# Tutorial

This page leads you through the main features of the applet.

#### The maps

The applet allows the iteration of a choice of two functions:
- Quadratic map:
*f(x)=a x (1-x)* with *0 < a < 4*
- Sine map:
*f(x)=(a/4) sin ( x )* with
*0 < a < 4*

The function is chosen from the list box at the bottom of the screen.
Choosing the function or hitting the **Reset** button brings up
the plot, and shows the first iteration from the starting value *x0*.
For now choose the default quadratic function.
The behavior depends on the value of *a* giving the "height" of the
function.

#### Fixed point

First try *a=2.5* for the quadratic map by typing this value into the
**a =** box and
hitting **Enter** or **Reset**. The function is replotted for this value.
The iteration
can be done step by step by repeatedly hitting the **Step** button, or by
hitting the **Start** button, when the iteration is repeated with a speed
set by the **Speed** scroll bar (roughly in iterations per second). You
should see iterations approaching a *fixed point* at around *x=0.6*.

#### Period 2

Now change the value of *a* to *a=3.1* and again iterate. Although
the values approach close to the fixed point value, which has moved to around
*x=0.7*,
rather than converging to this value the orbit moves away, and eventually
converges to an orbit that jumps between two values of *x*.
After watching this process a bit,
you can demonstrate the convergence to a *period two orbit* by restarting
the iteration after setting the value in the **trans** box to, say, 100
and hitting **Enter** or the **Reset** button -
this sets a number of iterations to be done to eliminate transient effects before
the first value is plotted.

#### Period 4 and higher

Increase *a* to *a=3.5* and iterate: you will see a *period 4
orbit*. As *a* is slightly increased further (note the jumps we
have taken in *a* are becoming smaller) higher and higher period orbits
are found until at about *a=3.57* a complex orbit with no apparent repetition
is obtained (try this!): this is the *onset of chaos*.

#### Functional composition

As the period of the orbit becomes higher, it is useful to look at a simpler
representation. We can do this by looking not at *f(x)* but at the
function *f(f(x))*. This is effectively looking at every second
iteration of *f(x)*. We will use the notation *f*^{(2)}(x)
for this. More generally we can look at *f*^{(m)}(x) with
*m=2*^{nf}, for example *nf=2* gives
*f*^{(4)}(x)=f(f(f(f(x)))) - now
you see the need for the new notation! Note, of course, that
*f*^{(2)}
is *not* the square of the function *f*. You can study
*f*^{(m)}
with *m=2*^{nf} by setting the value in the **nf** box.
For example, first reset *a* to the value *a=3.5* (remember this gave a
period 4 orbit). Now set *nf* to the value *nf=1*
(remember to hit **Enter** or **Reset**).
The function *f(f(x))* is plotted, and can be iterated in the usual way.
Since this corresponds to "strobing" the orbit with period 2, the period 4
orbit in *f* appears as a period 2 orbit in *f*^{(2)}.

#### Rescaling

Since the orbit of *f*^{(2)} is confined to a small region
near *x=0.5* (at least for the default value of the initial condition
*x0=0.35*), it is convenient to "blow up" this central region. This is
done by the rescaling parameter **nsc**: both *x* and *f* around
*0.5* are
rescaled by (-)^{nsc}
with = 2.502807876... and then plotted between 0 and 1.
(The reason for using this scale factor will become apparent later.) The minus
sign inverts the curve for odd *nsc* as well.
Setting *nsc=1* for *a=3.5*
and *nf=1* will show just the enlarged central region - which (iterate)
actually
looks quite similar to the original period 2 orbit at *a=3.1*. This is the
beginning of the *"scaling similarity"* that is investigated further later.
(To make the picture look even nicer, you can slightly adjust the rescaling
further using the parameter **scale**, which simply rescales *x* and
*f* by this additional factor, e.g. set *scale=1.08* so that the
intersection of the parabola and the diagonal line just fits in the plot region.

#### Geometric convergence

It turns out to be useful to look at a sequence of values of *a*,
*a*_{0} , a_{1} .... given by
a_{m} = a_{c} - C ^{ -m}
where *a*_{c} depends on the map function
(*a*_{c}=3.56995... for the quadratic map), but = 4.6692016091...
for all map functions. Such a
parameter that is
independent of the details of the map is known as a
*universal* parameter. The value of *m* can be used to
step through values of *a* yielding *2*^{m} orbits
(the symbol *I* in the list
box for m corresponds to infinity i.e. *a=a*_{c}) The value of
*C* can be chosen
to give particular "types" of orbit: the applet has preset values for
the "superstable" orbit, where *a* is adjusted so that one point on the
orbit is at the maximum of the function, the "instability" point where the
*2*^{m} just becomes unstable, and "bandmerging" values which
are useful to study the chaotic dynamics for *a > a*_{c}. Study
this for both the quadratic and sine map, choosing successive values of *m*
and using *nf* and *nsc* to investigate
*f*^{ 2nf} rescaled.
#### Putting it all together

Become familiar with the
appearance of the map itself at *a=a*_{m} i.e. (*nf=nsc=0*),
the *2*^{m} th
iterate of the map (*nf=m, nsc=0*) and other iterates, and the blown up
central portion
*(nf=nsc=m)*.

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Last modified Thursday, September 26, 1996

Michael Cross