# The 1:1 resonance in Hamiltonian systems

## TL;DR

This paper investigates the 1:1 resonance bifurcation in two-degree-of-freedom Hamiltonian systems, revealing a co-dimension five unfolding with two moduli parameters, advancing understanding of semisimple resonance cases.

## Contribution

It provides a detailed analysis of the 1:1 resonance in Hamiltonian systems, identifying the co-dimension and unfolding parameters, which was previously not well understood.

## Key findings

- Identified the co-dimension of 1:1 resonance as five.
- Discovered two moduli parameters in the unfolding.
- Connected the normal form symmetry to the frequency ratio.

## Abstract

Two-degree-of-freedom Hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearisation. Considering the frequencies as parameters, the system undergoes a bifurcation when the frequencies pass through a resonance. These bifurcations are well understood for most resonances k:l, but not the semisimple cases 1:1 and 1:-1. A two-degree-of-freedom Hamiltonian system can be approximated to any order by an integrable approximation. The reason is that the normal form of a Hamiltonian system has an additional integral due to the normal form symmetry. The latter is intimately related to the ratio of the frequencies. Thus we study $S^1$--symmetric systems. The question we wish to address is about the co-dimension of such a system in 1:1 resonance with respect to left-right-equivalence, where the right action is $S^1$--equivariant. The result is a co-dimension five unfolding of the central singularity. Two of the unfolding parameters are moduli and the remaining non-modal parameters are the ones found in the linear unfolding of this system.