The 1:+2 / 1:-2 resonance

TL;DR

This paper investigates the 1:2 and 1:-2 resonances in Hamiltonian systems, revealing critical points in the frequency ratio map and providing new insights into fractional monodromy near equilibria.

Contribution

It analyzes the non-degeneracy of the frequency map and the twist condition in 1:2 resonances, offering novel proofs and understanding of fractional monodromy.

Findings

The frequency ratio map has a critical point near the origin for the 1:-2 case.

The frequency map remains non-degenerate near the equilibrium.

Provides a new proof of fractional monodromy in the 1:-2 resonance.

Abstract

On the linear level elliptic equilibria of Hamiltonian systems are mere superpositions of harmonic oscillators. Non-linear terms can produce instability, if the ratio of frequencies is rational and the Hamiltonian is indefinite. In this paper we study the frequency ratio 1/2 and its unfolding. In particular we show that for the indefinite case (1:-2) the frequency ratio map in a neighbourhood of the origin has a critical point, i.e. the twist condition is violated for one torus on every energy surface near the energy of the equilibrium. In contrast, we show that the frequency map itself is non-degenerate (i.e. the Kolmogorov non-degeneracy condition holds) for every torus in a neighbourhood of the equilibrium point. As a byproduct we are able to obtain another proof of fractional monodromy in the 1:-2 resonance.