FRIB, East Lansing, Michigan, USA
BNL, Upton, New York, USA
The Hénon Map represents a linear lattice with a single sextupole kick. This map has been extensively studied due to its chaotic behavior. The case for the two dimensional phase space has recently been revisited using ideas from KAM theory to create an iterative process that transforms nonlinear perturbed trajectories into rigid rotations*. The convergence of this method relates to the resonance structure and can be used as an indicator of the dynamic aperture. The studies of this method have been extended to the four dimensional phase space case which introduces coupling between the transverse coordinates.
*Hao, Y., Anderson, K., & Yu, L. H. (2021, August). Revisit of Nonlinear Dynamics in Hénon Map Using Square Matrix Method. https://doi.org/10.18429/JACoW-IPAC2021-THPAB016
BNL, Upton, New York, USA
FRIB, East Lansing, Michigan, USA
We report progress on applying the square matrix method to obtain in high speed a "convergence map", which is similar but different from a frequency map. We give an example of applying the method to optimize a nonlinear lattice for the NSLS-II upgrade. The convergence map is obtained by solving the nonlinear dynamical equation by iteration of the perturbation method and studying the convergence. The map provides information about the stability border of the dynamical aperture. We compare the map with the frequency map from tracking. The result in our example of nonlinear optimization of the NSLS-II lattice shows the new method may be applied in nonlinear lattice optimization, taking advantage of the high speed (about 30~300 times faster) to explore x, y, and the off-momentum phase space.