非线性动力学分析的方阵法
Analysis of Nonlinear Dynamics by Square Matrix Method
摘要：
具有周期结构的系统的非线性动力学可以用方阵来分析。我们证明，由于非线性动力学所构造的方阵的特殊性质，我们可以在分析的第一步中将矩阵的维数从原来的高阶计算的大数目降到低维数。然后得到了一个稳定的Jordan分解，其维数大大降低.约旦分解导致一个新的变量，这是一个准确的动作角度变量，在良好的协议轨迹和从跟踪获得的调的变换。更重要的是，新的作用角变量的恒定性的偏差提供了相空间轨迹和调谐波动的稳定性的度量。因此，方阵理论显示出良好的潜力，在理论上理解一个复杂的动力学系统，以指导动态孔径的优化。通过理论与数值模拟的比较，说明了该方法的有效性.特别是，我们表明，方阵法可用于快速优化，以减少系统的非线性。
英文摘要：
The nonlinear dynamics of a system with periodic structure can be analyzed using a square matrix. 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 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. 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 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.
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