Dipartimento di Matematica Pura ed Applicata, Università degli Studi di Padova, Via Trieste 63-35121 Padova, Italy.
Chaos. 2011 Sep;21(3):033101. doi: 10.1063/1.3603819.
We investigate numerically a conjecture by N. N. Nekhoroshev about the influence of a geometric property, called steepness, on the long term stability of quasi-integrable systems. In a 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 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, and the convex case is definitely more stable than the steep one. For four dimensional maps (ν = 3), since we find that in the steep case D(ɛ) has large oscillations around an exponential behaviour, the agreement of our numerical experiments with the conjecture is not sharp, and it is found by considering a sup over different initial conditions.
我们通过数值方法研究了 N. N. Nekhoroshev 关于一个几何性质（称为陡峭度）对拟可积系统长期稳定性影响的猜想。在 Nekhoroshev 1977 年的论文中，他猜测在具有相同频率数 ν 的陡峭系统中，凸系统是最稳定的，并建议对此问题进行数值研究。根据这一建议，我们在很大的微扰参数 ɛ 变化范围内和对应于 ν = 3 和 ν = 4 的相空间不同维度（ν ≤ 2 对猜想不重要）下，对具有不同陡峭度性质的拟可积系统中的作用扩散进行了数值研究和比较。对于六维映射（ν = 4），我们的数值实验完全符合 Nekhoroshev 猜想：对于凸和非凸两种情况，作用的数值计算扩散系数 D 与指数拟合兼容，并且凸情况明显比陡峭情况更稳定。对于四维映射（ν = 3），由于我们发现在陡峭情况下 D(ɛ) 在指数行为周围有很大的波动，因此我们的数值实验与猜想的一致性不是很精确，需要通过考虑不同初始条件的上确界来发现。
