Asymptotic stabilization with phase of periodic orbits of three-dimensional Hamiltonian systems

TL;DR

This paper introduces a geometric method for asymptotic phase stabilization of any fixed periodic orbit in three-dimensional Hamiltonian systems, requiring only Hamiltonian and Casimir values, demonstrated on the Rikitake geomagnetic model.

Contribution

A novel geometric approach to stabilize periodic orbits in 3D Hamiltonian systems without needing orbit parameterization, using only Hamiltonian and Casimir values.

Findings

Successfully stabilized periodic orbit in Rikitake model

Method does not require orbit parameterization

Applicable to general 3D Hamiltonian systems

Abstract

We provide a geometric method to stabilize asymptotically with phase an arbitrary fixed periodic orbit of a locally generic three-dimensional Hamiltonian dynamical system. The main advantage of this method is that one needs not know a parameterization of the orbit to be stabilized, but only the values of the Hamiltonian and a fixed Casimir (of the Poisson configuration manifold) at that orbit. The stabilization procedure is illustrated in the case of the Rikitake model of geomagnetic reversal.