Stability of the phase motion in race-track microtons

TL;DR

This paper models phase oscillations in race-track microtrons using an area-preserving map, analyzing the nonlinear stability of the synchronous trajectory and how the stability domain evolves with the synchronous phase.

Contribution

It provides a detailed nonlinear stability analysis of the synchronous trajectory in race-track microtrons, including estimates of the stability domain and its evolution.

Findings

The stability domain is enclosed by the last rotational invariant curve.

The evolution of the stability domain depends on the synchronous phase.

Invariant curves are approximated by Hamiltonian level sets.

Abstract

We model the phase oscillations of electrons in race-track microtrons by means of an area preserving map with a fixed point at the origin, which represents the synchronous trajectory of a reference particle in the beam. We study the nonlinear stability of the origin in terms of the synchronous phase -- the phase of the synchronous particle at the injection. We estimate the size and shape of the stability domain around the origin, whose main connected component is enclosed by the last rotational invariant curve. We describe the evolution of the stability domain as the synchronous phase varies. Besides, we approximate some rotational invariant curves by level sets of certain Hamiltonians. Finally, we clarify the role of the stable and unstable invariant curves of some hyperbolic (fixed or periodic) points.