Logistic Map - Solution in Some Cases

Solution in Some Cases

The special case of r = 4 can in fact be solved exactly, as can the case with r = 2; however the general case can only be predicted statistically. The solution when r = 4 is,

where the initial condition parameter is given by . For rational, after a finite number of iterations maps into a periodic sequence. But almost all are irrational, and for irrational never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2n shows the exponential growth of stretching, which results in sensitive dependence on initial conditions, while the squared sine function keeps folded within the range .

By contrast, the solution when r=2 is

for . Since for any value of other than the unstable fixed point 0, the term goes to 0 as n goes to infinity, so goes to the stable fixed point