Our inability to predict precisely

Our inability to predict precisely the evolution of a chaotic system is due to its sensitive dependence on initial conditions, whereby “microscopic perturbations are amplified to affect macroscopic behavior”—the so-called butterfly effect.34 Crutchfield et al. illustrate this effect with an example of a billiard player “with perfect control over his or her stroke” trying to predict the trajectory of the ball: “If the player ignored an effect even as miniscule as the gravitational attraction of an electron at the edge of the galaxy, the prediction would become wrong after one minute!”35 The slightest variation in variables can lead to exponentially different result. When Lorenz, for example, rounded off the digits specifying initial conditions in an attempt to replicate earlier results, he obtained very different results over a period of time: “[T]wo states differing by imperceptible amounts may eventually evolve into two considerably different states.”