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A type of arrangement of the trajectories of an autonomous system of first-order ordinary differential equations

$$ \tag{* }
\dot{x}  =  f ( x),\ \ 
x = ( x _ {1} , x _ {2} ),\ \ 
f:  G \subset  \mathbf R  ^ {2} \rightarrow
\mathbf R  ^ {2} ,
$$

$  f \in C ( G) $, 
where  $  G $
is the domain of uniqueness, in a neighbourhood of a singular point  $  x _ {0} $(
cf. [[Equilibrium position|Equilibrium position]]). This type is characterized as follows. There is a neighbourhood  $  U $
of  $  x _ {0} $
such that for all trajectories of the system starting in  $  U \setminus  \{ x _ {0} \} $, 
the negative semi-trajectories are escaping (in the course of time they leave any compact set  $  V \subset  U $) 
and the positive semi-trajectories, without leaving  $  U $, 
tend to  $  x _ {0} $, 
winding round it like a [[Logarithmic spiral|logarithmic spiral]], or conversely. The point  $  x _ {0} $
itself is also called a focus. The nature of the approach of the trajectories of the system to  $  x _ {0} $
can be described more precisely if one introduces polar coordinates  $  r, \phi $
on the  $  ( x _ {1} , x _ {2} ) $-
plane with pole at  $  x _ {0} $. 
Then for any semi-trajectory  $  r = r ( t) $, 
$  \phi = \phi ( t) $, 
$  t \geq  0 $(
$  t \leq  0 $), 
that tends to  $  x _ {0} $, 
the polar angle of the variable point  $  \phi ( t) \rightarrow + \infty $(
a left focus) or  $  - \infty $(
a right focus) as  $  t \rightarrow \infty $.

A focus is either asymptotically Lyapunov stable or completely unstable (asymptotically stable as  $  t \rightarrow - \infty $). 
Its Poincaré index is 1. The figure depicts a right unstable focus at  $  x _ {0} = ( 0, 0) $.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/f040700a.gif" />

Figure: f040700a

For a system (*) of class  $  C  ^ {1} $(
$  f \in C  ^ {1} ( G) $) 
a singular point  $  x _ {0} $
is a focus in case the matrix  $  A = f ^ { \prime } ( x _ {0} ) $
has complex conjugate eigen values with non-zero real part, but it may also be a focus in case this matrix has purely imaginary or multiple real eigen values (see also [[Centre|Centre]]; [[Centre and focus problem|Centre and focus problem]]).

For references see [[Singular point|Singular point]] of a differential equation.

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Hartman,   "Ordinary differential equations" , Birkhäuser  (1982)</TD></TR></table>