Difference between revisions of "Centro-focus"
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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 an isolated singular point $ x _ {0} $, | |
| + | where | ||
| + | |||
| + | $$ \tag{* } | ||
| + | \dot{x} = f ( x),\ \ | ||
| + | x \in \mathbf R ^ {2} ,\ \ | ||
| + | f: G \rightarrow \mathbf R ^ {2} , | ||
| + | $$ | ||
| + | |||
| + | $ f \in C ( G) $ | ||
| + | and $ G $ | ||
| + | is a domain of uniqueness. This type is characterized as follows: In every neighbourhood $ U $ | ||
| + | of $ x _ {0} $ | ||
| + | there exist closed trajectories of the system going around $ x _ {0} $ | ||
| + | and also complete trajectories that are not closed; the latter fill out annulus-shaped domains contracting to $ x _ {0} $ | ||
| + | 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 $ x _ {0} $ | ||
| + | itself is also called a centro-focus. | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c021320a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/c021320a.gif" /> | ||
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Figure: c021320a | Figure: c021320a | ||
| − | In the figure, | + | In the figure, $ ( 0, 0) $ |
| + | is a centro-focus; the arrows indicate the direction of motion along the trajectories of the system with increasing $ t $( | ||
| + | they may also go in the opposite direction). | ||
A centro-focus is Lyapunov (but not asymptotically) stable. Its Poincaré index is 1. | A centro-focus is Lyapunov (but not asymptotically) stable. Its Poincaré index is 1. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |
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Latest revision as of 09:24, 26 March 2023
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 $ x _ {0} $,
where
$$ \tag{* } \dot{x} = f ( x),\ \ x \in \mathbf R ^ {2} ,\ \ f: G \rightarrow \mathbf R ^ {2} , $$
$ f \in C ( G) $ and $ G $ is a domain of uniqueness. This type is characterized as follows: In every neighbourhood $ U $ of $ x _ {0} $ there exist closed trajectories of the system going around $ x _ {0} $ and also complete trajectories that are not closed; the latter fill out annulus-shaped domains contracting to $ x _ {0} $ 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 $ x _ {0} $ itself is also called a centro-focus.
Figure: c021320a
In the figure, $ ( 0, 0) $ is a centro-focus; the arrows indicate the direction of motion along the trajectories of the system with increasing $ t $( they may also go in the opposite direction).
A centro-focus is Lyapunov (but not asymptotically) stable. Its Poincaré index is 1.
References
| [1] | V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) MR0121520 Zbl 0089.29502 |
| [2] | H. Dulac, "Sur les cycles limites" Bull. Soc. Math. France , 51 (1923) pp. 45–188 MR1504823 Zbl 49.0304.01 |
| [3] | 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 |
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How to Cite This Entry:
Centro-focus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centro-focus&oldid=13952
Centro-focus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Centro-focus&oldid=13952
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article