Details

This Demonstration shows how the distance between an orbit of the logistic map starting at and a perturbed orbit starting at evolves. The initial perturbation is the starting error . The error changes on average with a rate (the error amplification factor), so that the error after iterations is roughly given by (see [1, Section 10.1] for more details). The Lyapunov exponent is . The plot is of versus for an orbit starting at with perturbation and parameter . You can explore how changes with:

the starting value ,

the initial size of the perturbation ,

the parameter of the logistic map: .

The Lyapunov exponent is estimated by means of the slope of the linear fitting of . The expansion factor is , and . (This exponent is better estimated by for ; see the Related Link "Lyapunov Exponents for the Logistic Map" and [1]).

The iterations stop if or or . The starting point of the perturbed orbit is if .

Reference

[1] H.-O. Peitgen, H. Jürgens and D. Saupe, Chaos and Fractals: New Frontiers of Science, 2nd ed., New York: Springer, 2004.