Limit set


In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system. A system that has reached its limiting set is said to be at equilibrium.

Types

In general, limits sets can be very complicated as in the case of strange attractors, but for 2-dimensional dynamical systems the Poincaré–Bendixson theorem provides a simple characterization of all nonempty, compact -limit sets that contain at most finitely many fixed points as a fixed point, a periodic orbit, or a union of fixed points and homoclinic or heteroclinic orbits connecting those fixed points.

Definition for iterated functions

Let be a metric space, and let be a continuous function. The -limit set of, denoted by, is the set of cluster points of the forward orbit of the iterated function. Hence, if and only if there is a strictly increasing sequence of natural numbers such that as. Another way to express this is
where denotes the closure of set. The points in the limit set are non-wandering. This may also be formulated as the outer limit of a sequence of sets, such that
If is a homeomorphism, then the -limit set is defined in a similar fashion, but for the backward orbit; i.e..
Both sets are -invariant, and if is compact, they are compact and nonempty.

Definition for flows

Given a real dynamical system with flow, a point, we call a point an -limit point of ' if there exists a sequence in so that
For an orbit of, we say that is an -limit point of, if it is an -limit point of some point on the orbit.
Analogously we call ' an -limit point of ' if there exists a sequence in so that
For an orbit of, we say that ' is an -limit point of, if it is an -limit point of some point on the orbit.
The set of all -limit points for a given orbit is called -limit set for and denoted .
If the -limit set is disjoint from the orbit, that is , we call a ω-limit cycle.
Alternatively the limit sets can be defined as
and

Examples

  • For any periodic orbit of a dynamical system,
  • For any fixed point of a dynamical system,

    Properties

  • and are closed
  • if is compact then and are nonempty, compact and connected
  • and are -invariant, that is and