if λ > 0 (the system is chaotic) but still very small. No matter how small λ is, chaotic behavior will appear whenever t ≫ 1 : even a very small amount of divergence becomes significant on long enough time scales. Similarly, if t ≪ 1 then we can generally treat the system as if it is non-chaotic (as in the case of the orbits of planets in our solar system). The lesson to be drawn is that it isn’t the value of either t or λ that matters so much as the ratio between the two values.
5.1.4 Prediction and Chaos
It can be tempting to conclude from this that if we know λ, ε, and t, then we can put a meaningful and objective “horizon” on our prediction attempts. If we know the amount of uncertainty in the initial measurement of the system’s state (ε), the maximal rate at which two paths through the state space could diverge (λ), and the amount of time that has elapsed between the initial measurement and the time at which we’re trying to make our prediction (t), then shouldn’t we be able to design things to operate within the uncertainty by defining relevant macroconditions of our system as being uniformly smaller than ε(eλt) ? If this were true, it would be very exciting—it would let us deduce the best way to construct our models from the dynamics of the system under consideration, and would tell us how to carve up the state space of some system of interest optimally given the temporal scales involved.
Unfortunately, things are not this simple. In particular, this suggestion assumes that the state space can be neatly divided into continuously connected macroconditions, and that it is not possible for a single macrostate’s volume to be distributed across a number of isolated regions. It assumes, that is, that simple distance in state-space is always going to be the best measure of qualitative similarity between two states. This is manifestly not the case. Consider, for instance,
case with the damped pendulum discussed above—all initial conditions eventually converge on the rest state.