A look at some details of the growth of initial uncertainties

Because of the errors entailed in observing certain systems, the states that one might believe to be the true states form an ensemble, as do the states obtained from these states by forward extrapolation in time.We identify the uncertainty with the root-mean-square distance in state space of the ensemble members from their mean. We enumerate the properties of a special three-variable system that behaves chaotically, and we use the system to evaluate a logarithmic measure α (t₁, t₀) of the ratio of the uncertainty at a ‘verifying time’ t₁ to that at an cobserving time’ t₀. With t₀ and t₁ as coordinates, we construct diagrams displaying contours of α (t₁, t₀).We find that the details of the diagrams tend to line up in the horizontal and vertical directions, rather than parallel to the diagonal where t₁ = t₀, as they would if α (t₁, t₀) depended mainly on the forecast range t₁ — t₀. The implication is that states at certain times t₁ are highly predictable, i.e. α (t₁, t₀) < α (t, t₀) if t occurs somewhat before or after t₁, and that states at certain times t₀ are highly predictive, i.e. α (t₁, t₀)< α (t₁, t) if t occurs somewhat before or after t₀. When observations at times preceding t₀ are combined with those at t₀, the greatest resulting reductions in uncertainty at t₁ occur when the states at the additional times are highly predictive. We speculate as to the applicability of these findings to larger systems.