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A kind of pattern of the trajectories of an autonomous system of ordinary second-order differential equations (*) in a neighbourhood of a singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212301.png" />, where

<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/c/c021/c021230/c0212302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212304.png" /> is a domain of uniqueness. This kind of pattern is characterized as follows. There exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212305.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212306.png" /> such that all trajectories of the system starting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212307.png" /> are closed curves going around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212308.png" />. Here the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c0212309.png" /> itself is also called a centre. In the figure the point 0 is the centre. The motion along trajectories with increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123010.png" /> 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.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c021230a.gif" />

Figure: c021230a

A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123011.png" /> is a centre for a system (*), for example when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123013.png" /> 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|saddle]], a [[Node|node]] or a [[Focus|focus]]), a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123014.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123015.png" /> 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|Centre and focus problem]]). For a non-linear system (*) of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123016.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123017.png" />) a rest point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123018.png" /> can be a centre also in the case when the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021230/c02123019.png" /> has two zero eigen values.

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

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Amel'kin,   N.A. Lukashevich,   A.P. Sadovskii,   "Non-linear oscillations in second-order systems" , Minsk  (1982)  (In Russian)</TD></TR></table>



====Comments====
For a precise topological definition see [[#References|[a1]]], p. 71.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.V. Nemytskii,   V.V. Stepanov,   "Qualitative theory of differential equations" , Princeton Univ. Press  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  V.I. Arnol'd,   "Geometrical methods in the theory of ordinary differential equations" , Springer  (1983)  (Translated from Russian)</TD></TR></table>