An Example of Three-Dimensional Chaos

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration illustrates the periodic or chaotic behavior and the sensitivity to initial conditions of three-dimensional maps of the form:


x'(t)=f(x(t),y(t),z(t)) y'(t)=g(x(t),y(t),z(t)) z'(t)=h(x(t),y(t),z(t))

The functions , , and here are nonlinear differential equations with three parameters , , and . For a particular selection of these, you can observe periodic behavior, period doubling, or chaotic behavior. Both the Rössler (map 4) and Lorentz (map 5) systems can be considered as special cases. Try your own set of equations by changing the functions in the program.


Contributed by: Erik Mahieu (December 2011)
Open content licensed under CC BY-NC-SA



The systems considered here are as follows:

map 1:

map 2:

map 3 (Rössler):

map 4:

map 5 (Lorentz):


[1] J. L. Casti, "Order in Chaos," chap. 4 in Reality Rules: I, New York: John Wiley & Sons, 1992 p. 355.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.