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

Read more about this topic:  Logistic Map

Famous quotes containing the words solution and/or cases:

    To the questions of the officiously meddling police Falter replied absently and tersely; but, when he finally grew tired of this pestering, he pointed out that, having accidentally solved “the riddle of the universe,” he had yielded to artful exhortation and shared that solution with his inquisitive interlocutor, whereupon the latter had died of astonishment.
    Vladimir Nabokov (1899–1977)

    And in cases where profound conviction has been wrought, the eloquent man is he who is no beautiful speaker, but who is inwardly drunk with a certain belief. It agitates and tears him, and perhaps almost bereaves him of the power of articulation.
    Ralph Waldo Emerson (1803–1882)