Wandering set
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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
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