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**T**he movement of particles in turbulent water. The changeable state of our weather. Measuring either of these things, you will encounter non-linear patterns: the direction of each turbulent water particle or amount of rainfall per year appears to fluctuate at random around no fixed point. But if we look at weather patterns over a longer period we can indeed see patterns forming: ice-ages rise and fall as the climate shifts. Chaos theory searches for order in non-linear disorder.

The logistic equation is both the simplest and most significant equation in the field of chaos. This equation was used by ecologist Robert May to map the fluctuating population of various species over years. Basic ecological sequences assume next year’s population (Xn+1) is the product of this year’s population (Xn) and a constant (r) — i.e. r is multiplied by this year’s population to get next year’s. This creates an equation of the form:

Xn+1 = r(Xn)

[note initial population Xn is represented by a number between 0 and 1] If you then take Xn+1 and re-enter it in the equation (in place of Xn), you get Xn+2 (the population for the year after next) and so on. Continually iterating this process creates a linear sequence, rising steadily across time if r is larger than 1, falling steadily if r is smaller. But most populations do not grow in a linear manner — they bounce around from year to year: a large population increase one year may create starvation, resulting in a drop the next year, which leaves plentiful resources and a subsequent rise… overall trends can then only be seen from a long perspective.

The logistic equation is one way to represent such non-linear trends. The logistic equation takes the form:

Xn+1 = r(Xn)(1-Xn)

As Xn increases, 1-Xn decreases, so a rise one year creates a fall the next, and the population bounces around. However, over time, if r is smaller than 3, the sequence will eventually converge upon a single point. For each value of r this end point is always the same, no matter what the initial value of Xn.

If r is between 3.0 and 3.45, X will eventually converge upon two points which alternate with one another but are otherwise stable. With r above 3.45, X will eventually oscillate between 4 points; above 3.54, eight points; and so on. The increased value of X required for each jump in the number of eventual endpoints becomes gradually smaller and smaller (i.e. 3.54 – 3.45 is smaller than 3.45 – 3.0). Then, with an r of 3.569945672 or higher, X never resolves into a specific number of steady points — it simply continues to fluctuate at random forever. This point 3.569945672 is therefore the point at which chaos kicks in.

It is possible to replicate the logistic equation on a simple scientific calculator (e.g. a Casio fx-82W). Enter in an initial Xn (any number between 0 and 1, say for instance 0.3) and press =. Then pick a value for r between 2 and 3 (say 2.1). Type in 2.1 then press the Ans key (which enters 0.3 in the equation), then type (1-Ans) which means your screen should say: 2.1Ans(1-Ans). Press = and you’ll get 0.441. Press = again — the calculator will enter 0.441 (Xn+1) in the Ans slots where 0.3 (Xn) had been, and give you the answer 0.5176899. Keep pressing = and watch:

0.3 0.441

0.5176899

0.5243428

0.5237556

0.5238149

0.5238090

0.5238096

0.5238095

0.5238095

X rises three times, then oscillates before settling upon a singular end point. Try the equation again with r = 3.1: instead of resolving into one point, the sequence resolves into an oscillation between two points, 0.7645665 and 0.5580141. If you try r = 3.6, X will never resolve — hours of fun are to be had watching chaos approach as you jiggle r about!

Fixed points: I said above that each value of r will eventually lead the sequence to resolve at a certain point or set of points, regardless of what the initial value Xn is. However, if the initial Xn is equal to (r-1)/r then X will never change. As far as I am aware this oddity is of no significance.

For more information read James Gleick’s *Chaos*