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

TL;DR

This work extends the square-matrix convergence map (CM) method from 4-D to the full $6$-D phase space to capture synchro-betatron coupling driven by time-dependent perturbations such as crab cavities. It combines an eigen-decomposition of the one-turn map with a near-rigid-rotation transformation to construct approximated action-angle variables, using the convergence residual as a stability indicator. The approach is validated on a 6-D toy map, where CM detects known and additional high-order resonances beyond what frequency map analysis (FMA) reveals, and is then applied to the Electron-Ion Collider Hadron Storage Ring to evaluate dynamic aperture under crab-cavity nonlinearities, showing good agreement with FMA and revealing resonance structures across multiple phase-space slices. A key finding is that increasing crab-cavity sextupole components can significantly shrink the DA, underscoring the method’s utility for fast, accurate nonlinear beam-dynamics analysis and accelerator design decisions that require DA evaluation without long multi-turn tracking. The proposed 6-D CM offers a computationally efficient alternative to full particle tracking, with potential applications beyond accelerator physics to other nonlinear dynamical systems.

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.

Figures (6)

• Figure 1: CM and FMA results for the toy system in the $x$-$z$ plane. The left plot shows the minimum difference between two successive iterations of the CM method. A large value of the convergence metric indicates divergence of the convergence map. The color scale in the right plot represents tune diffusion of the trajectories.
• Figure 2: Results of CM (left) and FMA (right) in the frequency space. The black circles in the left plot correspond to the black circles in spatial space shown in Fig. 1(a). The x-axis represents $2\nu_x+2\nu_y$ and the y-axis represents $\nu_z$.
• Figure 3: CM analysis results for the EIC HSR in the $x$-$z$ (left) and $y$-$z$ (right) plane. The color scale from blue to red represents the convergence error to a rigid rotation increasing from small to large.
• Figure 4: Results of applying CM (left) and FMA (right) to the EIC HSR in the $x$-$y$ plane. The initial $z$ for all particles are fixed at 6 cm. The black points represent some selected trajectories crossing resonance lines.
• Figure 5: FMA results in frequency space. The left plot shows the result in $\nu_x$ and $\nu_y$ space, the middle one shows the $\nu_x$ and $\nu_y+\nu_z$ space, and the right one shows the $\nu_x$ and $2\nu_y+\nu_z$ space. The black cross in the plots correspond to the selected points in Fig. 4.