Bendixson transformation

The mapping

$$u=\frac{4x}{x^2+y^2},\quad v=\frac{4y}{x^2+y^2}$$

of the Euclidean $(x,y)$-plane punctured at the point $(0,0)$ into a similar Euclidean $(u,v)$-plane. It is the coordinate expression of the bijection $\phi$ generated by the Bendixson sphere. If the planes $(u,v)$ and $(x,y)$ coincide, the Bendixson transformation is the inversion of the plane $(x,y)$ with respect to the circle $x^2+y^2=4$.

References

[1]	A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)