Logistic Map Bifurcation Diagram
The following is an javascript interface that shows how the bifurcation diagram for the Logistic Map is constructed. The logistic map is a discrete dynamical system, that exhibits chaotic behavior for certain values of its parameter, r. The bifurcation diagram is a numerical method for showing the asymptotic behavior of the logistic map for various values of the parameter, r.
The algorithm works as follows:
Beginning with a certain values for x and r, find n points which represent the periodic attractors of the logistic map. These periodic attractors will be approximated numerically. The algorithm involves "running off" a large number of iterations and retaining the last n as the approximate values.
- x = x0
- for i = 1 to N - n; x = f(x); loop
- for i = 1 to n; plot x; x = f(x); loop
Suggestions
Enter a value for r between 0 and 4 and a number of iterations N of the logistic map to use. The bifurcation diagram will plot the last n iterations of the logistic map with your chosen parameter. If the periodicity of the logistic map of with the specified parameter is greater than n, not all periodic points will be displayed so choose n wisely. It is interesting to choose n to be at least double the periodicity of the map so that you can see how closely the periodic points are estimated. You can specify a list of values for r separated by spaces (e.g. 1 2 3 3.2). You can also specify ranges of values for r in MATLAB format (e.g. 0:0.1:3 is shorthand for the list 0 0.1 0.2 ... 2.9 3).
- n=2, N=10000, x0=0.2, r=2 (x=1/2 is an attractive fixed point)
- n=8, N=10000, x0=0.2, r=3.2 (x=0.5130,0.7995 are approximate period 2 points)
Notice: only two points are visible, even though we have asked for 8. This is evidence to suggest that the map is 2-periodic.
- n=8, N=10000, x0=0.2, r=3.54 (x=0.5218,0.8833,0.3648,0.8203 are approximate period 4 points)
- n=1000, N=50000, x0=0.2, r=4 (the chaotic attractor of this parameter value is the interval (0,1))
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