The so-called ‘butterfly effect’ is a very important scientific concept that the general public should be aware of, unfortunately it has been popularised using the obligatory oversimplified mental image of a butterfly causing a storm, etc.
Those kinds of oversimplifications-that has been popular since Sputnik made the headlines-are favoured by the journalists who copy them from Wikipedia without really understanding them and paste them into their articles to add some padding and give the appearance of understanding where there is none. If one does not understand the concept beforehand he will not be helped by these mental images that simplify too much and inform too little.
But the concept remains important and has to be convoyed to those who wouldn’t understand the mathematical definition, so what is the alternative to the oversimplified mental image? How about a simple image?
I have here a graph of a function that starts with an input between zero and one and for every step afterwards gives an output, also, between zero and one. The function is very sensitive to the input and exhibits the so-called ‘butterfly effect’.
So we start this function with two inputs the first=0.51000001, and the second=0.51000002. The difference between the inputs is about 0.000002%; this is extremely small difference, yet if we run the function for only 43 steps we get the following graph:
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For the first twenty steps the two inputs have almost the same output, from the twentieth step the difference becomes visible on our graph and grows quite quickly. Within ten steps the two outputs are completely separated not only the range of output they give but also in their behaviour (one is oscillating while the other declines from a high), at step 43 the difference between the two outputs is almost 90%.
Thus an initial difference of two millionth of a percent has been increased to a difference in output of ninety percent, which is a 45 million times increase!
The second graph shows roughly the same thing, but this function is a solution of the Lorenz Equations, which is the simplest way to describe the weather system using mathematical equations:
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So we see that no matter how good are our mathematical weather models and how accurate are our measurements of the daily weather we can’t predict future weather for more than few days ahead, as the most infinitesimal error in measurement will grow to an enormous difference very quickly.
