Long-time stability of hierarchical point vortex configurations

Abstract

In this paper, we investigate the dynamics and long-time stability of hierarchical configurations in an $N$-point vortex system on $\mathbb{R}^2$ and in simply connected domains, without the Cantor-set restriction inherent in KAM theory. Since the Hamiltonian perturbation lacks a standard power-series expansion in the usual normalized variables, we develop a modified Birkhoff normal form based on successive ratios of adjacent variables, tailored to the hierarchical structure. A key feature is the preservation of the resulting natural exponent structure throughout the normal form procedure. Consequently, for $n$ sufficiently large, the hierarchical configuration persists for times of order at least $\varepsilon^{-\left(n-2(N-1)\right)}$.