Difference between revisions of "Bendixson transformation"
From Encyclopedia of Mathematics
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The mapping | The mapping | ||
| − | + | $$u=\frac{4x}{x^2+y^2},\quad v=\frac{4y}{x^2+y^2}$$ | |
| − | of the Euclidean | + | 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
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