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'' $  \{ f ^ { t } x \} $ of a dynamical system  $  f ^ { t } $''

The set  $  A _ {x} $
of all  $  \alpha $-limit points (the  $  \alpha $-limit set) or the set  $  \Omega _ {x} $
of all  $  \omega $-limit points (the  $  \omega $-limit set) of this trajectory (cf. [[Limit point of a trajectory|Limit point of a trajectory]]). The  $  \alpha $-limit set ( $  \omega $-limit set) of a trajectory  $  \{ f ^ { t } x \} $
of a system (or, in other notation,  $  f ( t, x) $, 
cf. [[#References|[1]]]) is the same as the  $  \omega $-limit set (respectively,  $  \alpha $-limit set) of the trajectory  $  \{ f ^ { - t } x \} $
of the [[Dynamical system|dynamical system]]  $  f ^ { - t } $ (the system with reversed time). Therefore the properties of  $  \alpha $-limit sets are similar to those of  $  \omega $-limit sets.

The set  $  \Omega _ {x} $
is a closed invariant set. If  $  \Omega _ {x} = \emptyset $, 
then the trajectory  $  \{ f ^ { t } x \} $
is called divergent in the positive direction; if  $  A _ {x} = \emptyset $, 
divergent in the negative direction; if  $  \Omega _ {x} = A _ {x} = \emptyset $, 
the trajectory is called divergent. If  $  x \in \Omega _ {x} $, 
then  $  x $
is called positively Poisson stable; if  $  x \in A _ {x} $, 
negatively Poisson stable; and if  $  x \in A _ {x} \cap \Omega _ {x} $, 
then  $  x $
is called Poisson stable. If  $  x \notin \Omega _ {x} $
and  $  \Omega _ {x} \neq \emptyset $, 
then  $  x $
is called positively asymptotic; if  $  x \notin A _ {x} $
and  $  A _ {x} \neq \emptyset $, 
the point  $  x $
is called negatively asymptotic.

If  $  x $
is a positively Lagrange-stable point (cf. [[Lagrange stability|Lagrange stability]]), then  $  \Omega _ {x} $
is a non-empty connected set,

$$ 
\lim\limits _ {t \rightarrow + \infty } \ 
d ( f ^ { t } x, \Omega _ {x} )  =  0
$$

(where  $  d ( z, Y) $
is the distance from a point  $  z $
to a set  $  Y $) 
and there is a [[Recurrent point|recurrent point]] (trajectory) in  $  \Omega _ {x} $. 
If  $  x $
is a fixed point, then  $  \Omega _ {x} = \{ x \} $. 
If  $  x $
is a periodic point, then

$$ 
\Omega _ {x}  = \ 
\{ f ^ { t } x \} _ {t \in \mathbf R }   = \ 
\{ f ^ { t } x \} _ {t \in [ 0, T) }  ,
$$

where  $  T $
is the period. If  $  x $
is not a fixed point and not a periodic point, and if the underlying metric space of the dynamical system under consideration is complete, then the points in  $  \Omega _ {x} $
not on the trajectory  $  \{ f ^ { t } x \} $
are everywhere-dense in  $  \Omega _ {x} $.

If a dynamical system in the plane is given by an [[Autonomous system|autonomous system]] of differential equations

$$ 
\dot{x}  =  f ( x),\ \ 
x \in \mathbf R  ^ {2} ,\ \ 
f \in C  ^ {1}
$$

(with a smooth vector field  $  f  $), 
$  x $
is positively Lagrange stable but not periodic, and  $  f $
does not vanish on  $  \Omega _ {x} $ (i.e.  $  \Omega _ {x} $
does not contain fixed points), then  $  \Omega _ {x} $
is a cycle, i.e. a closed curve (the trajectory of a periodic point), while the trajectory  $  \{ f ^ { t } x \} $
winds, spiral-wise, around this cycle as  $  t \rightarrow \infty $. 
For dynamical systems in  $  \mathbf R  ^ {n} $, 
$  n > 2 $, 
or on a two-dimensional surface, e.g. a torus, the  $  \omega $-limit sets can have a different structure. E.g., for an irrational winding on a torus (the system  $  \dot \phi  = 1 $, 
$  \dot \psi  = \mu $, 
where  $  ( \phi , \psi ) (  \mathop{\rm mod}  1) $
are cyclic coordinates on the torus  $  T  ^ {2} $
and  $  \mu $
is an irrational number) the set  $  \Omega _ {x} $
coincides, for every  $  x = ( \phi , \psi ) $, 
with the torus.

====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><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,   "Ordinary differential equations" , Addison-Wesley  (1962)  (Translated from Russian)</TD></TR></table>

====Comments====
Instead of  "divergent in the positive direction" ,  "divergent in the negative direction"  and  "divergent" , also the terms positively receding, negatively receding and receding are used.

The statement above about the cyclic structure of certain limit sets in a dynamical system in the plane is part of the so-called Poincaré–Bendixson theorem (cf. [[Poincaré–Bendixson theory|Poincaré–Bendixson theory]] and also [[Limit cycle|Limit cycle]]). It is valid for arbitrary dynamical systems in the plane (not necessarily given by differential equations). See [[#References|[a3]]], Sect. VIII.1 or, for an approach avoiding local cross-sections, [[#References|[a1]]], Chapt. 2. It follows also from [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Beck,   "Continuous flows in the plane" , Springer  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Gutierrez,   "Smoothing continuous flows on two-manifolds and recurrences"  ''Ergodic Theory and Dynam. Syst.'' , '''6'''  (1986)  pp. 17–44</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  O. Hajek,   "Dynamical systems in the plane" , Acad. Press  (1968)</TD></TR></table>