Calculating the nonlinear tune shifts with amplitude using measured BPM data

An algorithm is proposed to calculate the approximate tune shifts with amplitude using only the linear transfer map of a circular accelerator and with little or no information on higher order nonlinearities. To extract information about the nonlinear dynamics, the decay rate of the average amplitude of the particle distribution after an instantaneous transversal kick is used. This method works when strong, low-order resonances are not present, that is where the linear lattice rather than the nonlinear driving terms dominates the machine dynamics. Nonlinear normal form transformation and differential algebra methods are employed to establish the connection between measurement results and the nonlinear tune shifts with amplitude. Proposed algorithm is applicable to a wide range of circular accelerators.