Logistic map: x_n+1 = rx_n(1 - x_n)

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Background: The logistic map is a simple nonlinear difference equation that models the population of a species with the passing of each generation (or similar fixed length of time). In the above equation x represents the ratio of the population to its maximum possible value, the first term (rx) represents reproduction, the second term (-rx²) represents starvation, and the growth-rate parameter r is between 0 and 4. Despite the equation's simplicity and depending upon the value of r, the solutions can be remarkably complex, giving rise to period doubling and chaos. The table below summarizes some of this. See Wikipedia for more information.

rmin	rmax	asymptotic behavior
0	1	converges to zero
1	3	converges to nonzero value
3	3.44949	two cycle
3.44949	3.54409	four cycle
3.54409	3.56441	eight cycle
3.56441	3.56995	more period doubling
3.56995	4	chaos

Instructions: After ...herokuapp.com above you should type either /json/ or /svg/ depending upon whether you want the results returned numerically or graphically. After that you should type the following two to four numerical inputs separated by hyphens:

1. growth-rate parameter r
2. number of generations to be calculated
3. (optional) initial value of x (If omitted this will be generated randomly.)
4. (optional) first generation to include in the results (If omitted this will include all generations.)
For instance the url https://logistic-map-ed5bb7ec94fe.herokuapp.com/svg/3.54-100-0.42 should generate a graphic of an unstable two-cycle which evolves to a stable four-cycle after about 25 generations.

creator: Peter Knipp