Homemade name.

 

Cartesian parametrization: .

The 2D basins are the curves with the above parametrization.
 

The 2D basins  are the projections on the planes containing Oz of the 3D basins , traced on a paraboloid of revolution, hence their basin-like shape.

The formula  shows that they are entirely traced on the parabola  and the line y = b.
Some examples:
 

n = 2, phi =0:, : rational quartic.	n = 3, phi = pi/2:, , rational sextic.
n = 3/2, phi =0:	n = 3/2, phi = pi/2:

 

Any basin  is associated to the Lissajous curve: ; opposite, the basin and its associated Lissajous curve for .

 

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© Robert FERRÉOL 2017