Difference between revisions of "Poincaré sphere"
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| − | The sphere in the space | + | The sphere in the space $\mathbf R^{3}$ |
| − | with diametrically-opposite points identified. The Poincaré sphere is diffeomorphic to the projective plane | + | 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{x} = P ( x , y ) ,\ \ | ||
\dot{y} = Q ( 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: | |
| − | |||
| − | |||
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| − | |||
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| − | An equivalent method of investigating the system | ||
a) | a) | ||
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The first (respectively, the second) is suitable outside a sector containing the $ y $- | The first (respectively, the second) is suitable outside a sector containing the $ y $- | ||
| − | axis ( $ x $- | + | axis ( $ x $-axis). For example, the transformation a) reduces the system \eqref{e1} to the form |
| − | axis). For example, the transformation a) reduces the system | ||
| − | + | \begin{equation} \label{e2} | |
\frac{du}{d \tau } | \frac{du}{d \tau } | ||
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\frac{dz}{d \tau } | \frac{dz}{d \tau } | ||
= Q ^ {*} ( u , z ) , | = Q ^ {*} ( u , z ) , | ||
| − | + | \end{equation} | |
where $ d t = z ^ {n} d \tau $ | where $ d t = z ^ {n} d \tau $ | ||
and $ n $ | and $ n $ | ||
is the largest of the degrees of $ P $, | is the largest of the degrees of $ P $, | ||
| − | $ Q $ | + | $ 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 $ | and $ Q $ | ||
are coprime, then the polynomials $ P ^ {*} $ | are coprime, then the polynomials $ P ^ {*} $ | ||
and $ Q ^ {*} $ | and $ Q ^ {*} $ | ||
| − | are also coprime and the system | + | are also coprime and the system \eqref{e1} has a finite number of singular points at infinity. |
====References==== | ====References==== | ||
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) |
Poincaré sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_sphere&oldid=53057