Fractal Nature of Chaos
Simulation
The fractal nature of chaos is most easily shown using the iteration of two variable maps, i.e.
xn+1=X(xn,yn)
yn+1=Y(xn,yn)
with some simple functions chosen for X and Y. This applet iterates these equations for a number
of well known choices for the functions. The first five choices (Henon, Duffing, Circle, Bakers' and Yorke
maps) show dissipative chaos with the
iterated points collapsing to a complicated and often beautiful fractal structure independent of the
coordinates of the initial point chosen to start the iteration. The sixth choice
(the "Standard Map") is area preserving and derives from Hamiltonian (non-dissipative) dynamics.
Here there is no collapse onto an attractor: instead every initial condition leads to a different orbit, and
starting with many initial conditions builds up an amazingly rich structure.
With this applet you can:
- display the attractor or orbits for the different 2d maps;
- continue the iteration with a new initial condition by clicking once on the running plot;
- zoom in on any portion of the plot (by clicking on two points of the stopped plot forming the corners
of the desired magnification) to see the structure within structure characteristic of a fractal;
- calculate the dimension of the attractor (or a magnified portion of the attractor) using a
simple "box counting" algorithm.
Last modified Friday, June 6, 1997
Michael Cross, mcc@caltech.edu