On the characterization of non-stationary chaotic systems: Autonomous and non-autonomous cases

Properties of non-stationary dynamical systems are studied using the concept of a snapshot attractor which takes into account a large number of initial conditions within the basin of attraction. In this paper, both autonomous and non-autonomous systems are considered. We have discussed three different ways of inducing non-stationarity in a chaotic system—(a) when the parameter changes in a linear fashion, (b) when the parameter changes in a random fashion and (c) when it is governed by an equation inducing a chaotic change in it. Here all the variations are with respect to time. Though the first and second situations have already been considered in Romeiras et al. (1990) [1] and Serquina et al. (2008) [2], the last one has never been considered before. In each case the basic fractal properties and the famous Pesin identity are proved numerically. The Lyapunov exponents, which are calculated via an averaging procedure, lead to the expected values.