McMillan map and nonlinear Twiss parameters

TL;DR

This paper provides a comprehensive analysis of McMillan sextupole and octupole maps, describing their stable trajectories, invariant curves, and introducing nonlinear Twiss parameters to better understand amplitude-dependent dynamics in accelerator physics.

Contribution

It offers the first complete description of stable trajectories and invariant structures of McMillan maps and introduces nonlinear Twiss parameters for amplitude-dependent phase advance analysis.

Findings

Complete characterization of stable trajectories and invariant curves.

Relation of McMillan maps to general chaotic maps in McMillan-Turaev form.

Introduction of nonlinear Twiss parameters for amplitude-dependent rotation numbers.

Abstract

In this article we consider two dynamical systems: the McMillan sextupole and octupole integrable mappings originally introduced by Edwin McMillan; the second one is also known as canonical McMillan map. Both of them are simplest symmetric McMillan maps with only one intrinsic parameter, the trace of the Jacobian at the fixed point. While these dynamical systems have numerous of applications and are used in many areas of math and physics, some of their dynamical properties have not been described yet. We fulfill the gap and provide complete description of all stable trajectories including parametrization of invariant curves, Pioncar\'e rotation numbers and canonical action-angle variables. In the second part we relate these maps with general chaotic map in McMillan-Turaev form. We show that McMillan sextupole and octupole mappings are first order approximations of dynamics around the fixed point, in a similar way as linear map and quadratic invariant (Courant-Snyder invariant in accelerator physics) is the zeroth order approximation (known as linearization). Finally we suggest the new formalism of nonlinear Twiss parameters which incorporate dependence of rotation number as a function of amplitude, in contrast to e.g. betatron phase advance used in accelerator physics which is independent of amplitude. Specifically in application to accelerator physics this new formalism is capable of predicting dynamical aperture around 1-st, 2-nd, 3-rd and 4-th order resonances for flat beams, which is critical for beam injection/extraction.