Lyapunov Exponents for the Logistic Map

This Demonstration plots the orbit diagram of the logistic map and the corresponding Lyapunov exponents for different ranges of the parameter .
The Lyapunov exponent is a parameter characterizing the behavior of a dynamical system. It gives the average rate of exponential divergence from nearby initial conditions. The Lyapunov exponent of the logistic map is given by .
If the Lyapunov exponent is positive, then the system is chaotic; if it is negative, the system will converge to a periodic state; and if it is zero, there is a bifurcation.
By dragging the locator to the left or right or clicking the plot, you can scroll through the whole range of -values (0.70–1.0), generate the bifurcation diagram, and plot the Lyapunov exponent over that range.
You can zoom to the position of the locator by using the zoom sliders. To replot the graphs at higher zoom scales, use the "detail" button to increase the number of values and the number of iterations to 5000.


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


Snapshot 1: period 4-8-16 bifurcations
Snapshot 2: period 3-6-12 bifurcations
Snapshot 3: period 5-10-20 bifurcations
    • 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.

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+