Bendixson sphere
The sphere in real analysis which is known as the Riemann sphere in the theory of functions of a complex variable.
Let be the unit sphere in the Euclidean
-space, and let
and
be its north and south pole, respectively; let
and
be planes tangent to
at the points
and
respectively; let
and
be coordinate systems in
and
with axes parallel to the corresponding axes of the system
in the plane
and pointing in the same directions; let
be the stereographic projection of
onto
from the centre
, and let
be the stereographic projection of
onto
from the centre
. Then
is the Bendixson sphere with respect to any one of the planes
,
. It generates the bijection
of the plane
(punctured at the point
) onto the plane
, which is punctured at the point
. 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
. 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) |
Bendixson sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bendixson_sphere&oldid=46010