Square matrix-based six-dimensional convergence map for nonlinear beam dynamics analysis

TL;DR

This paper extends the square matrix-based convergence map method to 6-D phase space for nonlinear beam dynamics, enabling efficient analysis of complex coupling and resonance structures in colliders.

Contribution

The paper introduces a 6-D CM method using eigen-decomposition and iterative procedures, improving nonlinear analysis in systems with time-dependent perturbations.

Findings

Successfully resolves high-order resonance structures.

Maintains computational efficiency with one-turn map.

Shows close agreement with frequency map analysis in EIC study.

Abstract

The square matrix-based convergence map (CM) method has proven effective in characterizing nonlinear dynamics in several 4-D dynamical systems. However, when time-dependent perturbations, such as crabbing kicks in colliders, are present, a comprehensive 6-D analysis becomes essential to accurately capture the coupling between transverse and longitudinal motions. In this work, we extend the CM method to the full 6-D phase space by employing an eigen-decomposition-based formulation of the square matrix combined with iterative procedures. The proposed 6-D CM approach is first validated using a simplified crabbing map. We demonstrate that the 6-D CM preserves computational efficiency by using only one-turn map, while successfully resolving high-order resonance structures that remain unresolved by conventional frequency map analysis (FMA). This method is subsequently applied to the dynamic aperture (DA) study of the future Electron-Ion Collider (EIC). The results obtained from the CM analysis exhibit close agreement with those derived from FMA, demonstrating its potential as a powerful tool for nonlinear beam dynamics analysis and DA evaluation, as well as for broader applications in other nonlinear dynamical systems.