# Centre

A kind of pattern of the trajectories of an autonomous system of ordinary second-order differential equations (*) in a neighbourhood of a singular point , where

(*) |

and is a domain of uniqueness. This kind of pattern is characterized as follows. There exists a neighbourhood of such that all trajectories of the system starting in are closed curves going around . Here the point itself is also called a centre. In the figure the point 0 is the centre. The motion along trajectories with increasing can proceed clockwise (indicated in the figure by arrows) or counterclockwise. A centre is Lyapunov stable (but not asymptotically stable). Its Poincaré index is 1.

Figure: c021230a

A point is a centre for a system (*), for example when , where is a constant matrix with pure imaginary eigen values. In contrast to simple rest points of other types that occur for linear second-order systems (a saddle, a node or a focus), a point of centre-type does not, generally speaking, remain a centre under a perturbation of the right-hand side of the linear system, whatever the order of smallness of the perturbations relative to and the order of their smoothness may be. It can then change into a focus (stable or unstable) or into a centre-focus (see Centre and focus problem). For a non-linear system (*) of class () a rest point can be a centre also in the case when the matrix has two zero eigen values.

See also the references to Singular point of a differential equation.

#### References

[1] | V.V. Amel'kin, N.A. Lukashevich, A.P. Sadovskii, "Non-linear oscillations in second-order systems" , Minsk (1982) (In Russian) |

#### Comments

For a precise topological definition see [a1], p. 71.

#### References

[a1] | V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) |

[a2] | A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Theory of bifurcations of dynamic systems on a plane" , Israel Program Sci. Transl. (1971) (Translated from Russian) |

[a3] | V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Springer (1983) (Translated from Russian) |

**How to Cite This Entry:**

Centre.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Centre&oldid=13006