28 1. Differential Equations and Their Solutions is that when we try to compute σp 0 (t), there will always be some tiny error in the initial condition, and in addition there will be system- atic rounding, discretization, and truncation errors in our numerical integration process, so we are always in essence computing σp 1 (t) for some p1 near p0 rather than computing σp 0 (t) itself. The important thing to remember is that even the most miniscule of deviations will get enormously amplified after the loss of correlation occurs. While there is no mathematical proof of the fact, it is generally believed that the fluid mechanics equations that govern the evolution of weather are chaotic. The betting is that accurate weather pre- dictions more than two weeks in advance will never be feasible, no matter how much computing power we throw at the problem. As the meteorologist Edward Lorenz once put it, “... the flap of a butterfly’s wings in Brazil can set off a tornado in Texas.” This metaphor has caught on, and you will often hear sensitive dependence on initial conditions referred to as the “butterfly effect”. In Figure 1.2 we show a representation of the so-called “Lorenz attractor”. This shows up in an ODE that Lorenz was studying as a highly over-simplified meteorological model . The Web Companion has a QuickTime Movie made with 3D-XplorMath that shows the Lorenz attractor being generated in real time. What is visible from the movie (and not in the static figure) is how two points of the orbit that are initially very close will moments later be far apart, on different “wings” of the attractor. (By the way, the fact that the Lorenz attractor resembles a butterfly is totally serendipitous!) Figure 1.2. The Lorenz attractor.

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