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 (- alpha

)^nsc with alpha

= 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, a₀ , a₁ .... given by a_m = a_c - C delta

^-m where a_c depends on the map function (a_c=3.56995... for the quadratic map), but delta

= 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^{2^nf} 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