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:


Last modified Friday, June 6, 1997
Michael Cross, mcc@caltech.edu