Study of fully coupled 3D envelope instability using automatic differentiation

Abstract

Auto-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 dynamics system that is governed by 21 ordinary differential equations. This second-order instability (named envelope instability) is driven by space-charge effects and has 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 auto-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 auto-differentiation in analyzing complicated dynamical systems involving a large number of ordinary differential equations.

Figures (4)

• Figure 1: Growth rate amplitudes of unstable envelope modes as a function of the horizontal tune depression, shown both without (embedded plot) and with six-dimensional (6D) coupling, for a longitudinal zero-current phase advance of $60^\circ$.
• Figure 2: Phases of unstable envelope modes as a function of the horizontal tune depression, with coupling and a longitudinal phase advance of $60^\circ$.
• Figure 3: Scan of the transverse $x$- and $y$-zero-current phase advances showing the maximum envelope mode growth rate amplitudes without coupling (top) and with coupling (bottom).
• Figure 4: Scan of the transverse-to-longitudinal zero-current phase advances showing the maximum envelope mode growth rate amplitudes without coupling (top) and with coupling (bottom).