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