First numerical investigation of a conjecture by N. N. Nekhoroshev about stability in quasi-integrable systems

The characterization of the long-term stability properties of Hamiltonian systems has a big relevance for scientific problems of Celestial Mechanics and Statistical Physics, and also for recent studies of stability in particle accelerators. The celebrated Kolmogorov-Arnold-Moser (KAM) and Nekhoroshev theorems provide fundamental informations: while the KAM theorem establishes the stability of the actions of the systems for infinite times in large measure sets of non resonant initial conditions, the Nekhoroshev theorem establishes the stability in open sets of initial conditions, including the resonances, for times which increase exponentially with respect to an inverse power of the perturbation parameter. The Nekhoroshev theorem applies to analytic perturbations of integrable systems whose Hamiltonian satisfies a non-degeneracy property, called steepness. The simplest examples of steep functions are the convex ones, but many applications to Physics are modeled by perturbation of steep non convex systems. In Nekhoroshev’s 1977 paper, it is conjectured that, among the steep systems with the same number ν of frequencies, the convex ones are the most stable, and it is suggested to investigate numerically the problem. Following this suggestion, we numerically study and compare the diffusion of the actions in quasi-integrable systems, precisely quasi-integrable symplectic maps, with different steepness properties in a large range of variation of the perturbation parameter ɛ and different dimensions of phase space corresponding to ν = 3 and ν = 4 (ν ≤ 2 is not significant for the conjecture). For six dimensional maps (ν = 4), our numerical experiments perfectly agree with the Nekhoroshev conjecture: for both convex and non convex cases, the numerically computed diffusion coefficient D of the actions is compatible with an exponential fit D(ɛ) ∼ exp − (ɛ0/ɛ)β, and the convex case is definitely more stable than the steep one. For four dimensional maps (ν = 3): in the convex case, we provide for the first time a fit of β; in the steep case, D(ɛ) has large oscillations around an exponential behaviour. In this case, the agreement of our numerical experiments with the conjecture is not sharp and requires to consider a sup over different initial conditions.