Abstract
The nonlinear dynamics of a system with periodic structure can be analyzed using a square matrix. We show that because of the special property of the square matrix constructed for nonlinear dynamics, we can reduce the dimension of the matrix from the original large number for high order calculations to a low dimension in the first step of the analysis. Then a stable Jordan decomposition is obtained with much lower dimension. The Jordan decomposition leads to a transformation to a new variable, which is an accurate action-angle variable, in good agreement with trajectories and tune obtained from tracking. More importantly, the deviation from constancy of the new action-angle variable provides a measure of the stability of the phase space trajectories and tune fluctuation. Thus the square matrix theory shows a good potential in theoretical understanding of a complicated dynamical system to guide the optimization of dynamical apertures. The method is illustrated by many examples of comparison between theory and numerical simulation. In particular, we show that the square matrix method can be used for fast optimization to reduce the nonlinearity of a system.
9 More- Received 1 August 2016
DOI:https://doi.org/10.1103/PhysRevAccelBeams.20.034001
Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society