Analysis of Nonlinear Dynamics by Square Matrix Method
- Brookhaven National Lab. (BNL), Upton, NY (United States). Energy and Photon Sciences Directorate. National Synchrotron Light Source II
The nonlinear dynamics of a system with periodic structure can be analyzed using a square matrix. In this paper, we show that because 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 calculation to low dimension in the first step of the analysis. Then a stable Jordan decomposition is obtained with much lower dimension. The transformation to Jordan form provides an excellent action-angle approximation to the solution of the nonlinear dynamics, in good agreement with trajectories and tune obtained from tracking. And more importantly, the deviation from constancy of the new action-angle variable provides a measure of the stability of the phase space trajectories and their tunes. Thus the square matrix provides a novel method to optimize the nonlinear dynamic system. The method is illustrated by many examples of comparison between theory and numerical simulation. Finally, in particular, we show that the square matrix method can be used for optimization to reduce the nonlinearity of a system.
- Research Organization:
- Brookhaven National Lab. (BNL), Upton, NY (United States)
- Sponsoring Organization:
- USDOE Office of Science (SC), Basic Energy Sciences (BES)
- DOE Contract Number:
- SC0012704
- OSTI ID:
- 1340371
- Report Number(s):
- BNL-112480-2016
- Country of Publication:
- United States
- Language:
- English
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