If enough paths are drawn, it will be. In general (in the mathematical sense of "in every case", not "usually"), all paths cover every point in the plane, and no two paths ever intersect.
Anyway, I didn't find any graphing programs for general systems of ordinary differential equations that would draw more than a small number of (usually unsymmetrical) paths. Since ODEs are pretty cool, I decided to start writing one.
The "black hole" system is described by
\\<br />
\frac{dy}{dt} = x + y (1-x^2-y^2)<br />
\end{cases}<br />
)
A few additional ones that I thought turned out pretty well:
Some lines are broken because it only plots systems in unidirectional time, for now.
Also, the odd helix-shape in the last image is due to a (very small) time dependency, which I didn't introduce into any of the other systems. The greater the time dependency, the more chaotic paths around the stationary point in the vortex become.
Time dependency in ODEs tends to make organized paths "fall apart", which is a kind of neat example of how time and entropy are strongly related, I think.
"Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today." - Stephen Wolfram
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