So what's the problem?

One point that needs to be addressed is the idea that the prediction time equals the Lyapunov Time—the time for nearby orbits to diverge by e (in phase space). In the solar system case, the orbits (original and perturbed) diverge linearly—which will happen in a linear system (and cannot be chaotic)—until 30 Myr, when the orbits diverge exponentially. [5] The Lyapunov Time IS STILL IN THE LINEAR RANGE. Does it really make sense to use the Lyapunov Time as a measure of the prediction time?

Two possibilities: we either abandon the Lyapunov Characteristic Exponent (LCE) as a measure of predictability time, or we retain it, perhaps with a modification.

1) It has been recently shown, by using the Lorenz system, the Ikeda map, and a class of generalized Baker's maps, that the LCE "need not reflect practical limits of prediction" [8] and "place no a priori limit on the predictability of a dynamical system" and should perhaps be replaced by the concept of "Epsilon-Shadowing". (The Anosov-Bowen Lemma states that any "good" model can be epsilon-shadowed.) [9]

2) The classical formula for prediction time holds only for infinitesimal perturbations and in non-intermittent** systems.

We prefer to retain the concept of the lyapunov exponent and we will examine in some detail the second issue above.

IMPORTANT:

• The LCE is associated with the smallest, fastest scales in the system which rule the initial exponential growth of infinitesimal errors (which seems like a VERY important point in fluid dynamics).

MAJOR QUESTION: When a numerical experiment is performed to examine the the predictability time, aren't the initial conditions perturbed by some finite amount (not really infinitesimal)?

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**Intermittency: In the vicinity of a bifurcation point, the presence of random forces always leads to an accelerated evolution (an "explosive instability") of the system and to the development of intermittent stochasticity. [10]