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                                         Sequences -- Chaos

The cell below examines the dynamical system

f[p_] := 4.0 (1 - 0.001 p) p

pop[1] := 50

pop[n_] := f[pop[n - 1]]

and counts how many of the first 1,000 terms in the sequence fall between 0 and 500.  If the sequence is truly "random" or "chaotic" we would expect roughly one-half of the terms to be between 0 and 500.  Evaluate the next cell now.  Notice the syntax.  The third line is strange.  This line forces Mathematica  to save the sequence as it computes it.  this saves a lot of time.  This line must be typed exactly as shown.
;[s]
9:0,2;59,0;107,2;176,0;382,1;411,4;476,5;487,4;598,3;599,-1;
6:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,65535,0,0;2,13,9,Times,1,12,0,0,0;1,13,9,Times,1,12,65535,0,65535;2,13,9,Times,1,12,0,0,65535;1,13,9,Times,3,12,0,0,65535;
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f[p_] := 4.0 (1 - 0.001 p) p
pop[1] :=50
pop[n_] := pop[n] = f[pop[n - 1]]
;[s]
3:0,0;41,1;74,0;75,-1;
2:2,12,10,Courier,1,12,0,0,0;1,12,10,Courier,1,12,0,0,65535;
:[font = input; preserveAspect; ]
Ct = 0;
For[i = 1, i <= 1000, i = i + 1, 
    If[pop[i] <= 500, Ct = Ct + 1]]
Print[Ct]
^*)