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) |
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) |
Limit set of a trajectory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Limit_set_of_a_trajectory&oldid=14411