A n iterative procedure is used to draw the Logistic Map.
Figure 1: The logistic map.
The sequence given by x(n+1) = r * x(n) * ( 1 - x(n) ), x(0) = random() 0<=r<=4 where random() is a uniform random number between 0 and 1 can be used to generate the Logistic Map. The x values after 200 iterations are plotted as a function of r to produce the logistic map. For r = 2.8, the process converges to a single value. For r = 3.3, the process oscillates between two values. For r = 3.5, the process oscillates between four values. For r = 3.8, the process is chaotic even though it is deterministic. Plotting pairs of points ( x(n), x(n+1) ) reveals a one dimensional curve (parabola). The logistic map is a fractal because if any part of it is expanded, it reveals a similar structure. The logistic map shows bifurcating behavior. Small changes in r can change the set of x values from a finite number to an infinite number. There are other variations of this algorithm which are also called the Logistic equation, e.g. x(n+1) = x(n) + r * x(n) * ( 1 - x(n) ), x(0) = random() 0<=r<=3.
Iterative equation for Logistic Map
The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.
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Bassingthwaighte JB, Liebovitch LS, and West BJ. Fractal Physiology. New York, London: Oxford University Press, 1994, 364 pp.
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The National Simulation Resource, Director J. B. Bassingthwaighte, Department of Bioengineering, University of Washington, Seattle WA 98195-5061.
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