Project Euclid

We review a wide variety of applications in different branches of sciences arising from the study of dynamical systems. The emergence of chaos and fractals from iterations of simple difference equations is discussed. Notions of phase space, contractive mapping, attractor, invariant density and the relevance of ergodic theory for studying dynamical systems are reviewed. Various concepts of dimensions and their relationships are studied, and their use in the measurement of chaotic phenomena is investigated. We discuss the implications of the growth of nonlinear science on paradigms of model building in the tradition of classical statistics. The role that statistical science can play in future developments of nonlinear science and its possible impact on the future development of statistical science itself are addressed.