Quantitative predictions with detuned normal forms

TL;DR

This paper develops a resonant detuned normal form to accurately predict the phase-space structure of galactic potentials, providing reliable analytical bifurcation predictions even beyond the series convergence radius.

Contribution

It introduces a novel application of resonant detuned normal forms with Lie transforms for galactic dynamics, enhancing predictive accuracy of bifurcations in Hamiltonian systems.

Findings

Analytical bifurcation expressions match numerical results.

Normal form predictions remain reliable outside convergence radius.

Resummation techniques improve series convergence analysis.

Abstract

The phase-space structure of two families of galactic potentials is approximated with a resonant detuned normal form. The normal form series is obtained by a Lie transform of the series expansion around the minimum of the original Hamiltonian. Attention is focused on the quantitative predictive ability of the normal form. We find analytical expressions for bifurcations of periodic orbits and compare them with other analytical approaches and with numerical results. The predictions are quite reliable even outside the convergence radius of the perturbation and we analyze this result using resummation techniques of asymptotic series.