Strange Attractors
An attractor is called a strange attractor if flow lines depend sensitively
on the initial values.
In a strange attractor, initial points that are arbitrarily close to each other
are macrocopically separated by the flow after long time intervals.
The sensitive dependence on initial conditions has practical consequences.
Small deviations of initial conditions are always present, therefore the position of the trajectory inside a strange attractor is not accurately predictable.
The longer the time interval of a prediction, the less that can be said about the future state. This affects, for example, the long-term weather forecast.
Integrating an ODE system that exhibits a strange attractor is to some extent a nondeterministic problem; limits of computability are reached.
The following picture illustrates these comments. From two starting points close
to each other, the two flows are quite different.
The equation of the Lorenz attractor is:
- x'=s(y-x)
- y'=rx-y-xz
- z'=-bz+xy
For s=10, r=28, and b=8/3, the solution set is:
(starting at point (1,1,1))
If r is set to 100, a limit cycle appears.
