Reconstruction of systems with impulses and delays from time series data

In this paper, we present an approach to identification of dynamical systems with irregular impulses and time delays. The suggested method enables one to reconstruct the underlying differential equations, using the l1-minimization technique in signal processing which takes advantage of the signal's sparseness. Based on the idea that irregular impulses can be regarded as sparse error in the fitting procedure, we obtain an efficient algorithm for reconstructions that separates the regular parts of dynamics from impulsive ones. From time series data sampled from an impulsive ecological models, the suggested method restores an essential dynamics of the original systems. The method also applies to chaotic systems perturbed by intermittent impacts and successfully captures dynamics reflecting qualitative behavior independent of impacts. In addition, we can identify a time-delay Lotka-Volterra model with no prior information on delay time given, to which conventional parameter estimate methods are hardly applicable.