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The property of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p0733501.png" /> (a trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p0733502.png" />) of a [[Dynamical system|dynamical system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p0733503.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p0733504.png" />, cf. [[#References|[2]]]), given in a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p0733505.png" />, consisting in the following: There are sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p0733506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p0733507.png" /> such that
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<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/p/p073/p073350/p0733508.png" /></td> </tr></table>
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In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p0733509.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p07335010.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p07335011.png" />-limit point (cf. [[Limit point of a trajectory|Limit point of a trajectory]]) of the trajectory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p07335012.png" />. The concept of Poisson stability was introduced by H. Poincaré [[#References|[1]]] on the basis of an analysis of results of Poisson on the stability of planetary orbits.
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The property of a point $  x $(
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a trajectory $  f ^ { t } x $)
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of a [[Dynamical system|dynamical system]] $  f ^ { t } $(
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or  $  f ( t , \cdot ) $,
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cf. [[#References|[2]]]), given in a topological space  $  S $,
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consisting in the following: There are sequences  $  t _ {k} \rightarrow + \infty $,
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$  \tau _ {k} \rightarrow - \infty $
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such that
  
Every Poisson-stable point is non-wandering; the converse is not true (cf. [[Wandering point|Wandering point]]). Every fixed and every periodic point, more generally, every [[Recurrent point|recurrent point]], is Poisson stable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p07335013.png" /> and the dynamical system is smooth (i.e. given by a vector field of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p07335014.png" />), then every Poisson-stable point is either fixed or periodic (cf. [[Poincaré–Bendixson theory|Poincaré–Bendixson theory]]).
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$$
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\lim\limits _ {k \rightarrow \infty }  f ^ { t _ {k} } x  = \
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\lim\limits _ {k \rightarrow \infty }  f ^ { \tau _ {k} } x  = x .
 +
$$
  
Poincaré's recurrence theorem (cf. [[Poincaré return theorem|Poincaré return theorem]]): If a dynamical system is given in a bounded domain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p07335015.png" /> and if Lebesgue measure is an [[Invariant measure|invariant measure]] of the system, then all points are Poisson stable, with the exception of a certain set of the first category of measure zero (cf. [[#References|[1]]], [[#References|[3]]]). A generalization of this theorem to dynamical systems given on a space of infinite measure is the Hopf recurrence theorem (cf. [[#References|[2]]]): If a dynamical system is given on an arbitrary domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p07335016.png" /> (e.g. on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073350/p07335017.png" /> itself) and if Lebesgue measure is an invariant measure of the system, then every point, with the exception of the points of a certain set of measure zero, is either Poisson stable or divergent, i.e.
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In other words,  $  x $
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is an  $  \alpha $-
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and  $  \omega $-
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limit point (cf. [[Limit point of a trajectory|Limit point of a trajectory]]) of the trajectory  $  f ^ { t } x $.  
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The concept of Poisson stability was introduced by H. Poincaré [[#References|[1]]] on the basis of an analysis of results of Poisson on the stability of planetary orbits.
  
<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/p/p073/p073350/p07335018.png" /></td> </tr></table>
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Every Poisson-stable point is non-wandering; the converse is not true (cf. [[Wandering point|Wandering point]]). Every fixed and every periodic point, more generally, every [[Recurrent point|recurrent point]], is Poisson stable. If  $  S = \mathbf R  ^ {2} $
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and the dynamical system is smooth (i.e. given by a vector field of class  $  C  ^ {1} $),
 +
then every Poisson-stable point is either fixed or periodic (cf. [[Poincaré–Bendixson theory|Poincaré–Bendixson theory]]).
 +
 
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Poincaré's recurrence theorem (cf. [[Poincaré return theorem|Poincaré return theorem]]): If a dynamical system is given in a bounded domain of  $  \mathbf R  ^ {n} $
 +
and if Lebesgue measure is an [[Invariant measure|invariant measure]] of the system, then all points are Poisson stable, with the exception of a certain set of the first category of measure zero (cf. [[#References|[1]]], [[#References|[3]]]). A generalization of this theorem to dynamical systems given on a space of infinite measure is the Hopf recurrence theorem (cf. [[#References|[2]]]): If a dynamical system is given on an arbitrary domain in  $  \mathbf R  ^ {n} $(
 +
e.g. on  $  \mathbf R  ^ {n} $
 +
itself) and if Lebesgue measure is an invariant measure of the system, then every point, with the exception of the points of a certain set of measure zero, is either Poisson stable or divergent, i.e.
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$$
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| f ^ { t } x |  \rightarrow  \infty \  \textrm{ as }  | t | \rightarrow \infty .
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$$
  
 
There are still more general formulations of the theorems of Poincaré and E. Hopf (cf. [[#References|[2]]]).
 
There are still more general formulations of the theorems of Poincaré and E. Hopf (cf. [[#References|[2]]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Les méthodes nouvelles de la mécanique céleste" , '''3''' , Gauthier-Villars  (1899)  pp. Chapt. 26</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  J.C. Oxtoby,  "Measure and category" , Springer  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  "Les méthodes nouvelles de la mécanique céleste" , '''3''' , Gauthier-Villars  (1899)  pp. Chapt. 26</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  J.C. Oxtoby,  "Measure and category" , Springer  (1971)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 08:06, 6 June 2020


The property of a point $ x $( a trajectory $ f ^ { t } x $) of a dynamical system $ f ^ { t } $( or $ f ( t , \cdot ) $, cf. [2]), given in a topological space $ S $, consisting in the following: There are sequences $ t _ {k} \rightarrow + \infty $, $ \tau _ {k} \rightarrow - \infty $ such that

$$ \lim\limits _ {k \rightarrow \infty } f ^ { t _ {k} } x = \ \lim\limits _ {k \rightarrow \infty } f ^ { \tau _ {k} } x = x . $$

In other words, $ x $ is an $ \alpha $- and $ \omega $- limit point (cf. Limit point of a trajectory) of the trajectory $ f ^ { t } x $. The concept of Poisson stability was introduced by H. Poincaré [1] on the basis of an analysis of results of Poisson on the stability of planetary orbits.

Every Poisson-stable point is non-wandering; the converse is not true (cf. Wandering point). Every fixed and every periodic point, more generally, every recurrent point, is Poisson stable. If $ S = \mathbf R ^ {2} $ and the dynamical system is smooth (i.e. given by a vector field of class $ C ^ {1} $), then every Poisson-stable point is either fixed or periodic (cf. Poincaré–Bendixson theory).

Poincaré's recurrence theorem (cf. Poincaré return theorem): If a dynamical system is given in a bounded domain of $ \mathbf R ^ {n} $ and if Lebesgue measure is an invariant measure of the system, then all points are Poisson stable, with the exception of a certain set of the first category of measure zero (cf. [1], [3]). A generalization of this theorem to dynamical systems given on a space of infinite measure is the Hopf recurrence theorem (cf. [2]): If a dynamical system is given on an arbitrary domain in $ \mathbf R ^ {n} $( e.g. on $ \mathbf R ^ {n} $ itself) and if Lebesgue measure is an invariant measure of the system, then every point, with the exception of the points of a certain set of measure zero, is either Poisson stable or divergent, i.e.

$$ | f ^ { t } x | \rightarrow \infty \ \textrm{ as } | t | \rightarrow \infty . $$

There are still more general formulations of the theorems of Poincaré and E. Hopf (cf. [2]).

References

[1] H. Poincaré, "Les méthodes nouvelles de la mécanique céleste" , 3 , Gauthier-Villars (1899) pp. Chapt. 26
[2] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[3] J.C. Oxtoby, "Measure and category" , Springer (1971)

Comments

In Western literature on (abstract) topological dynamics (as opposed to the qualitative theory of differential equations) often the term "recurrent" is used for Poisson stability; see [a1]. For further comments, see Recurrent point.

References

[a1] W.H. Gottschalk, G.A. Hedlund, "Topological dynamics" , Amer. Math. Soc. (1955)
How to Cite This Entry:
Poisson stability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_stability&oldid=48221
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article