When Small Differences Grow
Last timeMany Parts, One Total
Some rules multiply the distance between two nearby starts by a fixed factor every step. Nothing is random, yet the honest forecast horizon grows only with the logarithm of your precision.
Everything in this course so far has pointed towards being able to say what a system
will do. This lesson is about a limit on that, and the limit is not a shortage of
computing power or a flaw in the method. It is a property some rules have.
The same growth, applied to an error
Take the fading displacement from the third lesson and turn the factor the other way.
Instead of a gap between two nearby states shrinking by a fixed fraction each step,
let it be multiplied by something larger than one.
- how far apart the two histories are after k steps. It starts as the error in your knowledge of the starting state, which is never zero for anything measured
- the factor the gap is multiplied by each step. Below one the two histories come together and forecasting is easy. Above one they separate, and how far above one sets how fast
A rule that does it, and a formula that proves it is not random
Here is one of the simplest rules with this property: a population that breeds in
proportion to how many there are and to how much room is left, with the breeding
rate turned all the way up to four. And here is the thing worth seeing: this rule has
an exact closed form, so the figure below can be checked against a formula in the
same way every other run in this course has been.
The lesson stops here
3 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents