Dynamics near the p : -q Resonance

TL;DR

This paper analyzes the complex dynamics near the p:-q resonant Hamiltonian equilibrium, revealing fractional monodromy and twist vanishing, with explicit computations of key dynamical invariants using hyperelliptic integrals.

Contribution

It provides explicit formulas for the reduced period, rotation number, and action in the p:-q resonance, and demonstrates fractional monodromy and twist vanishing near the origin.

Findings

Fractional monodromy is present in the p:-q resonance.

Explicit hyperelliptic integral formulas for dynamical invariants.

Twist vanishes near the origin of the 1:-q resonance.

Abstract

We study the dynamics near the truncated p : +/- q resonant Hamiltonian equilibrium for p, q coprime. The critical values of the momentum map of the Liouville integrable system are found. The three basic objects reduced period, rotation number, and non-trivial action for the leading order dynamics are computed in terms of complete hyperelliptic integrals. A relation between the three functions that can be interpreted as a decomposition of the rotation number into geometric and dynamic phase is found. Using this relation we show that the p : -q resonance has fractional monodromy. Finally we prove that near the origin of the 1 : -q resonance the twist vanishes.