# Limit set of a trajectory

of a dynamical system

The set of all -limit points (the -limit set) or the set of all -limit points (the -limit set) of this trajectory (cf. Limit point of a trajectory). The -limit set (-limit set) of a trajectory of a system (or, in other notation, , cf. [1]) is the same as the -limit set (respectively, -limit set) of the trajectory of the dynamical system (the system with reversed time). Therefore the properties of -limit sets are similar to those of -limit sets.

The set is a closed invariant set. If , then the trajectory is called divergent in the positive direction; if , divergent in the negative direction; if , the trajectory is called divergent. If , then is called positively Poisson stable; if , negatively Poisson stable; and if , then is called Poisson stable. If and , then is called positively asymptotic; if and , the point is called negatively asymptotic.

If is a positively Lagrange-stable point (cf. Lagrange stability), then is a non-empty connected set,

(where is the distance from a point to a set ) and there is a recurrent point (trajectory) in . If is a fixed point, then . If is a periodic point, then

where is the period. If 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 not on the trajectory are everywhere-dense in .

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

(with a smooth vector field ), is positively Lagrange stable but not periodic, and does not vanish on (i.e. does not contain fixed points), then is a cycle, i.e. a closed curve (the trajectory of a periodic point), while the trajectory winds, spiral-wise, around this cycle as . For dynamical systems in , , or on a two-dimensional surface, e.g. a torus, the -limit sets can have a different structure. E.g., for an irrational winding on a torus (the system , , where are cyclic coordinates on the torus and is an irrational number) the set coincides, for every , 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)