Dynamics of McMillan Mappings I. McMillan Multipoles | Sciact - CRIS-system of Boreskov Institute of Catalysis

Dynamics of McMillan Mappings I. McMillan Multipoles Full article

Journal

Physica D: Nonlinear Phenomena
ISSN: 0167-2789

Output data

Year: 2025, Number: 476, Article number : 134620, Pages count : DOI: 10.1016/j.physd.2025.134620

Authors

Zolkin T. ¹ , Nagaitsev S. ^2,3 , Morozov I. ^4,5

Affiliations

1	Fermi National Accelerator Laboratory, PO Box 500, Batavia, 60510-5011, IL, USA
2	Brookhaven National Laboratory, Upton, 11973, NY, USA
3	Old Dominion University, Norfolk, 23529, VA, USA
4	Synchrotron Radiation Facility “SKIF”, Koltsovo, 630559, Russia
5	Novosibirsk State Technical University, Novosibirsk, 630073, Russia

In this article, we consider two dynamical systems: the McMillan sextupole and octupole integrable mappings, originally proposed by Edwin McMillan. Both represent the simplest symmetric McMillan maps, characterized by a single intrinsic parameter. While these systems find numerous applications across various domains of mathematics and physics, some of their dynamical properties remain unexplored. We aim to bridge this gap by providing a comprehensive description of all stable trajectories, including the parametrization of invariant curves, Poincaré rotation numbers, and canonical action–angle variables. In the second part, we establish connections between these maps and general chaotic maps in standard form. Our investigation reveals that the McMillan sextupole and octupole serve as first-order approximations of the dynamics around the fixed point, akin to the linear map and quadratic invariant (known as the Courant–Snyder invariant in accelerator physics), which represents zeroth-order approximations (referred to as linearization). Furthermore, we propose a novel formalism for nonlinear Twiss parameters, which accounts for the dependence of rotation number on amplitude. This stands in contrast to conventional betatron phase advance used in accelerator physics, which remains independent of amplitude. Notably, in the context of accelerator physics, this new formalism demonstrates its capability in predicting dynamical aperture around low-order resonances for flat beams, a critical aspect in beam injection/extraction scenarios.

Zolkin T. , Nagaitsev S. , Morozov I.
Dynamics of McMillan Mappings I. McMillan Multipoles
Physica D: Nonlinear Phenomena. 2025. N476. 134620 . DOI: 10.1016/j.physd.2025.134620 WOS Scopus OpenAlex publication_identifier_short.sciact_skif_identifier_type

Submitted:	Dec 12, 2024
Accepted:	Mar 4, 2025
Published online:	Mar 25, 2025
Published print:	Jun 1, 2025

Web of science:	WOS:001460999900001
Scopus:	2-s2.0-105001134351
OpenAlex:	W4396822254
publication_identifier.sciact_skif_identifier_type:	3820

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