Bendixson sphere

The sphere in real analysis which is known as the Riemann sphere in the theory of functions of a complex variable.

Let $ \Sigma : X ^ {2} + Y ^ {2} + Z ^ {2} = 1 $ be the unit sphere in the Euclidean $ (X, Y, Z) $- space, and let $ N(0, 0, 1) $ and $ S(0, 0, -1) $ be its north and south pole, respectively; let $ \nu $ and $ \sigma $ be planes tangent to $ \Sigma $ at the points $ N $ and $ S $ respectively; let $ xSy $ and $ uNv $ be coordinate systems in $ \sigma $ and $ \nu $ with axes parallel to the corresponding axes of the system $ XOY $ in the plane $ Z = 0 $ and pointing in the same directions; let $ \Pi $ be the stereographic projection of $ \Sigma $ onto $ \sigma $ from the centre $ N $, and let $ \Pi ^ { \prime } $ be the stereographic projection of $ \Sigma $ onto $ \nu $ from the centre $ S $. Then $ \Sigma $ is the Bendixson sphere with respect to any one of the planes $ \sigma $, $ \nu $. It generates the bijection $ \phi = \Pi ^ { \prime } \Pi ^ {-1} $ of the plane $ \sigma $( punctured at the point $ S $) onto the plane $ \nu $, which is punctured at the point $ N $. This bijection is employed in the study of the behaviour of the trajectories of an autonomous system of real algebraic ordinary differential equations of the second order in a neighbourhood of infinity in the phase plane (the right-hand sides of the equations are polynomials of the unknown functions). This bijection reduces the problem to a similar problem in a neighbourhood of the point $ (0, 0) $. Named after I. Bendixson.

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)