Experiment: Rates of Convergence

Goal: In this experiment you will investigate how quickly (or slowly) an orbit is attracted to a fixed point. Your goal is to relate this speed of convergence to the value of the derivative at the fixed point.

Procedure: Each of the functions listed below has a fixed point and the orbit of 0.123 is attracted to this point. For each function, compute the orbit of 0.123 until it "reaches" the fixed point, or else comes within some designated distance from the fixed point, say (10^(-3)).  We say "reaches" in quotes because, in actuality, the orbit will never reach the fixed point. However, because of computer round-off, it may appear that the orbit has landed on the fixed point.
For each function you should record:
a. The fixed point p.
b. |F' (p)|
c. Is p atracting or neutral?
d. The number of iterations necessary for the orbit of 0.123 to "reach" p.
The functions you should try are:
a. F(x)  = (x^2) + 0.25
b. F(x)  = (x^2)
c. F(x)  = (x^2) - 0.24
d. F(x)  = (x^2) - 0.75
e. F(x)  = 0.4  x(1 - x)
f. F(x)  = x(1 - x)
g. F(x)  = 1.6  x(1 - x)
h. F(x)  = 2  x(1 - x)
i. F(x)  = 2.4  x(1 - x)
j. F(x)  = 3  x(1 - x)
k. F(x)  = 0.4  Sin[x]
l. F(x)  = Sin[x]

Results: After compiling the data, compare the results for each function. In an essay, describe what you have observed. What is the relationship between  |F' (p)|and the speed of convergence of p? Which fixed points have fastest convergence and which have slowest?

Notes and Questions: Each of the following functions has a cycle of period 2 that attracts the orbit of 0.2. In each case, determine the cycle experimentally and estimate the derivative of (F^2)along the cycle. (You need not solve the equations explicitly to get exact values, although this can be done.) Again compare the attracting vs. neutral cases.
a. F(x)  = (x^2) - 1
b. F(x)  = (x^2) - 1.1
c. F(x)  = (x^2) - 1.25.
Hint: The 2-cycle is given by (-1±(Sqrt[2]))/2.

Define the function

You should change the definition of Equation for each particular case under analysis. Make sure you evaluate this cell before any other in the notebook.

Equation = Function[x, (x^2) - 0.75];

Find the attracting fixed points

This cell solves the equation f(x) = x to find the fixed points of Equation.

Reduce[Equation[x] - x == 0, x]

The next cell is used to find the type of each fixed point, (i.e., attracting, repelling or neutral). Replace the value of x found above in the following cell for p to find the absolute value of the derivative of f(x) at p, i.e., |f' (p)|.

p = - .5;
Abs[Equation'[p]]

Test speed of convergence

The next cell gives the number of iterations n for which the distance |(f^n)(seed) - p| is smaller than ϵ. Also, it shows the value of |(f^n)(seed) - p| and displays the time series. The previous evaluation defined the value of p to be used here. You should specify the value of ϵ and the value of InitialSeed. A horizontal solid red line shows the fixed point and two dashed red lines shows the limits given by p ±  ϵ.

ϵ = (10^(-1));
InitialSeed = 0.123;
Orbit = 
        NestWhileList[Equation, InitialSeed, 
          Abs[# - p] > ϵ &];
n = Length[Orbit] - 1
Nest[Equation, InitialSeed, n]
$TextStyle = {FontFamily -> "Times", 
          FontSize -> 17};
Fixedpointline = 
        Graphics[{RGBColor[1, 0, 0], 
            Line[{{0, p}, {n, p}}]}];
Limitlines = 
        Graphics[{Dashing[{0.05, 0.05}], RGBColor[1, 0, 0], 
            Line[{{0, p + ϵ}, {n, p + ϵ}}], 
            Line[{{0, p - ϵ}, {n, 
                  p - ϵ}}]}];
\(\(TimeSeries = 
        ListPlot[Orbit, PlotJoined -> True, Frame -> True, 
          FrameLabel -> {"n", \*""\!\(x\_n\)""}, 
          PlotLabel -> \ \*""Time series for \!\(x\_0\) ="" <> 
              ToString[N[InitialSeed]], RotateLabel -> False, 
          DisplayFunction -> Identity];\)\)
Show[TimeSeries, Fixedpointline, Limitlines, 
        DisplayFunction -> $DisplayFunction, 
        ImageSize -> 500];
Clear[n];