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followed by the states that are included in it will, given enough time, diverge significantly[1].

How much does this behavior actually limit the practice of predicting what chaotic systems will do in the future? Let’s keep exploring the mathematics and see what we can learn. Consider two limit cases of the inequality in 5(j). First:


This is just the limiting case of perfect measurement of the initial condition of the system—a case where there’s absolutely no uncertainty in our first measurement, and so the size of our “neighborhood” of possible initial conditions is zero. As the distance between the two points in the initial pair approaches zero, then the distance between the corresponding pair at time t will also shrink. Equivalently, if the size of the neighborhood is zero—if the neighborhood includes one and only one point—then we can be sure of the system’s position in its state space at any later time (assuming no stochasticity in our equations). This is why the point that chaotic dynamics are not the same thing as indeterministic dynamics is so important. However:


As the Lyapunov exponent approaches zero, the second term on the right side of the inequality in 5(j) approaches unity. This represents another limiting case—one which is perhaps even more interesting than the first one. Note that 5(k) is still valid for non-chaotic systems: the MLE is just set to zero, and so the distance between two trajectories will remain constant as those points are evolved forward in time[2]. More interestingly, think about what things look like

  1. of guaranteeing that the region at the later time will include all the trajectories.

  2. Attentive readers will note the use of what Wikipedia editors call a “weasel word” here. What counts as “significant” divergence? This is a very important question, and will be the object of our discussion for the next few pages. For now, it is enough to note that “significance” is clearly a goal-relative concept, a fact which ends up being a double-edged sword if we’re trying to predict the behavior of chaotic systems. We’ll see how very soon.
  3. If the Lyapunov exponent is negative, then the distance between two paths decreases exponentially with time. Intuitively, this represents the initial conditions all being “sucked” toward a single end-state. This is, for instance, the