Limit set of a trajectory

$\{ f ^ { t } x \}$ of a dynamical system $f ^ { t }$

The set $A _ {x}$ of all $\alpha$- limit points (the $\alpha$- limit set) or the set $\Omega _ {x}$ of all $\omega$- limit points (the $\omega$- limit set) of this trajectory (cf. Limit point of a trajectory). The $\alpha$- limit set ( $\omega$- limit set) of a trajectory $\{ f ^ { t } x \}$ of a system (or, in other notation, $f ( t, x)$, cf. [1]) is the same as the $\omega$- limit set (respectively, $\alpha$- limit set) of the trajectory $\{ f ^ { - t } x \}$ of the dynamical system $f ^ { - t }$( the system with reversed time). Therefore the properties of $\alpha$- limit sets are similar to those of $\omega$- limit sets.

The set $\Omega _ {x}$ is a closed invariant set. If $\Omega _ {x} = \emptyset$, then the trajectory $\{ f ^ { t } x \}$ is called divergent in the positive direction; if $A _ {x} = \emptyset$, divergent in the negative direction; if $\Omega _ {x} = A _ {x} = \emptyset$, the trajectory is called divergent. If $x \in \Omega _ {x}$, then $x$ is called positively Poisson stable; if $x \in A _ {x}$, negatively Poisson stable; and if $x \in A _ {x} \cap \Omega _ {x}$, then $x$ is called Poisson stable. If $x \notin \Omega _ {x}$ and $\Omega _ {x} \neq \emptyset$, then $x$ is called positively asymptotic; if $x \notin A _ {x}$ and $A _ {x} \neq \emptyset$, the point $x$ is called negatively asymptotic.

If $x$ is a positively Lagrange-stable point (cf. Lagrange stability), then $\Omega _ {x}$ is a non-empty connected set,

$$\lim\limits _ {t \rightarrow + \infty } \ d ( f ^ { t } x, \Omega _ {x} ) = 0$$

(where $d ( z, Y)$ is the distance from a point $z$ to a set $Y$) and there is a recurrent point (trajectory) in $\Omega _ {x}$. If $x$ is a fixed point, then $\Omega _ {x} = \{ x \}$. If $x$ is a periodic point, then

$$\Omega _ {x} = \ \{ f ^ { t } x \} _ {t \in \mathbf R } = \ \{ f ^ { t } x \} _ {t \in [ 0, T) } ,$$

where $T$ is the period. If $x$ is not a fixed point and not a periodic point, and if the underlying metric space of the dynamical system under consideration is complete, then the points in $\Omega _ {x}$ not on the trajectory $\{ f ^ { t } x \}$ are everywhere-dense in $\Omega _ {x}$.

If a dynamical system in the plane is given by an autonomous system of differential equations

$$\dot{x} = f ( x),\ \ x \in \mathbf R ^ {2} ,\ \ f \in C ^ {1}$$

(with a smooth vector field $f$), $x$ is positively Lagrange stable but not periodic, and $f$ does not vanish on $\Omega _ {x}$( i.e. $\Omega _ {x}$ does not contain fixed points), then $\Omega _ {x}$ is a cycle, i.e. a closed curve (the trajectory of a periodic point), while the trajectory $\{ f ^ { t } x \}$ winds, spiral-wise, around this cycle as $t \rightarrow \infty$. For dynamical systems in $\mathbf R ^ {n}$, $n > 2$, or on a two-dimensional surface, e.g. a torus, the $\omega$- limit sets can have a different structure. E.g., for an irrational winding on a torus (the system $\dot \phi = 1$, $\dot \psi = \mu$, where $( \phi , \psi ) ( \mathop{\rm mod} 1)$ are cyclic coordinates on the torus $T ^ {2}$ and $\mu$ is an irrational number) the set $\Omega _ {x}$ coincides, for every $x = ( \phi , \psi )$, with the torus.

References

[1]	V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[2]	L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)

Comments

Instead of "divergent in the positive direction" , "divergent in the negative direction" and "divergent" , also the terms positively receding, negatively receding and receding are used.

The statement above about the cyclic structure of certain limit sets in a dynamical system in the plane is part of the so-called Poincaré–Bendixson theorem (cf. Poincaré–Bendixson theory and also Limit cycle). It is valid for arbitrary dynamical systems in the plane (not necessarily given by differential equations). See [a3], Sect. VIII.1 or, for an approach avoiding local cross-sections, [a1], Chapt. 2. It follows also from [a2].

References

[a1]	A. Beck, "Continuous flows in the plane" , Springer (1974)
[a2]	C. Gutierrez, "Smoothing continuous flows on two-manifolds and recurrences" Ergodic Theory and Dynam. Syst. , 6 (1986) pp. 17–44
[a3]	O. Hajek, "Dynamical systems in the plane" , Acad. Press (1968)