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. ) 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.
|||V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)|
|||L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962) (Translated from Russian)|
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].
|[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)|
Limit set of a trajectory. V.M. Millionshchikov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Limit_set_of_a_trajectory&oldid=14411