We have studied the effect of the β solar wind parameter (equal to the ratio of the plasma pressure to the magnetic pressure) on statistical characteristics of the Ap index reflecting the triggering behavior of the activity of Earth’s magnetosphere. The trigger effect of the dynamics of magnetospheric activity consists in the abrupt transition from the periodic mode to the chaotic mode in the solar activity cycle. It is shown that cumulative amplitude distribution functions and power spectra of the Ap index of both the periodic and chaotic modes are well approximated by power and exponential functions respectively. At the same time, the indices of power functions and the indices characterizing the slope of the Ap index spectrum differ significantly in magnitude for the periodic and chaotic modes. We have found that Ap nonlinearly depends on β for both the modes of magnetospheric dynamics. The maximum of the Ap index amplitude for periodic modes is observed when β>1; and for chaotic ones, when β<1. In almost every cycle of solar activity, the energy of the Ap index fluctuations of chaotic modes is higher than that of periodic ones. The results indicate intermittency and its associated turbulence of magnetospheric activity. The exponential character of the spectral density of the Ap index suggests that the behavior of magnetospheric activity is determined by its internal dynamics, which can be described by a finite number of deterministic equations. The trigger effect of magnetospheric activity is assumed to be due to the angle of inclination of the axis of the solar magnetic dipole to the ecliptic plane, on which the dynamics of the β parameter in the solar activity cycle depends.