Summary: We develop a model of functioning of complex information systems (in particular, cognitive systems) in that information states are coded by \(p\)-adic integers. An information state evolves by iterations of a discrete \(p\)-adic dynamical system. The \(p\)-adic utrametric on the space of information states (\(p\)-adic homogeneous tree) describes the ability of an information system to operate with associations. The main attention is paid to the collective dynamics of families of associations.