Monday, October 09, 2006

Chaos: Fractal Basin Boundaries

Chaos always makes nice background images for your desktop systems. This is my current image:

Basin boundaries arise in dissipative dynamical systems when two, or more, attractors are present. In such situations each attractor has a basin of initial conditions which lead asymptotically to that attractor. The basin boundaries are the sets which separate different basins. It is very common for basin boundaries to contain unstable chaotic sets. In such cases the basin boundaries can have very complicated fractal structure. Because of this complicated very fine-scaled structure, fractal basin boundaries can pose an impediment to predicting long-term behavior. In particular, if an initial condition is specified with only finite precision, it may be very difficult a priori to determine in which basin it lies if the boundaries are fractal.

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