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The emergence of chaotic dynamics from simpler behavior may be observed in this example called the logistic map.
The iterative equation:
x:=r*x*(1-x)
exhibits a variety of behaviors, depending upon its initial value, and the value of the parameter r.
To explore the behavior of this equation, set values for r and x, return to the home cell, and select Evaluate from the Special menu.
For a value of the parameter r of 0.4, and an initial x value of 0.7, successive values of x approach zero and stay there.
For an r value of 2.4, and initial x value 0f 0.7, successive values of x approach a constant 0.583.
For r=3.0, and an initial x, of 0.5; an extended damped oscillation takes place between between two numbers.
For r=3.5, initial x=0.7; an extended oscillation among four numbers takes place.
For r=3.8, initial x=0.7; a chaotic sequence of numbers is the result.
For more information on the period doubling approach to chaos, see the book:
Creating Artificial Life, by Edward Rietman; from which this example was taken.
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