Beam–beam interaction models under narrow-band random excitation

The aim of this paper is to continue our previous investigations on beam–beam interaction models in particle accelerators by considering these models under narrow-band random excitation. In this investigation the multiple scale and the moment methods are used. The equations of modulation of the amplitude and the phase are derived and perturbation techniques are used to seek approximate steady-state solutions. Two special cases are considered to illustrate this study and excellent agreement is found between analytical and numerical results. Other cases can be similarly studied. The effects of noise and detuning parameters are examined numerically and theoretically and we find that the system demonstrates a diffused limit cycle as the intensity of noise increases. The numerical results show that the multiple scale method is effective for these stochastic systems.