I will talk about the establishment of Exponentially-long time stability for multi-scale, nearly integrable Hamiltonian systems.

We focus on two specific scenarios: (I) the integrable component at each scale satisfies (α, K) complete non-resonance; and (II) resonance occurs exclusively in the highest scale of the integrable part. I will show the Nekhoroshev estimates under these two cases and I will also mention the characteristics  and the differences between multi-scale systems and classical systems.This topic is inspired by celestial mechanics and N-vortex systems. As a consequence, I will illustrate our results through explaining the stability of the singularity point of the (1+1) restricted vortex systems.