Analysis of Nonlinear Dynamics by Square Matrix Method

TL;DR

This paper introduces a square matrix method for analyzing nonlinear dynamics that simplifies high-order calculations, provides accurate action-angle variables, and aids in system optimization and stability assessment.

Contribution

The paper presents a novel square matrix approach that reduces computational complexity and improves stability analysis in nonlinear dynamical systems.

Findings

Effective dimension reduction in high-order calculations

Accurate action-angle variables matching trajectories

Potential for fast nonlinear system optimization

Abstract

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.