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Difference between revisions of "Bendixson transformation"

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The mapping
 
The mapping
  
<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/b/b015/b015550/b0155501.png" /></td> </tr></table>
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$$u=\frac{4x}{x^2+y^2},\quad v=\frac{4y}{x^2+y^2}$$
  
of the Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015550/b0155502.png" />-plane punctured at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015550/b0155503.png" /> into a similar Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015550/b0155504.png" />-plane. It is the coordinate expression of the bijection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015550/b0155505.png" /> generated by the [[Bendixson sphere|Bendixson sphere]]. If the planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015550/b0155506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015550/b0155507.png" /> coincide, the Bendixson transformation is the inversion of the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015550/b0155508.png" /> with respect to the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015550/b0155509.png" />.
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of the Euclidean $(x,y)$-plane punctured at the point $(0,0)$ into a similar Euclidean $(u,v)$-plane. It is the coordinate expression of the bijection $\phi$ generated by the [[Bendixson sphere|Bendixson sphere]]. If the planes $(u,v)$ and $(x,y)$ coincide, the Bendixson transformation is the inversion of the plane $(x,y)$ with respect to the circle $x^2+y^2=4$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Andronov,  E.A. Leontovich,  I.I. Gordon,  A.G. Maier,  "Qualitative theory of second-order dynamic systems" , Wiley  (1973)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Andronov,  E.A. Leontovich,  I.I. Gordon,  A.G. Maier,  "Qualitative theory of second-order dynamic systems" , Wiley  (1973)  (Translated from Russian)</TD></TR></table>

Latest revision as of 15:19, 17 July 2014

The mapping

$$u=\frac{4x}{x^2+y^2},\quad v=\frac{4y}{x^2+y^2}$$

of the Euclidean $(x,y)$-plane punctured at the point $(0,0)$ into a similar Euclidean $(u,v)$-plane. It is the coordinate expression of the bijection $\phi$ generated by the Bendixson sphere. If the planes $(u,v)$ and $(x,y)$ coincide, the Bendixson transformation is the inversion of the plane $(x,y)$ with respect to the circle $x^2+y^2=4$.

References

[1] A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier, "Qualitative theory of second-order dynamic systems" , Wiley (1973) (Translated from Russian)
How to Cite This Entry:
Bendixson transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bendixson_transformation&oldid=32478
This article was adapted from an original article by A.F. Andreev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article