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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l0589101.png" /> of a dynamical system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l0589102.png" />''
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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l0589103.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l0589104.png" />-limit points (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l0589106.png" />-limit set) or the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l0589107.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l0589108.png" />-limit points (the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891010.png" />-limit set) of this trajectory (cf. [[Limit point of a trajectory|Limit point of a trajectory]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891011.png" />-limit set (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891012.png" />-limit set) of a trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891013.png" /> of a system (or, in other notation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891014.png" />, cf. [[#References|[1]]]) is the same as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891015.png" />-limit set (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891016.png" />-limit set) of the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891017.png" /> of the [[Dynamical system|dynamical system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891018.png" /> (the system with reversed time). Therefore the properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891019.png" />-limit sets are similar to those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891020.png" />-limit sets.
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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891021.png" /> is a closed invariant set. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891022.png" />, then the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891023.png" /> is called divergent in the positive direction; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891024.png" />, divergent in the negative direction; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891025.png" />, the trajectory is called divergent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891027.png" /> is called positively Poisson stable; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891028.png" />, negatively Poisson stable; and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891030.png" /> is called Poisson stable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891033.png" /> is called positively asymptotic; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891035.png" />, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891036.png" /> is called negatively asymptotic.
+
'' $  \{ f ^ { t } x \} $ of a dynamical system  $  f ^ { t } $''
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891037.png" /> is a positively Lagrange-stable point (cf. [[Lagrange stability|Lagrange stability]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891038.png" /> is a non-empty connected set,
+
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.
  
<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/l/l058/l058910/l05891039.png" /></td> </tr></table>
+
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.
  
(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891040.png" /> is the distance from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891041.png" /> to a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891042.png" />) and there is a [[Recurrent point|recurrent point]] (trajectory) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891043.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891044.png" /> is a fixed point, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891045.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891046.png" /> is a periodic point, then
+
If  $  x $
 +
is a positively Lagrange-stable point (cf. [[Lagrange stability|Lagrange stability]]), then $  \Omega _ {x} $
 +
is a non-empty connected set,
  
<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/l/l058/l058910/l05891047.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {t \rightarrow + \infty } \
 +
d ( f ^ { t } x, \Omega _ {x} )  = 0
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891048.png" /> is the period. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891049.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891050.png" /> not on the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891051.png" /> are everywhere-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891052.png" />.
+
(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
 
If a dynamical system in the plane is given by an [[Autonomous system|autonomous system]] of differential equations
  
<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/l/l058/l058910/l05891053.png" /></td> </tr></table>
+
$$
 +
\dot{x}  = f ( x),\ \
 +
x \in \mathbf R  ^ {2} ,\ \
 +
f \in C  ^ {1}
 +
$$
  
(with a smooth vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891054.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891055.png" /> is positively Lagrange stable but not periodic, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891056.png" /> does not vanish on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891057.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891058.png" /> does not contain fixed points), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891059.png" /> is a cycle, i.e. a closed curve (the trajectory of a periodic point), while the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891060.png" /> winds, spiral-wise, around this cycle as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891061.png" />. For dynamical systems in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891063.png" />, or on a two-dimensional surface, e.g. a torus, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891064.png" />-limit sets can have a different structure. E.g., for an irrational winding on a torus (the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891066.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891067.png" /> are cyclic coordinates on the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891069.png" /> is an irrational number) the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891070.png" /> coincides, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058910/l05891071.png" />, with the torus.
+
(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====
 
====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>
 
<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====
 
====Comments====

Latest revision as of 04:31, 14 September 2022


$ \{ 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). The $ \alpha $-limit set ( $ \omega $-limit set) of a trajectory $ \{ f ^ { t } x \} $ of a system (or, in other notation, $ f ( t, x) $, cf. [1]) is the same as the $ \omega $-limit set (respectively, $ \alpha $-limit set) of the trajectory $ \{ f ^ { - t } x \} $ of the 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), 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 (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 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

[1] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[2] L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)

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 and also Limit cycle). It is valid for arbitrary dynamical systems in the plane (not necessarily given by differential equations). See [a3], Sect. VIII.1 or, for an approach avoiding local cross-sections, [a1], Chapt. 2. It follows also from [a2].

References

[a1] A. Beck, "Continuous flows in the plane" , Springer (1974)
[a2] C. Gutierrez, "Smoothing continuous flows on two-manifolds and recurrences" Ergodic Theory and Dynam. Syst. , 6 (1986) pp. 17–44
[a3] O. Hajek, "Dynamical systems in the plane" , Acad. Press (1968)
How to Cite This Entry:
Limit set of a trajectory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Limit_set_of_a_trajectory&oldid=14411
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article