Theory of Square Matrix for Nonlinear Optimization, and Application to NSLS-II

Nonlinear dynamics has an important role when designing modern synchrotron lattices. In this letter, we introduce a new method of using a square matrix to analyze periodic nonlinear dynamical systems 1,2. Applying the method to the National Synchrotron Light Source II storage ring lattice has helped to mitigate the chaotic motion within its dynamic aperture. For a given dynamical system, the vector space of a square matrix can be separated into different low dimension invariant subspaces according to their eigenvalues. When Jordan decomposition is applied to one of the eigenspaces, it yields a set of accurate action-angle variables. The distortion of the new action-angle variables provides a measure of the nonlinearity. Our studies show that the common convention of confining the tune-shift with amplitude to avoid the crossing of resonance lines may not be absolutely necessary. We demonstrate that the third order resonance can be almost perfectly compensated with this technique. The method itself is general, and could be applied to other nonlinear systems.