The second-order tune shift with amplitude for octupole-induced resonances in storage ring

The purpose of this note is to analyze the octupole-induced resonances, to lowest order, in a synchrotron and storage ring. When the Hamiltonian with octupole term is transformed to action-angle variables, it is found that the amplitude-dependent tune shift terms are composed of two types: terms of second-order in betatron oscillation amplitude of a particle and terms of fourth-order in oscillation amplitude. Obtaining fourth-order terms requires complicated analysis even with the first-order perturbation theory employed. Treatment of this analysis will be the subject of a subsequent note. Second-order terms are straightforward and simple to calculate, and therefore they treat them here first