# Emergence of Chaos

(There is order in chaos)An iterative process with a simple quadratic equation ^{2} + c

A study of iterations with another quadratic equation *chaos*. This is a well known *logisitic* equation. The process _{n+1} = kx_{n}(1 - x_{n})

In the real domain, iterative processes admit quite a transparent graphical representation. Start with x_{0}. Draw a vertical line until it intersects the graph of _{0})._{1} = f(x_{0})._{1} you obtain x_{2} in a similar way. A vertical line up to the graph, then a horizontal line towards the diagonal. There is no need to continue the lines until they meet the axes. Just move intermittently between the graph and the diagonal.

On the diagram, subsequent iterates become closer and closer and each closer to the point of intersection of the graph and the diagonal. The point of intersection is a solution of the equation *stable* because iterates {x_{n}} tend
to converge there. Not all solutions of _{0}, x_{1} will be father away from 0 than x_{0}, and the next iterate x_{2} will be even farther on and so on.

Below, the applet's panel is split into two parts - left and right. The horizontal
axis on the right corresponds to a parameter *a* changing from .7 through 1. When you
select a parameter by clicking on the right graph, on the left, the function
*a*x(1-x)_{0}. I replaced k with 4*a* so that the top of the graph (attained for *a*.

Now let's experiment with the iterations as *a* changes. For a while the equation *a*x(1-x)*a* = 0.75,*a* = 0.75.

After *a* = 0.86237... the two curves split further into four whereas the iteration oscillate between four different points. Next four points are replaced by 8 and 8 by 16 and so on. Note too that the horizontal distance between the split points (points of *bifurcation*) grows shorter and shorter. Until the bifurcation becomes so fast at the point *a*=.892 that iterates race all over a segment instead of
alternating between a few fixed points. The behavior is chaotic in the sense that it's absolutely impossible
to predict where the next iterate will appear. Ian Stewart (p. 58) gives the following definition: *Chaos* is apparently random behavior with purely deterministic causes.

Interestingly, for some values of *a* even after the point 0.86237... there appears some regularity. You can easily find a window where iterates oscillate between three different points forming a 3-cycle. If you are patient you'll be able to detect a 6-cycle as well. Of course there are 12- and 24-cycles and others. The program is just not accurate enough to support deeper investigation.

The reason for the bifurcation may be surmised from the graph of the second iterate function *stable* so that depending on where they start the iterates of *repeller*. However close to B you start iterations they will escape from there and converge to either A or C.

(To start iterations, click above in the right portion of the applet. Wait, then click at a different point... or in the left part.)

Instead of the applet you can download and run locally an application that is performing exactly the same job.

The terminology is as follows.

- The point x is
*stationary*(or*fixed*) for y=f(x) iff x=f(x). - A stationary point x is
*attractive*iff for x_{0}sufficiently close to x the iteration {x_{n}} converge to x. - A stationary point x is
*repelling*iff no iterations {x_{n}} with x_{0}≠x ever converge to x. - A finite sequence of points {x
_{n}, x_{n+1}, ..., x_{n+p-1}} withp > 1, is a*p-cycle*iffx and_{n}= f(x_{n+p-1})x for_{n}≠ f(x_{n+q-1})2 ≤ q < p. Cycles may be both attractive and repelling.

You can observe from the graphs that their slopes behave differently at the attractive and repelling points. A point x is attractive if |slope(f(x))|<1. A point x is repelling if |slope(f(x))|>1.

Another thing is worth noting. With the parameter *a* near but less than 0.75, the slope of the graph at the stationary point is less than 1 in absolute value. As *a* approaches 0.75 the absolute value of the slop grows and, at 0.75, becomes 1. This makes the point unstable and eventually causes the iterations split into a 2-cycle. First the 2-cycle is attractive. Then, as *a* grows, it loses its stability and, eventually, splits into a 4-cycle.

### References

- J. Gleick,
*Chaos*, Viking, 1987 - D. R. Hofstadter,
*Metamagical Themas*, Basic Books, Inc., 1985, Chapter 16. - B. Mandelbrot,
*The Fractal Geometry of Nature*, W.H.Freeman and Co., NY, 1977. *Chaos and Fractals*, R. L. Devaney and Linda Keen, eds., AMS, 1989.- Heinz-Otto Peitgen et al,
*Chaos and Fractals: New Frontiers of Science*, Springer, 2nd edition, 2004 - G. Strang,
*Introduction to Applied Mathematics*, Wellesley-Cambridge Press, MA, 1986. - I. Stewart,
*Nature's Numbers*, Basic Books, 1997.

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