View source for Centro-focus

A kind of pattern of the trajectories of an autonomous system of ordinary second-order differential equations (*) in a neighbourhood of an isolated singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021320/c0213201.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/c021320/c0213202.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/c021320/c0213203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021320/c0213204.png" /> is a domain of uniqueness. This type is characterized as follows: In every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021320/c0213205.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021320/c0213206.png" /> there exist closed trajectories of the system going around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021320/c0213207.png" /> and also complete trajectories that are not closed; the latter fill out annulus-shaped domains contracting to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021320/c0213208.png" /> and bounded by the closed trajectories; the trajectories in the interiors of an arbitrary annulus are spirals for which one end asymptotically approximates the outer boundary, while the other end asymptotically approximates the inner boundary of the annulus. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021320/c0213209.png" /> itself is also called a centro-focus.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c021320a.gif" />

Figure: c021320a

In the figure, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021320/c02132010.png" /> is a centro-focus; the arrows indicate the direction of motion along the trajectories of the system with increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021320/c02132011.png" /> (they may also go in the opposite direction).

A centro-focus is Lyapunov (but not asymptotically) stable. Its Poincaré index is 1.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) {{MR|0121520}} {{ZBL|0089.29502}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> H. Dulac, "Sur les cycles limites" ''Bull. Soc. Math. France'' , '''51''' (1923) pp. 45–188 {{MR|1504823}} {{ZBL|49.0304.01}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> Yu.S. Il'yashenko, "Dulac's memoir "On limit cycles" and related problems of the local theory of differential equations" ''Russian Math. Surveys'' , '''40''' : 6 (1985) pp. 1–49 ''Uspekhi Mat. Nauk'' , '''40''' : 6 (1985) pp. 41–78</TD></TR></table>