# Bendixson sphere

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.