Non-uniform decay of predictability and return of skill in stochastic oscillatory models

We examine the dynamical mechanisms that lead to the loss of predictability in low-dimensional stochastic models that exhibit three main types of oscillatory behavior: damped, self-sustained, and heteroclinic. We show that the information that an initial ensemble provides about the state of the system decays non-uniformly with time. Long intervals during which the forecast provided by the ensemble does not loose any of its power are typical in all the three cases. Moreover, the information that the forecast provides about the individual variables in the model may increase, despite the fact that information about the entire system always decreases. We analyze the fully solvable case of the linear oscillator, and use it to provide a general heuristic explanation for the phenomenon. We also show that during the intervals during which the forecast loses little of its power, there is a flow of information between the marginal and conditional distributions.