A Chaotic Solution Having observed a strange relationship in his weather model, Lorenz went to work in search of a more satisfying explanation for the bizarre difference in patterns he observed. This question would lead him to establish a previously undeveloped branch of mathematics.

Stripping down an existing model of atmospheric convection to it most simple, bear bones, Lorenz present a system of three, nonlinear, differential equations. To the causal observer with some mathematical ability, the system appears simple enough to solve- and many with would propose they could do so. In response, Lorenz would smile and reply "Yes, there is a tendency to think that when you see them. There are some non-linear terms in them, but you think there must be a way to get around them. But you can't." Just like the three body problem, the system refuses to be solved analytically.

Here is the system of non-linear, differencial equations:

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Below is an applet that models two iterations of the system above where alpha, beta, and rho are constants. They have the same initial conditions except for rho which differs between the two systems by one-hundredth. Press play and observe how the trajectories compare.
Initially it may appear as if one trajectory merely falls behind the other. That is, however, an incomplete assessment of the systems of equations. Important to note is how the trajectories diverge in behavior. Watch it again. After they deviate, how many times will each trajectory loop before it splits to the other side? Later, do the paths ever really cross or repeat themselves?

Many meteorologist of the time had great hopes that the advent of computers would allow them to forecast the weather further and further in the future. Lorenz, however, is able to use this mathematical model to demonstrate how nonlinear, dynamical systems- ones similar to the three body problem that have no apparent analytic solution- possess a sensitivity to initial conditions which result in vastly different outcomes over relatively short periods of time. Why can we not predict the weather far in the future? Because whether is a function of so many different factors all interwoven together such that a small variation in one variable can have compounding effects over other variables within a relativly small duration in time. To quote lorenz " When a butterfly flutters its wings in one part of the world, it can eventually cause a hurricane in another". Unfortunately, there are just too many minute variations that can’t be accounted for in nature such that the smallest change in one will lead to dramatically different outcomes in very short spans of time. It is, therefore, unlikely we will ever have a better model for predicting the weather.

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