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A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w0970701.png" /> in the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w0970702.png" /> of a [[Dynamical system|dynamical system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w0970703.png" /> with a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w0970704.png" /> for which there exists a moment in time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w0970705.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w0970706.png" /> has no common points with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w0970707.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w0970708.png" /> (all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w0970709.png" />, from some moment on, leave the neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707010.png" />). A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707011.png" /> without such a neighbourhood is said to be non-wandering. This property of a point — to be wandering or non-wandering — is two-sided: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707012.png" /> has no common points with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707014.png" /> has no common points with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707015.png" />. A wandering point may become non-wandering if the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707016.png" /> is extended. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707017.png" /> is a circle with one rest point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707018.png" />, all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707019.png" /> are wandering points. They become non-wandering if the points of some spiral without rest points, winding itself around this circle from the outside or from the inside, are added to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707020.png" />.
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A point $q$ in the phase space $R$ of a [[Dynamical system|dynamical system]] $f(p,t)$ with a neighbourhood $U(q)$ for which there exists a moment in time $T$ such that $f(U(q),t)$ has no common points with $U(q)$ for all $t\geq T$ (all points of $U(q)$, from some moment on, leave the neighbourhood $U(q)$). A point $q$ without such a neighbourhood is said to be non-wandering. This property of a point — to be wandering or non-wandering — is two-sided: If $f(U(q),t)$ has no common points with $U(q)$, then $U(q)$ has no common points with $f(U(q),-t)$. A wandering point may become non-wandering if the space $R$ is extended. For instance, if $R$ is a circle with one rest point $r$, all points of $R\setminus r$ are wandering points. They become non-wandering if the points of some spiral without rest points, winding itself around this circle from the outside or from the inside, are added to $R$.
  
  
  
 
====Comments====
 
====Comments====
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707021.png" /> is positively recursive with respect to a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707022.png" /> if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707023.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707025.png" />. Negatively recursive is defined analogously. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707026.png" /> is then non-wandering if every neighbourhood of it is positively recursive with respect to itself (self-positively recursive). A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707027.png" /> is positively Poisson stable (negatively Poisson stable) if every neighbourhood of it is positively recursive (negatively recursive) with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707028.png" />. A point is Poisson stable if it is both positively and negatively Poisson stable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707029.png" /> is such that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707030.png" /> is positively or negatively Poisson stable, then all points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097070/w09707031.png" /> are non-wandering. See also [[Wandering set|Wandering set]].
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A set $A\subset R$ is positively recursive with respect to a set $B\subset R$ if for all $T$ there is a $t>T$ such that $f(B,t)\cap A\neq\emptyset$. Negatively recursive is defined analogously. A point $x$ is then non-wandering if every neighbourhood of it is positively recursive with respect to itself (self-positively recursive). A point $x$ is positively Poisson stable (negatively Poisson stable) if every neighbourhood of it is positively recursive (negatively recursive) with respect to $\{x\}$. A point is Poisson stable if it is both positively and negatively Poisson stable. If $P\subset R$ is such that every $x\in P$ is positively or negatively Poisson stable, then all points of $\bar P$ are non-wandering. See also [[Wandering set|Wandering set]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.P. Bhatia,  G.P. Szegö,  "Stability theory of dynamical systems" , Springer  (1970)  pp. 30–36</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.P. Bhatia,  G.P. Szegö,  "Stability theory of dynamical systems" , Springer  (1970)  pp. 30–36</TD></TR></table>

Latest revision as of 08:09, 21 June 2014

A point $q$ in the phase space $R$ of a dynamical system $f(p,t)$ with a neighbourhood $U(q)$ for which there exists a moment in time $T$ such that $f(U(q),t)$ has no common points with $U(q)$ for all $t\geq T$ (all points of $U(q)$, from some moment on, leave the neighbourhood $U(q)$). A point $q$ without such a neighbourhood is said to be non-wandering. This property of a point — to be wandering or non-wandering — is two-sided: If $f(U(q),t)$ has no common points with $U(q)$, then $U(q)$ has no common points with $f(U(q),-t)$. A wandering point may become non-wandering if the space $R$ is extended. For instance, if $R$ is a circle with one rest point $r$, all points of $R\setminus r$ are wandering points. They become non-wandering if the points of some spiral without rest points, winding itself around this circle from the outside or from the inside, are added to $R$.


Comments

A set $A\subset R$ is positively recursive with respect to a set $B\subset R$ if for all $T$ there is a $t>T$ such that $f(B,t)\cap A\neq\emptyset$. Negatively recursive is defined analogously. A point $x$ is then non-wandering if every neighbourhood of it is positively recursive with respect to itself (self-positively recursive). A point $x$ is positively Poisson stable (negatively Poisson stable) if every neighbourhood of it is positively recursive (negatively recursive) with respect to $\{x\}$. A point is Poisson stable if it is both positively and negatively Poisson stable. If $P\subset R$ is such that every $x\in P$ is positively or negatively Poisson stable, then all points of $\bar P$ are non-wandering. See also Wandering set.

References

[a1] N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) pp. 30–36
How to Cite This Entry:
Wandering point. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wandering_point&oldid=32283
This article was adapted from an original article by K.S. Sibirskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article