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In the numerical study of classical dynamical systems presenting stochastic behaviour one frequently makes use, in an explicit or an implicit way, of the Birkhoff ergodic theorem. The correct interpretation of the obtained results presents some delicate problems related to the coexistence of many mutually singular invariant measures. In this paper we study this question in an experimental way on some simple model examples.

In the present paper we study by numerical methods a Hamiltonian model describing a chain of weakly coupled rotators. The existence of local phenomena of ordered and chaotic motions is illustrated and is interpreted, within classical perturbation theory, in terms of resonances.

We show how to construct a classical perturbation theory at any finite order for an Hamiltonian system describing a chain of weakly coupled rotators, both for the nonresonant and for the resonant cases. In particular, by means of a suitable algebraic scheme, we show how small denominantors and propagation of harmonics can be controlled.

In this paper a proof is given of Kolmogorov’s theorem on the existence of invariant tori in nearly integrable Hamiltonian systems. The scheme of proof is that of Kolmogorov, the only difference being in the way canonical transformations near the identity are defined. Precisely, use is made of the Lie method, which avoids any inversion and thus any use of the implicit-function theorem. This technical fact eliminates a spurious ingredient and simplifies the establishment of a central estimate.

For an autonomous nearly integrable Hamiltonian system ofndegrees of freedom withn> 1 it was shown by Poincaré that, in general, no integrals of motion exist which are independent of the Hamiltonian. This result was generalized by Fermi, who showed that in general not even single invariant (2n- 1)-dimensional manifolds exist, apart from constant-energy surfaces. On the other hand, the Kolmogorov-Amold-Moser theorem guarantees the existence ofn-dimensional invariant tori. In this paper we discuss the possible existence of invariant manifolds of intermediate dimensions and conclude that, apart from very well-defined exceptions (namely, manifolds of the so-called resonant type and (n+ 1)-dimensional families ofntori with mutually proportional frequencies), in general such invariant manifolds do not exist.

The stochastic properties of classical dynamical systems are often studied by means of numerical computations of orbits up to very large times, so that the accumulation of numerical errors would appear to destroy the reliability of the computations. We discuss this problem on the basis of a theorem of Anosov and Bowen which implies that, if the errors at each step are small enough, for Anosov systems the computations of time averages are reliable even for infinite times. We test numerically from this point of view three classical examples.