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The sphere in real analysis which is known as the [[Riemann sphere|Riemann sphere]] in the theory of functions of a complex variable.
 
The sphere in real analysis which is known as the [[Riemann sphere|Riemann sphere]] in the theory of functions of a complex variable.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155401.png" /> be the unit sphere in the Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155402.png" />-space, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155403.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155404.png" /> be its north and south pole, respectively; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155405.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155406.png" /> be planes tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155407.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155408.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b0155409.png" /> respectively; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554011.png" /> be coordinate systems in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554013.png" /> with axes parallel to the corresponding axes of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554014.png" /> in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554015.png" /> and pointing in the same directions; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554016.png" /> be the [[Stereographic projection|stereographic projection]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554017.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554018.png" /> from the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554019.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554020.png" /> be the stereographic projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554021.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554022.png" /> from the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554023.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554024.png" /> is the Bendixson sphere with respect to any one of the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554026.png" />. It generates the bijection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554027.png" /> of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554028.png" /> (punctured at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554029.png" />) onto the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554030.png" />, which is punctured at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554031.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015540/b01554032.png" />. Named after I. Bendixson.
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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|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Andronov,  E.A. Leontovich,  I.I. Gordon,  A.G. Maier,  "Qualitative theory of second-order dynamic systems" , Wiley  (1973)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Andronov,  E.A. Leontovich,  I.I. Gordon,  A.G. Maier,  "Qualitative theory of second-order dynamic systems" , Wiley  (1973)  (Translated from Russian)</TD></TR></table>

Latest revision as of 10:58, 29 May 2020


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)
How to Cite This Entry:
Bendixson sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bendixson_sphere&oldid=19212
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article