Citation information:

Luebeck, R. (2017, Feb 2). An Elementary Tour of the Logistic Map.


An Elementary Tour of the Logistic Map:  X = rX(1 - X)

Iterating this equation produces regions of distinct values, 
involving period doubling, as well as regions of chaos.  The 
rate of convergence of period doubling is identical to that
found in other chaotic systems.

This iterated equation is here graphed in a 
static manner in four ways.  Program code 
for all four graph types are presented.

  1. In a bifurcation diagram, the values of X are 
     plotted against the values of the parameter r.

  2. Plotting X against the previous value of X for 
     a given value of r produces a parabolic curve, 
     always in segments corresponding to the range
     of values that X acquires for that value of r.

     Of special interest is the question of whether 
     a point lands to the left or to the right of 
     the previous point.  Various LRLRRRL patterns 
     are found to repeat.

     We present a table showing some L-R patterning 
     and a table showing L shift : R shift ratios for 
     various values of the parameter r.

  3. Plotting each value of X against each iteration 
     produces a straight line graph which reveals how 
     many iterations are needed for convergence to 
     distinct values for X for a given value of r.  
     Anywhere from just a few iterations to four 
     million iterations prove to be sufficient, 
     depending on the value of r.

  4. Graphing in a circular manner:  Plotting the 
     sine and cosine of an incrementally increasing 
     angle times the ever evolving X value reveals 
     hidden patterns amidst the chaotic zones.