Dynamics of McMillan mappings III. Symmetric map with mixed nonlinearity

TL;DR

This paper analyzes the symmetric McMillan map's dynamics, revealing key parameters, classifying motion regimes, providing exact solutions, and demonstrating its accuracy in modeling complex nonlinear systems.

Contribution

It identifies only two essential parameters for the symmetric McMillan map and connects it with standard mappings, enhancing understanding of nonlinear dynamics.

Findings

Two irreducible parameters govern the map's dynamics.

Exact solutions and action-angle variables are derived.

The map accurately models a wide range of nonlinear systems.

Abstract

This article extends the study of the dynamical properties of the symmetric McMillan map, emphasizing its utility in understanding and modeling complex nonlinear systems. Although the map features six parameters, we demonstrate that only two are irreducible: the linearized rotation number at the fixed point and a nonlinear parameter representing the ratio of terms in the biquadratic invariant. Through a detailed analysis, we classify regimes of stable motion, provide exact solutions to the mapping equations, and derive a canonical set of action-angle variables, offering analytical expressions for the rotation number and nonlinear tune shift. We further establish connections between general standard-form mappings and the symmetric McMillan map, using the area-preserving H\'enon map and accelerator lattices with thin sextupole magnet as representative case studies. Our results show that, despite being a second-order approximation, the symmetric McMillan map provides a highly accurate depiction of dynamics across a wide range of system parameters, demonstrating its practical relevance in both theoretical and applied contexts.