11562

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.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+