Citation information: Luebeck, R. (2017, Feb 2). An Elementary Tour of the Logistic Map. https://chaos-equations.com/logistic-map Annotation: 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. |