Introduction to the Modeling and Analysis of Complex Systems

154 CHAPTER 9. CHAOS2.52.01.51.00.50.00.50 20 40 60 80 100Figure 9.1: Example of chaotic behavior, generated using Eq. (8.37) with r = 1.8 andthe initial condition x 0 = 0.1.• is a prevalent phenomenon that can be found everywhere in nature, as well asin social and engineered environments.The sensitivity of chaotic systems to initial conditions is particularly well known underthe moniker of the “butterfly effect,” which is a metaphorical illustration of the chaoticnature of the weather system in which “a flap of a butterfly’s wings in Brazil could setoff a tornado in Texas.” The meaning of this expression is that, in a chaotic system,a small perturbation could eventually cause very large-scale difference in the long run.Figure 9.2 shows two simulation runs of Eq. (8.37) with r = 1.8 and two slightly differentinitial conditions, x 0 = 0.1 and x 0 = 0.100001. The two simulations are fairly similarfor the first several steps, because the system is fully deterministic (this is why weatherforecasts for just a few days work pretty well). But the “flap of the butterfly’s wings” (the0.000001 difference) grows eventually so big that it separates the long-term fates of thetwo simulation runs. Such extreme sensitivity of chaotic systems makes it practicallyimpossible for us to predict exactly their long-term behaviors (this is why there are notwo-month weather forecasts 1 ).1 But this doesn’t necessarily mean we can’t predict climate change over longer time scales. What is notpossible with a chaotic system is the prediction of the exact long-term behavior, e.g., when, where, and howmuch it will rain over the next 12 months. It is possible, though, to model and predict long-term changesof a system’s statistical properties, e.g., the average temperature of the global climate, because it can bedescribed well in a much simpler, non-chaotic model. We shouldn’t use chaos as an excuse to avoid making