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''singular saddle point''

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

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

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