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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**.

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.

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 .^{[1]} 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 (but may not be *recurrent points*). This may also be formulated as the outer limit (limsup) of a sequence of sets, such that

If is a homeomorphism (that is, a bicontinuous bijection), 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.

Given a real dynamical system with flow , a point , we call a point *y* 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 (-limit points) for a given orbit is called -**limit set** (-**limit set**) for and denoted ().

If the -limit set (-limit set) is disjoint from the orbit , that is (), we call () a **ω-limit cycle** (**α-limit cycle**).

Alternatively the limit sets can be defined as

and

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