Poincaré sphere

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)