Fast chaos indicator from auto-differentiation for dynamic aperture optimization

TL;DR

The paper addresses the challenge of efficiently identifying chaotic regions that bound dynamic aperture in circular accelerators. It introduces a fast chaos indicator based on the norm of the tangent map $||M(s)||$, computed from differentiable tracking with forward-mode automatic differentiation (TPSA), and notes that chaotic growth follows $||\Delta\zeta(s)|| \sim ||\Delta\zeta(0)|| e^{\lambda s}$. The approach is demonstrated on a 4D Hénon–Heiles system and applied to ALS-U lattice optimization, where the one-turn log Frobenius norm guides the objective and yields a larger dynamic aperture (validated by long-term tracking). The contribution offers a practical, computation-efficient tool for nonlinear beam dynamics and lattice design.

Abstract

Automatic differentiation provides an efficient means of computing derivatives of complex functions with machine precision, thereby enabling differentiable simulation. In this work, we propose the use of the norm of the tangent map, obtained from differentiable tracking of particle trajectories, as a computationally efficient indicator of chaotic behavior in phase space. In many cases, a one-turn or few-turn tangent map is sufficient for this purpose, significantly reducing the computational cost associated with dynamic aperture optimization. As an illustrative application, the proposed indicator is employed in the dynamic aperture optimization of an ALS-U lattice design.

Figures (6)

• Figure 1: Dynamic aperture in the $y$--$p_y$ plane obtained using FMA (top left) and REM (top right) indicators hwang, and using the $p=1$ (bottom left), $p=\infty$ (bottom middle), and Frobenius (bottom right) norms of the tangent map computed via differentiable tracking.
• Figure 2: Evolution of the horizontal beta function (left), vertical beta function (middle), and dispersion (right) over one of the $12$ cells in the ALS-U ring, as computed with JuTrack and Elegant. The positions of the two quadrupoles---QF and QD---used to control the machine working tunes are indicated.
• Figure 3: Left: boundary of surviving particles in the $x$--$y$ plane after 1, 10, 100, and 1000 turns of tracking. Right: spatial distribution of the log Frobenius norm of the tangent map after one-turn differentiable tracking with zero momentum deviation in the nominal ALS-U lattice.
• Figure 4: Spatial distribution of the log of the maximum eigenvalue of the one-turn tangent map from differentiable tracking with zero momentum deviation in the nominal ALS-U lattice.
• Figure 5: Left: spatial distribution of the log Frobenius norm of the tangent map after one-turn differentiable tracking. Right: log tune diffusion rate from FMA, both for the optimized ALS-U lattice with zero momentum deviation.