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A type of arrangement of the trajectories in a neighbourhood of a singular point  $  x _ {0} $
of an [[Autonomous system|autonomous system]] of second-order ordinary differential equations

$$ \tag{* }
\dot{x}  =  f ( x),\ \ 
x \in \mathbf R  ^ {2} ,\ \ 
f:  G \rightarrow \mathbf R  ^ {2} ,
$$

$  f \in C ( G) $, 
where  $  G $
is the domain of uniqueness. This type is characterized as follows. Suppose that a certain neighbourhood  $  U $
of  $  x _ {0} $
is partitioned into  $  m $(
$  3 \leq  m < + \infty $) 
curvilinear sectors (cf. [[Sector in the theory of ordinary differential equations|Sector in the theory of ordinary differential equations]]) by semi-trajectories (the separatrices of the saddle node) approaching  $  x _ {0} $. 
Suppose that  $  h $
of these sectors,  $  2 \leq  h < m $, 
are saddle sectors and that the others are open nodal sectors, and suppose also that each semi-trajectory approaching  $  x _ {0} $, 
completed with  $  x _ {0} $, 
touches it in a definite direction. Then  $  x _ {0} $
is called a saddle node.

A saddle node is unstable in the sense of Lyapunov (cf. [[Lyapunov stability|Lyapunov stability]]). Its Poincaré index is  $  1 - ( h/2) $(
cf. [[Singular point|Singular point]]). If  $  f \in C  ^ {1} ( G) $
and the matrix  $  A = f ^ { \prime } ( x _ {0} ) \neq 0 $, 
then the singular point  $  x _ {0} $
can be a saddle node for (*) only when the eigenvalues  $  \lambda _ {1} , \lambda _ {2} $
of  $  A $
satisfy one of the following conditions:

a)  $  \lambda _ {1} = 0 \neq \lambda _ {2} $;

b)  $  \lambda _ {1} = \lambda _ {2} = 0 $.

In any of these cases  $  x _ {0} $
can also be a [[Saddle|saddle]] or a [[Node|node]] for (*), and in case b), also a point of another type. If it is a saddle node, then  $  m = 3 $, 
$  h = 2 $, 
and all the semi-trajectories of the system that approach  $  x _ {0} $
touch at this point the directions defined by the eigenvectors of  $  A $(
see Fig. aand Fig. b, where the heavy lines are the separatrices at the saddle node  $  x _ {0} = 0 $, 
and the arrows indicate the direction of motion along the trajectories of the system as  $  t $
increases; they can also be in the opposite direction).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083040a.gif" />

Figure: s083040a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s083040b.gif" />

Figure: s083040b

====Comments====
The flow near a saddle node does not enjoy structural stability: If  $  x _ {0} $
is a saddle node for (*), there is a neighbourhood  $  N $
of  $  x _ {0} $
in  $  \mathbf R  ^ {2} $
such that for any  $  \epsilon > 0 $
there is a system  $  \dot{x} = y( x) $
having no equilibrium in  $  N $, 
such that  $  | f- g | < \epsilon $
and  $  | ( \partial  f / \partial  x _ {i} ) - ( \partial  g/ \partial  x _ {i} ) | < \epsilon $, 
$  i = 1, 2 $. 
However, the saddle node [[Bifurcation|bifurcation]] is robust and cannot be perturbed away ([[#References|[a1]]]) (cf. also [[Rough system|Rough system]]).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Bautin, E.A. Leontovich, "Methods and means for a qualitative investigation of dynamical systems on the plane" , Moscow  (1976)  (In Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer  (1983)</TD></TR>
<TR><TD valign="top">[a2]</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>

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