# Chaos

This chapter describes some of the facts about the behavior of the maps
for parameters for which the dynamics is **chaotic**.
To begin, restart
the applet to return the parameters to their default values.
You can eliminate transients by setting *tr an* to some high value e.g.
*tr an=200*, and choose a fast *Speed* consistent with
the speed of your computer (but see bugs).

## Full band chaos

Set the parameter *a* to the maximum value *a=4* consistent with iterated
values remaining in the range *0< x <1*, and start
the iteration. Successive values of *x*_{n}
appear quite "random", and the values eventually fill up the whole interval
*0 < x < 1*.
This chaotic motion can in fact be completely understood by making a transformation
of variables. Define

*x*_{n}=sin^{2}( z_{n}/2)
The iteration of *z* for *a=4* is then just
*z*_{n+1} = 2 z_{n} for 0 < z_{n} < 1/2

*z*_{n+1} = 2 (1 - z_{n}) for 1/2 < z_{n} < 0
This is known as the tent map, and the dynamics is equivalent to an even simpler map:
*z*_{n+1} = 2 z_{n} mod 1
Successive values of *z*_{n} from a random initial value are as
random as a coin toss (and in fact in binary representation answering the question
of whether the *n* th value is to the right or left of 0.5 just gives the
binary representation of the initial value, which is random for a random initial value).
## Multi-band chaos

For other values of *a* the chaos is more complicated. In fact between the onset
of chaos at *a=a*_{c} and the full band chaos at *a=4* there are
many different chaotic points interspersed with points giving periodic orbits of *any*
period you want, each of which has its own period doubling cascade to chaos!
We can, however, find a sequence of values of *a* for which the chaos mimics, in reverse,
the *2*^{n} cycle period doubling cascade.

For example set *a=3.6* and iterate. Successive values of *x* appear
"random" but there is also a definite pattern. Check that successive values of *x*
always appear on opposite sides of the intersection of the diagonal with the curve.
So there is a completely deterministic period-2 motion jumping backward and forward
between two *bands* of values, with values spread "randomly" within the bands.
This can be made clearer by looking at every second iterate setting *nf=1*. Now
the *x* values appear on just one side of the intersection point. (Which side
depends on the initial value: for *x0=0.35* we get points on the lower side;
setting e.g. *x0=0.75*, gives points on the greater side.)

As you probably expect by now, moving *a* down closer to *a*_{c}
will give 4 chaotic bands, and then 8 bands, 16 ..... For example using values

a_{m} = a_{c} + 0.15 ^{ -m}
(set with the boxes at the bottom of the screen)
will give *2*^{m} chaotic bands. Again you can check this by
looking at the *2*^{m}th iteration setting *nf=m*.

## Band merging

The transition points at which the chaotic bands merge yield particularly simple
chaotic dynamics. These band merging points can be set with the drop down boxes
at the bottom of the screen. (Choose the "**+**" sign and "*BandMerging*".)
Then for
a_{m} = a_{c} + 0.485 ^{ -m}
the *2*^{m} band just merge to give *2*^{m-1} bands.
Within each of the *2*^{m} bands the distribution of the chaos is
*exactly* the same as the full band chaos at *a=4*. To show this set
*nf=m*, and *nsc=m* to blow up the central portion making up one band,
and compare with the full band chaos at *a=4*, *nf=nsc=0*.

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Last modified 18 August, 2009

Michael Cross