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  • Letter
  • Open Access

Study of fully coupled three-dimensional envelope instability using automatic differentiation

Ji Qiang

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Phys. Rev. Accel. Beams 29, L024201 – Published 6 February, 2026

DOI: https://doi.org/10.1103/bsvv-cdwt

Abstract

Automatic differentiation is a powerful tool for computing derivatives of simulation results with respect to given parameters. In this Letter, we have applied this tool to investigate the instability of a dynamical system governed by 21 ordinary differential equations. This second-order instability (named envelope instability) is driven by space-charge effects and has a significant impact on the operational regimes of particle accelerators. Our study delves into the three-dimensional envelope instability, incorporating both transverse and longitudinal coupling. Conventionally, analyzing this complex system would necessitate solving 441 ordinary differential equations, which is computationally intractable. However, by employing automatic differentiation, we were able to track only 21 equations. This approach allowed us to uncover an additional instability stopband, which arises from space-charge-induced coupling and has not been reported in previous studies. This research highlights the significant advantages of automatic differentiation in analyzing complicated dynamical systems involving a large number of ordinary differential equations.

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Phys. Rev. Accel. Beams 29, L024201– Published 6 February, 2026
  • Received 29 September 2025
  • Accepted 13 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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