Sensitivity to Initial Conditions for the Logistic Map

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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.

Contributed by: Santos Bravo Yuste (January 2018)
Open content licensed under CC BY-NC-SA


Snapshots


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.



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