# Unfolding of nilpotent equilibria of degree 4 in Hamiltonian systems with 2 degrees of freedom

## TL;DR

This paper analyzes the unfolding of nilpotent equilibria of degree 4 in two-degree-of-freedom Hamiltonian systems, identifying key bifurcations and providing a universal unfolding under certain conditions.

## Contribution

It presents the first universal unfolding for such nilpotent equilibria, detailing the bifurcation types that occur in the process.

## Key findings

- Identification of the universal unfolding under non-degeneracy conditions
- Characterization of bifurcations as hyperbolic, elliptic, or Hamiltonian-Hopf types
- Clarification of the bifurcation structure in these Hamiltonian systems

## Abstract

We consider Hamiltonian systems of two degrees of freedome having a nilpotent equilibrium point with only one eigenvector. We provide the universal unfolding of such equilibrium, provided a non-degeneracy condition holds. We show that the only co-dimension 1 bifurcations that happen in the unfolding are of two types: the normally hyperbolic or elliptic centre-saddle bifurcations and the supercritical Hamiltonian-Hopf bifurcation.