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

• fixed pointss

• periodic orbits

• limit cycles

• attractors

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-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

be a metric space be a continuous function, is the set of cluster points of the forward orbit of the iterated function. if and only if. Another way to express this is

denotes the closure. The points in the limit set are non-wandering (but may not be recurrent points). This may also be formulated as the outer limit (limsup) of a sequence of sets, such that

is a homeomorphism-limit set is defined in a similar fashion, but for the backward orbit; i.e..

is compact, they are compact and nonempty.

Definition for flows

Given a real dynamical system with flow-limit point of so that

1. .

For an orbit-limit point-limit point of some point on the orbit.

Analogously we call -limit point of so that

1. .

For an orbit, we say that -limit point-limit point of some point on the orbit.

-limit set-limit set).

) a ω-limit cycle (α-limit cycle).

Alternatively the limit sets can be defined as

and

Examples

• For any periodic orbit

• For any fixed point

Properties

• are closed

• are nonempty, compact and connected

See also

• Julia set

• Stable set

• Limit cycle

• Periodic point

• Non-wandering set

• Kleinian group

References

Further reading