle arbitrarily small perturbations, is to show that the process can be detected even for perturbations which are small, but not extremely small, and on times which are long, but not extremely long, making it relevant also for Physics.
Part of the computations have been done on the “Mesocentre SIGAMM” machine, hosted by the Observatoire de la Côte d'Azur.
II. NUMERICAL DETECTION OF THE ARNOLD WEB WITH DYNAMICAL INDICATORS
A. The FLI for different orbits
B. Sensitivity of the FLI
III. DIFFUSION THROUGH THE WEB OF RESONANCES
A. Topological properties of the diffusion
B. Statistical properties of the diffusion
C. Computation of diffusion coefficients
IV. WEAKLY DISSIPATIVE PERTURBATIONS OF QUASI–INTEGRABLE SYSTEMS
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