Analysis of nonlinear dynamics by square matrix method
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...
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2017-03-01
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Series: | Physical Review Accelerators and Beams |
Online Access: | http://doi.org/10.1103/PhysRevAccelBeams.20.034001 |
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doaj-e6cbaeff137d490ca3dfebedfe4d682f2020-11-24T22:39:13ZengAmerican Physical SocietyPhysical Review Accelerators and Beams2469-98882017-03-0120303400110.1103/PhysRevAccelBeams.20.034001Analysis of nonlinear dynamics by square matrix methodLi Hua YuThe 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.http://doi.org/10.1103/PhysRevAccelBeams.20.034001 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Li Hua Yu |
spellingShingle |
Li Hua Yu Analysis of nonlinear dynamics by square matrix method Physical Review Accelerators and Beams |
author_facet |
Li Hua Yu |
author_sort |
Li Hua Yu |
title |
Analysis of nonlinear dynamics by square matrix method |
title_short |
Analysis of nonlinear dynamics by square matrix method |
title_full |
Analysis of nonlinear dynamics by square matrix method |
title_fullStr |
Analysis of nonlinear dynamics by square matrix method |
title_full_unstemmed |
Analysis of nonlinear dynamics by square matrix method |
title_sort |
analysis of nonlinear dynamics by square matrix method |
publisher |
American Physical Society |
series |
Physical Review Accelerators and Beams |
issn |
2469-9888 |
publishDate |
2017-03-01 |
description |
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. |
url |
http://doi.org/10.1103/PhysRevAccelBeams.20.034001 |
work_keys_str_mv |
AT lihuayu analysisofnonlineardynamicsbysquarematrixmethod |
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1725710107506900992 |