Convergence map with action-angle variables based on square matrix for nonlinear lattice optimization

To analyze nonlinear dynamic systems, we developed a new technique based on the square matrix method. We propose this called \convergence map" for generating particle stability diagrams similar to frequency maps widely used in accelerator physics estimate aperture. The convergence map provides information as but much shorter computing time. equation can be rewritten terms of action-angle variables provided by derived from lattice. is obtained solving exact iteratively perturbation method using Fourier transform and studying convergence. When iteration convergent, solution expressed quasi-periodic analytical function highly accurate approximation, hence motion stable. border stable determines dynamical As an example, applied optimization NSLS-II storage ring demonstrated aperture comparable or larger than nominal one tracking. computation speed 30 300 times faster tracking, depending size lattice (number superperiods). ratio complex lattices with low symmetry, such colliders.