combining (5) and (6) lets us deduce

5(j) |

In other (English) words, if the space is a state space for some dynamical system with chaotic behavior, then for all times after the initialization time, the size of the smallest neighborhood that *must* include the successors to some collection of states that started off arbitrarily close together will increase as a function of the fastest rate at which any two trajectories in the system could diverge (i.e. the MLE) and the amount of time that has passed (whew!). That’s a mouthful, but the concepts behind the mathematics are actually fairly straightforward. In chaotic systems, the distance between two trajectories through the state space of the system increases exponentially as time goes by—two states that start off very close together will eventually evolve into states that are quite far apart. How quickly this divergence takes place is captured by the value of the Lyapunov exponent for the trajectories under consideration (with the “worst-case” rate of divergence defining the MLE). Generalizing from particular pairs of trajectories, we can think about defining a *region* in the state space. Since regions are just sets of points, we can think about the relationship between our region’s volume at one time and the smallest region encompassing the end-state of all the trajectories that started in that region at some later time. This size increase will be straightforwardly related to the rate at which individual trajectories in the region diverge, so the size of the later region will depend on three things: the size of the initial region, the rate at which paths through the system diverge, and the amount of time elapsed^{[1]}. If our system is chaotic, then no matter how small we make our region the trajectories

- ↑ If we have some way of determining the largest Lyapunov exponent that appears in D, then that can stand in for the global MLE in our equations here. If not, then we must use the MLE for the system as a whole, as that is the only way

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