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)