An example of an analytic system of differential equations in \(\mathbb{R}^{6}\) with an equilibrium formally stable and stable for most initial conditions is presented. By means of a divergent formal transformation this system is reduced to a Hamiltonian system with three degrees of freedom. Almost all its phase space is foliated by three-dimensional invariant tori carrying quasi-periodic trajectories. These tori do not fill all phase space. Though the “gap” between these tori has zero measure, this set is everywhere dense in \(\mathbb{R}^{6}\) and unbounded phase trajectories are dense in this gap. In particular, the formally stable equilibrium is Lyapunov unstable. This behavior of phase trajectories is quite consistent with the diffusion in nearly integrable systems. The proofs are based on the Poincaré – Dulac theorem, the theory of almost periodic functions, and on some facts from the theory of inhomogeneous Diophantine approximations. Some open problems related to the example are presented.

给出了\(\mathbb{R}^{6}\)中微分方程的解析系统的一个例子，其平衡形式上是稳定的并且对于大多数初始条件是稳定的。通过发散形式变换，该系统被简化为具有三个自由度的哈密顿系统。它的几乎所有相空间都由带有准周期轨迹的三维不变环面组成。这些圆环不会填满所有相空间。尽管这些 tori 之间的“间隙”测度为零，但这个集合在\(\mathbb{R}^{6}\)中处处密集和无界相轨迹在这个间隙中是密集的。特别地，形式上稳定的平衡是李雅普诺夫不稳定的。相轨迹的这种行为与几乎可积系统中的扩散非常一致。证明基于庞加莱-杜拉克定理，即几乎周期函数的理论，以及非齐次丢番图近似理论中的一些事实。提出了与该示例相关的一些未解决的问题。