Equivalent equations and incoherent lifetime calculated from e+eeam-beam simulation

For given tunes (ν_x, ν_y, ν_s ), the influence of different terms x^ky^ls ^m in the nonlinear beam-beam forces is different, so only some of them are important as sources of regular (nonstochastic) effects. As for irregular effects, it seems that in e⁺e^- colliders the simultaneous action of both quantum fluctuations and three-dimensional nonlinearities creates effectively an additional stochastic force. The developed program of beam-beam simulation including Fourier analysis of different moments M^klm(t)=x^k(t)y^l(t)s^m(t) permits us to identify both regular and stochastic effective forces in the equivalent equations of the particles´ motion. These differential equations can be used, for example, for the calculation of the particles´ lifetime