Page:Lawhead columbia 0054D 12326.pdf/175

From Wikisource
Jump to: navigation, search
This page has been proofread, but needs to be validated.

emphasizing: whatever a chaotic system is, it is not a system where every small change immediately “blows up” into a big change after a short time. We’ll need to get more precise.

Let’s stick with the butterfly effect as our paradigm case, but now consider things from the perspective of DyST. Suppose we’ve represented the Earth’s atmosphere in a state space that takes into account the position and velocity of every gas molecule on the planet. First, consider the trajectory in which the nefarious butterfly doesn’t flap its wings at some time t1, and the hurricane doesn’t develop at a later time t2. This is a perfectly well-defined path through the state space of the system that can be picked out by giving an initial condition (starting point in the space), along with the differential equations describing the behavior of the air molecules. Next, consider the trajectory in which the butterfly does flap its wings at t1, and the hurricane does develop at t2. What’s the relationship between these two cases? Here’s one obvious feature: the two trajectories will be very close together in the state space at t1—they’ll differ only with respect to the position of the few molecules of air that have been displaced by the butterfly’s wings—but they’ll be very far apart at t2. Whatever else a hurricane does, it surely changes the position and velocity of a lot of air molecules (to say the least!). This is an interesting observation: given the right conditions, two trajectories through state space can start off very close together, then diverge as time goes on. This simple observation is the foundation of chaos theory.

Contrast this case with the case of a clearly non-chaotic system: a pendulum, like the arm on a grandfather clock. Suppose we define a state space where each point represents a particular angular velocity and displacement angle from the vertical position for the pendulum. Now, look at the trajectory that the pendulum takes through the state space based on different initial