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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. | + | 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