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''singular saddle point''
 
''singular saddle point''
  
A type of arrangement of the trajectories of a [[Dynamical system|dynamical system]]. A dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083030/s0830301.png" /> (or, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083030/s0830302.png" />, see [[#References|[1]]]) defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083030/s0830303.png" /> is said to have a saddle at infinity if there are points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083030/s0830304.png" /> and numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083030/s0830305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083030/s0830306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083030/s0830307.png" />, such that the sequences
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A type of arrangement of the trajectories of a [[Dynamical system|dynamical system]]. A dynamical system $  f ^ { t } $(
 +
or, $  f ( \cdot , t) $,  
 +
see [[#References|[1]]]) defined on $  \mathbf R  ^ {n} $
 +
is said to have a saddle at infinity if there are points $  x _ {k} $
 +
and numbers $  \tau _ {k} < 0 $
 +
and  $  \theta _ {k} > 0 $,  
 +
$  k \in \mathbf N $,  
 +
such that the sequences
  
<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/s/s083/s083030/s0830308.png" /></td> </tr></table>
+
$$
 +
\{ f ^ { \tau _ {k} } x _ {k} \} _ {k \in \mathbf N }  ,\ \
 +
\{ f ^ { \theta _ {k} } x _ {k} \} _ {k \in \mathbf N }
 +
$$
  
are convergent and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083030/s0830309.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083030/s08303010.png" />. This definition, which is due to V.V. Nemytskii, was generalized by M.V. Bebutov to dynamical systems defined on an arbitrary metric space; here the condition  "xk∞ as k∞"  is replaced by  "the sequence xkkN does not contain any convergent subsequence" .
+
are convergent and $  | x _ {k} | \rightarrow \infty $
 +
as $  k \rightarrow \infty $.  
 +
This definition, which is due to V.V. Nemytskii, was generalized by M.V. Bebutov to dynamical systems defined on an arbitrary metric space; here the condition  "xk∞ as k∞"  is replaced by  "the sequence xkkN does not contain any convergent subsequence" .
  
 
The absence of a saddle at infinity is a necessary condition for the possibility of global rectification of the dynamical system (see [[Complete instability|Complete instability]]). For a completely-unstable dynamical system defined on a metric space not to have a saddle at infinity it is necessary and sufficient that the quotient space of the dynamical system be Hausdorff.
 
The absence of a saddle at infinity is a necessary condition for the possibility of global rectification of the dynamical system (see [[Complete instability|Complete instability]]). For a completely-unstable dynamical system defined on a metric space not to have a saddle at infinity it is necessary and sufficient that the quotient space of the dynamical system be Hausdorff.
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====References====
 
====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)</TD></TR></table>
 
<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)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The  "quotient space of a dynamical systemquotient space of a dynamical system"  is usually called its orbit space.
 
The  "quotient space of a dynamical systemquotient space of a dynamical system"  is usually called its orbit space.

Latest revision as of 08:12, 6 June 2020


singular saddle point

A type of arrangement of the trajectories of a dynamical system. A dynamical system $ f ^ { t } $( or, $ f ( \cdot , t) $, see [1]) defined on $ \mathbf R ^ {n} $ is said to have a saddle at infinity if there are points $ x _ {k} $ and numbers $ \tau _ {k} < 0 $ and $ \theta _ {k} > 0 $, $ k \in \mathbf N $, such that the sequences

$$ \{ f ^ { \tau _ {k} } x _ {k} \} _ {k \in \mathbf N } ,\ \ \{ f ^ { \theta _ {k} } x _ {k} \} _ {k \in \mathbf N } $$

are convergent and $ | x _ {k} | \rightarrow \infty $ as $ k \rightarrow \infty $. This definition, which is due to V.V. Nemytskii, was generalized by M.V. Bebutov to dynamical systems defined on an arbitrary metric space; here the condition "xk∞ as k∞" is replaced by "the sequence xkkN does not contain any convergent subsequence" .

The absence of a saddle at infinity is a necessary condition for the possibility of global rectification of the dynamical system (see Complete instability). For a completely-unstable dynamical system defined on a metric space not to have a saddle at infinity it is necessary and sufficient that the quotient space of the dynamical system be Hausdorff.

References

[1] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)

Comments

The "quotient space of a dynamical systemquotient space of a dynamical system" is usually called its orbit space.

How to Cite This Entry:
Saddle at infinity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Saddle_at_infinity&oldid=48602
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article