Earlier studies have found characteristics of self-similarity and self-organized criticality in AL time series. In this paper we have investigated various aspects and dynamical consequences of the nonlinear dynamical properties of AL time series. A smoothed time series is created by averaging an AL time series over a neighborhood of size T. The negative peaks are automatically detected as minima of the smoothed time series, and the time intervals between successive peaks, called interpeak intervals (IPI) are computed. The analysis is repeated with different averaging window sizes T to examine timescale dependence. The number of peaks increases with decreasing window size, showing power law scaling behavior with exponent 3/5, due to the self-similarity of the time series. The distribution function of the IPI is found to be well fitted to an analytic function which arises from the barrier crossing problem in stochastic dynamics. Also, the shape of the distribution function is scale invariant after a suitable time rescaling, which agrees with the scale-free property of self-organized criticality. A scale invariant parameter γ which determines the shape of the universal distribution function is identified. Examination of the annual values of γ reveals a good correlation with the solar cycle. However, the peak statistics for storm time and non--storm time data do not exhibit any significant differences.