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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w0970801.png" /> of all wandering points (cf. [[Wandering point|Wandering point]]) of some [[Dynamical system|dynamical system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w0970802.png" />. Since for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w0970803.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w0970804.png" /> contains all points of the neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w0970805.png" />, it is open in the phase space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w0970806.png" />. Accordingly, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w0970807.png" /> of all non-wandering points is closed. The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w0970808.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w0970809.png" /> are invariant, i.e. with each of their points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w09708010.png" /> they contain the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w09708011.png" /> for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w09708012.png" />. In a compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w09708013.png" /> each wandering point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w09708014.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w09708015.png" /> both when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w09708016.png" /> and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097080/w09708017.png" />.
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The set $W$ of all wandering points (cf. [[Wandering point|Wandering point]]) of some [[Dynamical system|dynamical system]] $f(p,t)$. Since for each point $q$ the set $W$ contains all points of the neighbourhood $U(q)$, it is open in the phase space $R$. Accordingly, the set $M=R\setminus W$ of all non-wandering points is closed. The sets $W$ and $M$ are invariant, i.e. with each of their points $q$ they contain the point $f(q,t)$ for an arbitrary $t$. In a compact space $R$ each wandering point $f(q,t)$ tends to $M$ both when $t\to\infty$ and when $t\to-\infty$.
  
 
====References====
 
====References====

Latest revision as of 10:45, 15 April 2014

The set $W$ of all wandering points (cf. Wandering point) of some dynamical system $f(p,t)$. Since for each point $q$ the set $W$ contains all points of the neighbourhood $U(q)$, it is open in the phase space $R$. Accordingly, the set $M=R\setminus W$ of all non-wandering points is closed. The sets $W$ and $M$ are invariant, i.e. with each of their points $q$ they contain the point $f(q,t)$ for an arbitrary $t$. In a compact space $R$ each wandering point $f(q,t)$ tends to $M$ both when $t\to\infty$ and when $t\to-\infty$.

References

[1] G.D. Birkhoff, "Dynamical systems" , Amer. Math. Soc. (1927)
[2] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[3] K.S. Sibirskii, "Introduction to topological dynamics" , Noordhoff (1975) (Translated from Russian)


Comments

References

[a1] S. Smale, "Differentiable dynamical systems" Bull. Amer. Math. Soc. , 73 (1967) pp. 747–817
[a2] N.P. Bhatia, G.P. Szegö, "Stability theory of dynamical systems" , Springer (1970) pp. 30–36
How to Cite This Entry:
Wandering set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wandering_set&oldid=31719
This article was adapted from an original article by K.S. Sibirskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article