Sensitivity to Initial Conditions for the Logistic Map

This Demonstration shows the evolution of the distance between two orbits of the logistic map , where or . The two orbits are initially separated by a perturbation of size . The plot is of versus for an orbit starting at , perturbation and parameter . An estimate of the error amplification factor and the Lyapunov exponent are shown at the top of the graphic.

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.