Poincaré sphere

The sphere in the space with diametrically-opposite points identified. The Poincaré sphere is diffeomorphic to the projective plane ; it was introduced by H. Poincaré (see ) to investigate the behaviour at infinity of the phase trajectories of a two-dimensional autonomous system

(1)

when and are polynomials. The Poincaré sphere is usually depicted so that it touches the -plane; the projection from the centre of the Poincaré sphere gives a one-to-one mapping onto , 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.

An equivalent method of investigating the system (1) is to apply a Poincaré transformation:

a)

or

b)

The first (respectively, the second) is suitable outside a sector containing the -axis (-axis). For example, the transformation a) reduces the system (1) to the form

(1prm)

where and is the largest of the degrees of , ; the singular points of the system (1prm) are called the singular points at infinity of the system (1). If the polynomials and are coprime, then the polynomials and are also coprime and the system (1) has a finite number of singular points at infinity.

References

[1a]	H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 7 (1881) pp. 375–422
[1b]	H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 8 (1882) pp. 251–296
[1c]	H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" J. de Math. , 1 (1885) pp. 167–244
[1d]	H. Poincaré, "Mémoire sur les courbes définiés par une équation differentielle" 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)