View source for Limit point of a trajectory

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''$\{f^tx\}$ of a dynamical system $f^t$''

A point

$$x_\alpha=\lim_{k\to\infty}f^{t_k}x\label{1}\tag{1}$$

(an $\alpha$-limit point) or

$$x_\omega=\lim_{k\to\infty}f^{t_k}x\label{2}\tag{2}$$

(an $\omega$-limit point), where $\{t_k\}_{k\in\mathbf N}$ is a sequence such that $t_k\to-\infty$ as $k\to\infty$ in \eqref{1}, or $t_k\to+\infty$ as $k\to\infty$ in \eqref{2}, and for which the limits in \eqref{1} or \eqref{2} exist.

For a trajectory $\{f^tx\}$ of a [[Dynamical system|dynamical system]] $f^t$ (or, in other words, for $f(t,x)$, cf. [[#References|[1]]]), an $\alpha$-limit point ($\omega$-limit point) is the same as an $\omega$-limit point ($\alpha$-limit point) of the trajectory $\{f^{-t}x\}$ of the dynamical system $f^{-t}$ (the system with reverse time). The set $\Omega_x$ ($A_x$) of all $\omega$-limit points ($\alpha$-limit points) of a trajectory $\{f^tx\}$ is called the $\omega$-limit set ($\alpha$-limit set) of this trajectory (cf. [[Limit set of a trajectory|Limit set of a trajectory]]).

====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></table>



====Comments====
For a dynamical system with discrete time (or, a [[Cascade|cascade]]) similar definitions and the same terminology are used (now in the above the sequences $\{t_k\}$ have to be in $\mathbf Z$).