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I am looking for a differential equation whose solutions are what I call "open" vortices. These vortices are not closed in themselves, but sort of "absorb" the surrounding "fluid" and also "emit" it. I know that the Gross–Pitaevskii equation has vortices as solutions, but these are closed vortices as far as I know.

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  • $\begingroup$ Do you mean a differential equation that has points that are simultaneous spiral sinks and sources? You cannot have that in 2D or 3D, but in 4D you could have a fixed point with two complex eigenvalues of the Jacobian with negative real part and two with positive real part. In 3D you could have a fixed line along which the real part shifted from negative (flow attracted to curve) over zero (center) to positive (flow repelled by line). $\endgroup$ – Anders Sandberg Apr 15 '18 at 20:33
  • $\begingroup$ No a point need not be a sink and source simultaneously. But there should be points that are sources and points that are sinks. $\endgroup$ – user139383 Apr 16 '18 at 8:43
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It is not too hard to come up with dynamical systems with both attractive and repulsive fixed points. Here is a simple one: $$x'=(1-k)\sin(x)+k\sin(y)$$ $$y'=(1-k)\sin(y)-k\cos(x)$$ where $k$ is a mixing constant. Here is the vector field for $k=0.8$.

Vector field with numerous sources and sinks

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