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The sphere in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p0731201.png" /> with diametrically-opposite points identified. The Poincaré sphere is diffeomorphic to the projective plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p0731202.png" />; it was introduced by H. Poincaré (see ) to investigate the behaviour at infinity of the phase trajectories of a two-dimensional autonomous system
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p0731203.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|auto}}{{TEX|done}}
  
when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p0731204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p0731205.png" /> are polynomials. The Poincaré sphere is usually depicted so that it touches the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p0731206.png" />-plane; the projection from the centre of the Poincaré sphere gives a one-to-one mapping onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p0731207.png" />, and, moreover, a point at infinity corresponds to a pair of diametrically-opposite points on the equator. Accordingly the phase trajectories of the system (1) map onto curves on the sphere.
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The sphere in the space $\mathbf R^{3}$
 +
with diametrically-opposite points identified. The Poincaré sphere is diffeomorphic to the projective plane $\mathbf R P  ^ {2} $.  
 +
It was introduced by H. Poincaré to investigate the behaviour at infinity of the phase trajectories of a two-dimensional autonomous system
 +
\begin{equation}\label{e1}
 +
\dot{x}  = P ( x , y ) ,\ \
 +
\dot{y}  = Q ( x , y )
 +
\end{equation}
 +
when $P$ and $Q$ are polynomials. The Poincaré sphere is usually depicted so that it touches the $  ( x , y ) $-plane; the projection from the centre of the Poincaré sphere gives a one-to-one mapping onto $  \mathbf R P  ^ {2} $, and, moreover, a point at infinity corresponds to a pair of diametrically-opposite points on the equator. Accordingly the phase trajectories of the system \eqref{e1} map onto curves on the sphere.
  
An equivalent method of investigating the system (1) is to apply a Poincaré transformation:
+
An equivalent method of investigating the system \eqref{e1} is to apply a Poincaré transformation:
  
 
a)
 
a)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p0731208.png" /></td> </tr></table>
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$$
 +
x =  
 +
\frac{1}{z}
 +
,\  y =  
 +
\frac{u}{z}
 +
,
 +
$$
  
 
or
 
or
Line 15: Line 35:
 
b)
 
b)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p0731209.png" /></td> </tr></table>
+
$$
 +
x =  
 +
\frac{u}{z}
 +
,\  y =  
 +
\frac{1}{z}
 +
.
 +
$$
  
The first (respectively, the second) is suitable outside a sector containing the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p07312010.png" />-axis (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p07312011.png" />-axis). For example, the transformation a) reduces the system (1) to the form
+
The first (respectively, the second) is suitable outside a sector containing the $  y $-
 +
axis ( $  x $-axis). For example, the transformation a) reduces the system \eqref{e1} to the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p07312012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1prm)</td></tr></table>
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\begin{equation} \label{e2}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p07312013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p07312014.png" /> is the largest of the degrees of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p07312015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p07312016.png" />; the singular points of the system (1prm) are called the singular points at infinity of the system (1). If the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p07312017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p07312018.png" /> are coprime, then the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p07312019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073120/p07312020.png" /> are also coprime and the system (1) has a finite number of singular points at infinity.
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\frac{du}{d \tau }
 +
  = P  ^ {*} ( u , z ) ,\ \
 +
 
 +
\frac{dz}{d \tau }
 +
  = Q  ^ {*} ( u , z ) ,
 +
\end{equation}
 +
 
 +
where  $  d t = z  ^ {n}  d \tau $
 +
and $  n $
 +
is the largest of the degrees of $  P $,  
 +
$  Q $. The singular points of the system \eqref{e2} are called the singular points at infinity of the system \eqref{e1}. If the polynomials $  P $
 +
and $  Q $
 +
are coprime, then the polynomials $  P  ^ {*} $
 +
and $  Q  ^ {*} $
 +
are also coprime and the system \eqref{e1} has a finite number of singular points at infinity.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> H. Poincaré,   "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''7'''  (1881)  pp. 375–422</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> H. Poincaré,   "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''8'''  (1882)  pp. 251–296</TD></TR><TR><TD valign="top">[1c]</TD> <TD valign="top"> H. Poincaré,   "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''1'''  (1885)  pp. 167–244</TD></TR><TR><TD valign="top">[1d]</TD> <TD valign="top"> H. Poincaré,   "Mémoire sur les courbes définiés par une équation differentielle"  ''J. de Math.'' , '''2'''  (1886)  pp. 151–217</TD></TR><TR><TD valign="top">[2]</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><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Lefschetz,  "Differential equations: geometric theory" , Interscience  (1957)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1a]</TD> <TD valign="top"> H. Poincaré, "Mémoire sur les courbes définies par une équation différentielle"  ''J. de Math.'' , '''7'''  (1881)  pp. 375–422</TD></TR>
 +
<TR><TD valign="top">[1b]</TD> <TD valign="top"> H. Poincaré, "Mémoire sur les courbes définies par une équation différentielle"  ''J. de Math.'' , '''8'''  (1882)  pp. 251–296</TD></TR>
 +
<TR><TD valign="top">[1c]</TD> <TD valign="top"> H. Poincaré, "Mémoire sur les courbes définies par une équation différentielle"  ''J. de Math.'' , '''1'''  (1885)  pp. 167–244</TD></TR>
 +
<TR><TD valign="top">[1d]</TD> <TD valign="top"> H. Poincaré, "Mémoire sur les courbes définies par une équation différentielle"  ''J. de Math.'' , '''2'''  (1886)  pp. 151–217</TD></TR>
 +
<TR><TD valign="top">[2]</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>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  S. Lefschetz,  "Differential equations: geometric theory" , Interscience  (1957)</TD></TR>
 +
</table>

Latest revision as of 07:59, 21 March 2023


The sphere in the space $\mathbf R^{3}$ with diametrically-opposite points identified. The Poincaré sphere is diffeomorphic to the projective plane $\mathbf R P ^ {2} $. It was introduced by H. Poincaré to investigate the behaviour at infinity of the phase trajectories of a two-dimensional autonomous system \begin{equation}\label{e1} \dot{x} = P ( x , y ) ,\ \ \dot{y} = Q ( x , y ) \end{equation} when $P$ and $Q$ are polynomials. The Poincaré sphere is usually depicted so that it touches the $ ( x , y ) $-plane; the projection from the centre of the Poincaré sphere gives a one-to-one mapping onto $ \mathbf R P ^ {2} $, and, moreover, a point at infinity corresponds to a pair of diametrically-opposite points on the equator. Accordingly the phase trajectories of the system \eqref{e1} map onto curves on the sphere.

An equivalent method of investigating the system \eqref{e1} is to apply a Poincaré transformation:

a)

$$ x = \frac{1}{z} ,\ y = \frac{u}{z} , $$

or

b)

$$ x = \frac{u}{z} ,\ y = \frac{1}{z} . $$

The first (respectively, the second) is suitable outside a sector containing the $ y $- axis ( $ x $-axis). For example, the transformation a) reduces the system \eqref{e1} to the form

\begin{equation} \label{e2} \frac{du}{d \tau } = P ^ {*} ( u , z ) ,\ \ \frac{dz}{d \tau } = Q ^ {*} ( u , z ) , \end{equation}

where $ d t = z ^ {n} d \tau $ and $ n $ is the largest of the degrees of $ P $, $ Q $. The singular points of the system \eqref{e2} are called the singular points at infinity of the system \eqref{e1}. If the polynomials $ P $ and $ Q $ are coprime, then the polynomials $ P ^ {*} $ and $ Q ^ {*} $ are also coprime and the system \eqref{e1} has a finite number of singular points at infinity.

References

[1a] H. Poincaré, "Mémoire sur les courbes définies par une équation différentielle" J. de Math. , 7 (1881) pp. 375–422
[1b] H. Poincaré, "Mémoire sur les courbes définies par une équation différentielle" J. de Math. , 8 (1882) pp. 251–296
[1c] H. Poincaré, "Mémoire sur les courbes définies par une équation différentielle" J. de Math. , 1 (1885) pp. 167–244
[1d] H. Poincaré, "Mémoire sur les courbes définies par une équation différentielle" J. de Math. , 2 (1886) pp. 151–217
[2] A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)
[3] S. Lefschetz, "Differential equations: geometric theory" , Interscience (1957)
How to Cite This Entry:
Poincaré sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_sphere&oldid=16511
This article was adapted from an original article by M.V. Fedoryuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article